Sapienza Universita di Roma

Dottorato di Ricerca in Fisica

Scuola di dottorato ”Vito Volterra”

Einstein-Maxwell equations:

soliton solutions, equilibrium configurations

and related aspects.

Thesis submitted to obtain the degree of

”Dottore di Ricerca” - Doctor PhilosophiæPhD in Physics - XX cycle - October 2008

by

Armando Paolino

Program Coordinator Thesis Advisors

Prof. Enzo Marinari Prof. Vladimir Belinski

Prof. Antonio Degasperis

to Ayumi and Lisa

i

Table of contents

Table of contents ii

Abstract 1

Acknowledgements 2

Introduction 3

1 Soliton solutions of Einstein equations: Belinski-Zakharov technique 81.1 The integrable ansatz in general relativity . . . . . . . . . . . . . . . . . 81.2 The integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Construction of n-soliton solution . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 The physical metric components gab . . . . . . . . . . . . . . . . 161.3.2 The physical metric components f . . . . . . . . . . . . . . . . . 18

1.4 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Solitonic solutions of Einstein-Maxwell equations: Alekseev technique 202.1 The Einstein-Maxwell field equations . . . . . . . . . . . . . . . . . . . . 202.2 The spectral problem for Einstein-Maxwell fields . . . . . . . . . . . . . . 242.3 The component gab and the potentials Aa . . . . . . . . . . . . . . . . . 29

2.3.1 The n-soliton solution of the spectral problem . . . . . . . . . . . 292.3.2 The matrix X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.3 Verifications of the constraints . . . . . . . . . . . . . . . . . . . . 35

2.4 The metric component f . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Summary of prescriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Exact stationary axially symmetric one-soliton solution on Minkowskybackground 41

Index notation and form of the line elements. . . . . . . . . . . . . . . . 423.1 Application of the first nine steps of the generating procedure. . . . . . . 42

3.1.1 Step-1: Background Einstein-Maxwell solution. . . . . . . . . . . 423.1.2 Step-2: Background value of the complex electromagnetic potential

Φ(0)a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.3 Step-3: Calculus of X(0) and X(0)−1. . . . . . . . . . . . . . . . . 43

ii

3.1.4 Step-4: Calculus of U(0)µ . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.5 Step-5: Deduction of β(xµ). . . . . . . . . . . . . . . . . . . . . . 453.1.6 Step-6: Calculus of W (0). . . . . . . . . . . . . . . . . . . . . . . . 453.1.7 Step-7: Deduction of background generating matrix ϕ(0) (w, xµ)

and its normalization. . . . . . . . . . . . . . . . . . . . . . . . . 463.1.8 Step-8: Costruction of m(k)A and p

(k)A vectors. . . . . . . . . . . . 48

3.1.9 Step-9: Costruction of Tkl matrix. . . . . . . . . . . . . . . . . . . 493.2 Determination of the mathematical parameters in terms of the physical one. 49

3.2.1 The McGuire-Ruffini one-body solution. . . . . . . . . . . . . . . 513.2.2 Calculus and comparison of the f conformal factors. . . . . . . . 533.2.3 Step-10 & 11: Determination of S matrix and calculus of gab

components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.4 Components of the electromagnetic potential. . . . . . . . . . . . 58

4 A perturbative approach for stationary axially symmetric soliton solu-tions 604.1 Some preliminary remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 A proposal for a perturbative approach. . . . . . . . . . . . . . . . . . . . 624.3 Outline of the perturbative solitonic generating technique. . . . . . . . . 64

4.3.1 Expansion of Einstein-Maxwell Equations. . . . . . . . . . . . . . 644.3.2 Lax Pair expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.3 Expanded dressing procedure. . . . . . . . . . . . . . . . . . . . . 67

5 Generation of approximate soliton solutions over a flat background. 695.1 Perturbative building block quantities . . . . . . . . . . . . . . . . . . . . 705.2 One-soliton approximate solution. . . . . . . . . . . . . . . . . . . . . . . 755.3 Two-soliton lowest order approximate solution . . . . . . . . . . . . . . . 81

6 Electric force lines of the double Reissner-Nordstrom exact solution 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Some Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Summary of the Alekseev-Belinski formulas . . . . . . . . . . . . . . . . . 896.4 Some further details of the solution . . . . . . . . . . . . . . . . . . . . . 906.5 Electric force lines definition . . . . . . . . . . . . . . . . . . . . . . . . . 946.6 Plots of the force lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.6.1 Two charges of equal sign ( e1e2 > 0 ) . . . . . . . . . . . . . . . . 966.6.2 Two charges of opposite sign ( e1e2 < 0 ) . . . . . . . . . . . . . . 986.6.3 Cases with only one charge . . . . . . . . . . . . . . . . . . . . . . 100

6.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 A stability analysis of the double Reissner-Nordstrom Alekseev-Belinskiexact solution 1057.1 Force of the strut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.2 Analysis of equilibrium in the AB solution . . . . . . . . . . . . . . . . . 1097.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

iii

Conclusions and prospects 113

Additional contributions 115

I. Charged membrane as a source for repulsive gravity 116

II. Intersection of self-gravitating charged shells in a Reissner-Nordstromfield 132

Personal works 153

Bibliography 154

iv

Abstract

This thesis includes two parts, both concerning the balance problem of two charged

compact objects in General Relativity.

In the first part, chapters 1-5, we review the Belinsky-Zakharov and Alekseev solitonic

dressing techniques to generate respectively exact solutions for the Einstein equations in

vacuum and the Einstein-Maxwell equations in electro-vacuum. We recall those prob-

lems that these generating techniques have in solving the equilibrium problem of interest.

Then we construct a procedure to calculate the approximate terms of solitonic solutions

in respect to a control parameter interpretable as the Newtonian constant. We apply

this method to re-derive the field of two charged and rotating sources on Minkowski

background, giving the reparameterization between the mathematical constant and the

physical one.

In the second section of our work, chapters 6-7, we analyze the configurations of the

Alekseev-Belinski solution [5] for two Ressner-Nordstrom like sources in static equilib-

rium. As regards a qualitative classification in respect to the physical parameters we

show: a) which configurations are forbidden in respect to the equilibrium conditions; b)

the pictures of the force lines of the electric field; c) that most of the permitted config-

urations are stable in respect to displacements from the distance of equilibrium; d) that

some results, found in the past by means of test particle approximation, agree with this

more general exact solution.

Two additional contributions are enclosed at the end as a separate body.

The first one deals with the construction of an everywhere regular membrane model for

a Reissner-Nordstrom naked singularity. It is shown that such model is stable and allows

an external repulsive-gravity region. This toy model contributes to give a more sensible

physical meaning also to the Alekseev-Belinski solution.

The second one regards the intersection of two charged spherical thin shells in a central

Reissner-Nordstrom field. We give the energy-exchange formula, and compare the ejec-

tion mechanism of one of the two shells for these charged configurations, with the one

studied in a previous work of Barkov et al [7] for neutral shells.

1

Acknowledgements

I am very grateful to my supervisor, Vladimir Belinski, for his continued guidance and

support; to Remo Ruffini and George Alekseev for the stimulating discussions; to Decio

Levi for the useful suggestions about the drafting of this thesis.

A special thank to Antonio Degasperis, for his teachings, his constant presence and very

kind helpful, but, overall, for his dear friendship.

I would like to thank Marco Pizzi, since to work with him has been a very interesting

and beautiful experience.

Thanks to Monica Rizzo, Patrizia Mezzabotta, Francesca Paolino, Chiara Milano and

Roberto Ricci for their useful, and necessary, linguistic hints.

I wish to remember all those friends with whom I spent lovely moments during this

period of study, in particular Antonino Marciano, Fabio Briscese, Calogero Tornese and

Riccardo Benini, and all my old friends, which I feel close to me in each moment of my

life and that gave me a great help during these last years: Stefano Granata, Giangiacomo

Gandolfi, Fabrizio Ferrante, Daniela Bonetti, Mario Rosati, Paolo Branchesi.

I am grateful to my mother, my father and my brothers, for their help and encouragement.

Finally to Ayumi and to Lisa chan: the light of our eyes.

2

Introduction

In the Einstein theory of gravitation, as in any nonlinear theory, the construction and

investigation of diverse families of exact solutions of the field equations is of considerable

interest in studying the complicated nonlinear character of interaction of the gravita-

tional fields of different sources with each other and with other forms of matter. The

characterizations of simple model configurations among these solutions and their investi-

gations yield explanations for many qualitative features of their fields and of the process

of their interaction. Having a clear physical content, these model configurations can

turn out to be very useful in discussing many questions in the theory of gravitation and

various applications of the theory in cosmology, the physics of compact cosmic objects,

and the theory of radiation, propagation, and interaction of waves.

The developments in the area of exact solutions have ran through various phases (See as

review work examples [3], [19], [20], [22], [83]). In a first period, the principal way of ob-

taining exact solutions to Einstein’s field equations was to impose sufficient symmetries

in order to reduce the integrability problem of the system to a rather easily tractable

set of equations. Later, the 60s saw an increase of the number of exact solutions thanks

to the development of techniques to treat geometries with particular algebraic prop-

erties. In 1968 Ernst [34, 35] introduced complex potentials which turned out to be

useful in treating stationary vacuum and electro-vacuum fields through a system of two

complex second-order equations. In the beginning of the 70s, Geroch [40, 41] showed

that for empty space-times admitting a two-parameter commutating group of motions it

appeared possible (Geroch conjecture) to generate new exact solutions from flat space-

time through the action of a hidden symmetry group associated with the field equations.

At the end of the same decade Belinski and Zakahrov [14, 15] discovered that Einstein’s

vacuum equations for space-times, possessing those symmetries as for the Geroch conjec-

ture, can be solved by means of the inverse scattering methods (ISM)1. These techniques

1It is dutiful to mention also other independent works of Maison [61, 62], Harrison [44, 45] andNeugebauer [69], which appeared at about the same time. A wide review of references and abstractsconcerning the subsequent developments of these area of Einstein’s field equations generating techniquescan be found in http://members.localnet.com/∼atheneum/bib/bib.html .

3

4

handle nonlinear equations as compatibility conditions of an over-determined system of

linear equations and enable to obtain new exact solutions, starting from exact back-

ground solutions, giving algebraic algorithms for their computation.

Despite the essential restrictions involved in the consideration of fields configurations

depending on only two coordinates, this class of two-dimensional fields is still very inter-

esting and rich in its physical content. It includes fields created by all sorts of stationary

axially symmetric sources; various kinds of wave fields (with planar, cylindrical, and

other symmetries); some fields having an explicitly expressed dynamical nature and de-

scribing a uniformly accelerated motion of different sources, with consideration of their

gravitational and electromagnetic radiation; sets of solutions of cosmological type (both

homogeneous and non homogeneous). The study of this class of fields can answer many

questions in gravitation theory often yielding new and unexpected results. Moreover it

can be the basis for the investigation of more general but also considerably more com-

plicated situations.

The first part (chapters 1-5) of this dissertation deals with the application of such ISM

generating techniques used to find solutions for fields of two compact electrically charged

objects in stationary equilibrium. In particular we will take into consideration only the

Alekseev soliton method [1, 3]. This is a direct generalization of Belinski-Zakahrov

technique (which generates solutions for Einstein equations in vacuum) and enables to

generate electro-vacuum exact solutions of the Einstein-Maxwell equations by means of

a dressing (Darboux) transformation performed on background known solutions. This

method enables to construct solutions for the fields of systems of a number of compact

sources aligned on a common axis. Anyway, as well as other solitonic generating tech-

niques, it presents some limitations concerning difficulties in finding easily a physical

interpretation for them (that is to give a representation of the mathematical parameters

in terms of physical one) and the possibility of yielding regular solutions. For example,

considering the equilibrium problem for two massive, non rotating sources, it produces

non elementary-flat solutions, i.e. with conic-like singularities on the axis. These irregu-

larities are a consequence of some limitations which are specific of the solitonic generating

tools. In Newtonian physics this two-body system can stay in equilibrium if (in geomet-

rical units) the product of the masses is equal to the product of the charges. The limit

to this classical condition shows that, in the relativistic regime, static equilibrium should

subsist only if this two-body system is composed either by a under-extreme (mass greater

than the charge) and a super-extreme (mass smaller than the charge) Reissner-Nordstom

like sources, or by two extreme (mass equal to the charge) sources. The experiences on

solitonic generating techniques tell us that they can generate only solutions relevant to

5

super-extreme sources; hence up to the present they are not able to yield those equilib-

rium conditions to remove the conic singularities from the axis.

In 2007, Alekseev and Belinski [5, 4] found the general solution (hereafter denoted with

AB) for the static balancing problem of two charged non rotating sources, giving the

general equilibrium conditions to remove each axial irregularity. They reached this re-

sult not by means of the Alekseev dressing solitonic technique, but using the Alekseev

“Monodromy Transform Approach”. Nevertheless, it should be possible to reach the

same result through the dressing technique; how to do this is still an open question, but

if this is the case, it could yield a more direct and simple way to obtain such solutions.

In regard to this, it is worth to recall that the problem of finding a similarly general

result concerning the regular solution for two charged and rotating sources, as well as

the relative explicit general equilibrium conditions, is still open2.

The original work, related to the Alekseev solitonic technique, presented in the first

part of this thesis (chapters 4-5), deals with the construction of the procedure to gener-

ate perturbations of the exact solitonic solutions respect to the Newtonian gravitational

constant. The five chapters relevant to this part are organized as follows.

In chapter 1, we recall the Belinski-Zakahrov dressing generating technique for vacuum

solutions.

In chapter 2, we describe the Alekseev technique which generalizes the previous one to

Einstein-Maxwell equations for electro-vacuum solutions.

The third chapter is devoted to show as the Alekseev dressing procedure works taking, as

an example, the one-soliton stationary solution. The results, presented in this chapter,

are not new; its utility is to give the reader a description of all basic aspects concerning

this generating technique such as, in particular, those concerning the difficulties incurred

by working on multi-soliton solutions.

In the chapter 4, the construction of the procedure to generate perturbative solutions

in respect to the Newtonian gravitational constant γ is presented. Its contextual task

is to investigate some aspects of the two-body problem mentioned above, as regards the

Alekseev exact generating technique in the framework of a weak field approximation.

Nevertheless, it posses independent interests; in fact this tool permits to generate weak

solutions for electromagnetic solitons on flat or fixed curved space-time, since it enables

to handle separately electromagnetic and gravitational fields.

The chapter 5, after the presentation of all the formulas by means of which the perturba-

tive terms can be calculated, describes a first applications of this perturbative approach.

We give: first the approximate solutions up to the first order in γ of one-soliton solution

2A review relevant to the particular results obtained up to 2003 can be found on [83].

6

with the aim to check the procedure; then the zero order approximate terms of two-

soliton solution, finding a first reparameterization between the mathematical parameters

and the physical one. The work relevant to the successive order of approximation of the

two-soliton solution, is not given here, since it is still in progress.

The second part of this thesis, constituted by chapters 6 and 7, regards respectively

two works, [71] and [74], relevant to a first analysis of the AB exact solution3.

In chapter 6, after a brief summary of the AB solution describing two Reissner-Nordstrom

sources in reciprocal equilibrium, we study in some detail the coordinate systems used

and the main features of the gravitational and electric fields, as well as the different con-

figurations permitted by it. Many of these configurations are forbidden by Newtonian

physics; we find that the equilibrium is possible for two opposite charged sources too.

We graph the plots of the electric force lines in three qualitatively different situations:

equal-signed charges, opposite charges and the case of a naked singularity near a neutral

black hole.

The chapter 7 is devoted to the analysis of the stability of the AB solution with respect

to displacements from the equilibrium distance between the two sources. To do this

we define the force of the conic singularity, and assume that the force between the two

bodies is equal and opposite. Then we analyze the stability of the equilibria in the three

qualitatively different situations: equal-signed charges, opposite-signed charges and the

case of a naked singularity near a neutral black hole. We show that most of such con-

figurations result to be stable and in agreement with the analog stability classification

given by Bonnor [21] for test charged particles in a Reissner-Nordstrom field.

At the end of the thesis, as a separate part, the preprint version of two additional

works ( [13], [75])4 are attached. They deal with some spherically symmetric solutions

of Einstein-Maxwell equations relevant to shell like configurations of matter.

The first one, [13], is about a membrane model (i.e. a thin shell with tension) for a

naked singularity. The link with the AB solution is rather natural. As specified above,

the equilibrium is allowed only by the presence of a naked singularity and of its repulsive

region near the center. Therefore, in order to give a more sensible physical meaning

to this solution, we construct a model, at least a toy model, for a Reissner-Nordstrom

naked source presenting a repulsive region around it. Such model consists in a charged

membrane with a dark-matter-like equation of state, ε = −p . Obviously this is a very

simple model and it does not resolve all the problems concerning the naked singularity

3All these works were produced in collaboration with Marco Pizzi.4Both these two works were produced, during the last two years in collaboration with Marco Pizzi;

the first one reported, together with Vladimir Belinski too.

7

(i.e. how to arrive to such configuration). Anyway, it gives al lest a hint that such a

configuration is physically possible (and we know that something like this should exist

because the electron has the parameters corresponding to a naked singularity). The

static configuration we present is stable with respect to the radial displacements. It is

important to recall that such kind of naked-singularity model already exist in literature;

however, we re-derive such results using a method more habitual for physicist, that is by

means the direct integration of the field equations with appropriate δ-shaped sources.

The last work, [75], is about the motion of two spherical intersecting thin charged shells

with positive tangential pressure in a Reissner-Nordstrom field. It is a direct general-

ization to the case of charged shells of the work of Barkov et al [7] dealing with neutral

shells. The motion of each shell is independent from the other until their intersect;

indeed, until the intersection, the outer shell feels the inside shell as a simple Reissner-

Nordstrom source. We obtain the exchanging energy formula between the two shells

due to their intersection. Finally, we describe the ejection mechanism, for which one of

the two bounded shell can acquire enough energy to be driven out to infinity. We show

that, because the energy transfer is larger due to the Coulomb interaction, the ejection

mechanism is more efficient in the charged case than in the neutral one if the charges

have opposite sign.

The original contributions presented in this thesis, which, as mentioned above, has been

subject matter of publications, were produced in collaboration with other authors. We

worked in parallel, comparing the results and discussing what had to be included or

excluded, how to explain it. Therefore, it is not possible to define clearly each different

personal contribution.

Chapter 1

Soliton solutions of Einsteinequations: Belinski-Zakharovtechnique

In this chapter we resume the scheme of the soliton technique of Belinski-Zakharov [14]

[15] to generate exact solutions of the Einstein Equations in vacuum for metric tensors

depending only on two coordinates. A more detailed and comprehensive description,

together with references to other different approaches, can be found in [9].

1.1 The integrable ansatz in general relativity

The Belinski-Zakharov soliton technique generates exact solutions for the Einstein Equa-

tions in vacuum,

Rij = 0 , (1.1)

where Rij is the Ricci tensor, for metric tensors g4ij depending on two variables only1.

These correspond to space-times that admit two Killing vectors fields, i.e an Abelian

two-parameter group of isometries. Moreover it is necessary to impose 2 the existence of

2-surfaces orthogonal to the group orbits. The metric tensors belonging to this class can

assume the following block diagonal form:

ds2 = gab(xρ) dxadxb + f(xρ) ηµν dxµdxν , (1.2)

where Latin letters from the first part of the alphabet take only 0, 1 values, while the

Greek ones 2, 3 ; f > 0 and ηµν = diag (−e, 1) , with e = ±1 . If e = 1 (1.2) describes

1Here g4 denotes the four-dimensional metric tensor; we denote the four-dimensional index withi, j, ... = 0, 1, 2, 3.

2It is unknown at present whether the Inverse Scattering Methods can be applied without thisfurther physical restriction.

8

9

non-stationary solutions and xi = (ρ, ϕ, t, z) ; while if e = −1 it refers to stationary

solutions and, in this case, xi = (t, ϕ, ρ, z) . More explicitly, this geometrical reduction

reads;

ds2 = g00(t, z)dρ2 + 2g01(t, z)dρ dϕ + g11(t, z)dϕ2 + f(t, z)(−dt2 + dz2

), if e = 1 ,

ds2 = g00(ρ, ϕ)dt2 + 2g01(ρ, ϕ)dt dz + g11(ρ, ϕ)dz2 + f(ρ, ϕ)(dρ2 + dϕ2

), if e = −1 ,

where signature(g) = (e, 1) . In what follows we shall always denote by g the two-

dimensional real and symmetric matrix with components gab . For the determinant of

this matrix it is convenient to introduce the notation:

det g = e α2 (1.3)

and we shall always consider that α is nonnegative. Hereafter, for the sake of simplicity,

we will consider the non-stationary case and hence we will assume e = 1 . It is convenient,

to write the field equations, to introduce a pair of null-coordinates defined by: ζ =

(√

e x2 + x3)/2 , η = −(√

e x2 − x3)/2 3. The system (1.1) for the metric tensor (1.2),

implies that Raµ ≡ 0 . The remaining equations can be decomposed into two sets. The

first one follows from the equations Rab ≡ 0 and can be written in the form of a single

matrix equation: (α g,ζg

−1)

,η+

(α g,ηg

−1)

,ζ= 0 . (1.4)

The second set follows from the equations R22 + R33 = 0 and R23 = 0 , and gives the

metric coefficient f(xρ) in terms of the matrix g , as obtained by the (1.4):

(ln f),ζ(ln α),ζ = (ln α),ζζ +1

4 α2Tr A2 , (1.5)

(ln f),η(ln α),η = (ln α),ηη +1

4 α2Tr B2 , (1.6)

where the matrices A and B are defined by

A = −α g,ζ g−1 , B = α g,η g−1 . (1.7)

The integrability condition for (1.5) and (1.6), with respect to f , is automatically sat-

isfied if g satisfies (1.4). The equation R22 −R33 = 0 can be written in the form:

(ln f),ζη =1

4 α2Tr AB − (ln α),ζη . (1.8)

This last equation does not add anything new, since it is a consequence of the system

(1.3)-(1.7) when α is not a constant. While the special case in which it is constant

3Note that in the case e = −1 , these coordinates are complex variables, namely ζ = (z + iρ)/2 andη = ζ = (z − iρ)/2 .

10

corresponds to flat Minkowsky spacetime.

The basic point on which the generating technique is constructed lies on the fact that the

principal set of the field equations, i.e. (1.4), is very similar to the field equations for some

integrable relativistic invariant model called principal chiral field [89], to which it reduces

if α is constant. The general idea of the method is based on the study of the analytic

structure of the eigenfunctions of two operators (as functions of a complex parameter λ ),

which are associated to the system (1.3)-(1.4). In particular, for the soliton solutions of

(1.3)-(1.4), the structure of the poles, λn , of the corresponding functions in the λ-plane

plays a fundamental role. As shown in the next section, if α is not constant, (1.3)-(1.4)

require the introduction of generalized differential operators entering into the Lax-Pair

(see below). These operators depend on the function α(ζ, η) and contain derivatives

with respect to the spectral parameter λ . For soliton solutions this leads, instead of

stationary poles as for the principal chiral field, to pole trajectories, since now, the poles

will depend on the coordinates: λn = λn(ζ, η) .

1.2 The integration scheme

From the trace of (1.4), it is immediate that

αζη = 0 , (1.9)

namely, α satisfies the two-dimensional wave equation which as the general solution

α = a(ζ) + b(η) , (1.10)

where a(ζ) and b(η) are arbitrary functions. For later use we define a second indepen-

dent solution of (1.9)4

β = a(ζ)− b(η) . (1.11)

The equation (1.4) is equivalent to a system consisting of the definitions (1.7) and two

first order matrix equations for the matrices A and B .

A,η −B,ζ = 0 (1.12)

A,η + B,ζ + α−1 [A,B]− α,η α−1A− α,ζ α−1B = 0 (1.13)

Where this last equation, in which the square brackets denote the commutator, repre-

sents an integrability condition of (1.7) with respect to g . The main step now consists

4It is to recall that the metric (1.2) admits arbitrary coordinate transformations x′ 2 = f1(ζ)+f2(η) ,x′ 3 = f1(ζ)− f2(η) , which leave unchanged its conformally flat block. By an appropriate choice of thefunctions f1 and f2 , it is possible to bring the functions a(ζ) and b(η) , in (1.10)-(1.11), into aprescribed form. When this freedom is used to write (α, β) as spacetime coordinates, we say that themetric (1.2) has the canonical form and (α, β) are called canonical coordinates.

11

in representing (1.12) and (1.13) in the form of compatibility conditions of a more gen-

eral overdetermined system of matrix equations related to an eigenvalue-eigenfunction

problem for some linear differential operators. Such a system will depend on a complex

spectral parameter λ , and the solutions of the original equations for the matrices g , A

and B will be determined by the possible types of analytic structure of the eigenfunctions

in the λ-plane. Therefore, introducing the differential operators5

D1 = ∂ζ − 2 α,ζ λ

λ− α∂λ , D2 = ∂η +

2 α,η λ

λ + α∂λ , (1.14)

we consider the linear system6

D1 ψ =A

λ− αψ , D2 ψ =

B

λ + αψ , (1.15)

for the complex matrix function ψ(λ, ζ, η) , which in this context is usually called the

generating matrix. It is easy to see that if α is solution of the wave equation (1.9), then:

[D1, D2] = 0 . (1.16)

By means of this property, it results that the compatibility conditions for (1.15) reproduce

exactly the (1.12)-(1.13) equations for the matrices A and B . Moreover, putting λ = 0

in (1.15), we obtain

∂ζψ = − 1

αAψ = g ζ g−1 ψ , ∂ηψ =

1

αB ψ = g η g−1 ψ . (1.17)

Hence we can choose ψ such that

ψ(0, ζ, η) ≡ ψ(0) = g(ζ, η) . (1.18)

The method we apply assumes the a priori knowledge of a particular solution, g0(ζ, η) ,

of the system (1.3)-(1.4). From it, by means of (1.7) one can calculate the correspond-

ing A0(ζ, η) and B0(ζ, η) , and integrating (1.15), obtain the corresponding generating

matrix ψ0(λ, ζ, η) 7.

We then introduce the dressing matrix χ(λ, ζ, η) according to the definition:

ψ = χ ψ0 , (1.19)

where ψ , is the solution of (1.15) corresponding to the new solution g we want to con-

struct. This way, through the restriction (1.18), we obtain a new solution g = χ(0, ζ, η)g0

5The symbol ∂ with a subscript denotes partial differentiation with respect to the correspondingvariable and λ is a complex parameter independent of the coordinates ζ and η .

6The matrices A and B together with α are real and independent on the spectral parameter λ .7It is worth noting that this is the only step of this generating procedure where it is necessary to

perform the integration of a differential system, which, however, is linear.

12

for the equations (1.4). Substituting (1.19) in (1.15) (and assuming that this transfor-

mation g0 → g does not change the determinant, detg = detg0 , see (1.3)), we have the

following differential constraints for the matrix χ :

D1 χ =1

λ− α(Aχ− χA0) , D2 χ =

1

λ + α(B χ− χB0) . (1.20)

Anyway, we recall that the matrix g(ζ, η) must be real and symmetric. Hence, it is

necessary to perform a reduction on the dressing and generating matrices. This can be

done through the imposition of two restrictions. The first consists of requiring the reality

of these matrices on the real axis of the λ-plane, that is:

χ(λ) = χ(λ) , ψ(λ) = ψ(λ) , (1.21)

where the overset bar denotes complex conjugation 8. The second condition is based on

the invariance of the solutions of the system (1.20) under the substitution λ → α2/λ .

Introducing the matrix

χ′(λ) = g χ−1(α2/λ) g−10 , (1.22)

where the tilde denotes transposition of the matrix, it follows that χ′ is solution of (1.20)

if g is symmetric. Then we can choose χ′(λ) = χ(λ) , and hence

g = χ(λ) g0 χ(α2/λ) . (1.23)

When λ →∞ , g = χ(∞) g0 χ(0) , which, because of (1.18) and (1.19), gives the asymp-

totic behavior

χ(∞) = I . (1.24)

Thus, the problem now consists of solving (1.20) and determining the dressing matrix χ

in accordance with these restrictions. In this way we have for the dressed solution:

g = χ(0) g0 . (1.25)

It is important to keep in mind that the dressing procedure must preserve the determinant

of g , that is det g = α2 = det g0 . Therefore it is necessary to impose on matrix χ , as

follows from (1.25), the further restriction: det χ(0) = 1 . However, it is more convenient

to renormalize the final results in order to obtain the correct functions. These (correct)

functions will be called physical functions and denoted with the small up set suffix (ph) .

8For the sake of brevity, when it is not necessary, we do not indicate the arguments ζ and η ofsome functions.

13

To legitimate this procedure it is sufficient to see that, if we obtain a solution of (1.4)

with det g 6= α2 , the trace of this equation implies that det g satisfies the equation

(α(ln det g),ζ),η + (α(ln det g),η),ζ = 0 . (1.26)

It is easy to see that the matrix

g(ph) = α (det g)−1/2g , (1.27)

satisfies both (1.4) and the condition det g(ph) = α2 . Correspondingly

A(ph) = A− α ln[ α (det g)−1/2 ] ,ζ I , (1.28)

B(ph) = B + α ln[ α (det g)−1/2 ] ,η I , (1.29)

where A and B are defined in terms of g according to (1.7) and A(ph) and B(ph) are

defined by the same formulas but in terms of the matrix g(ph) .

1.3 Construction of n-soliton solution

According to the Inverse Scattering Theory, the general solution for χ is given by the sum

of a solitonic and a nonsolitonic parts. Here only the solitonic part will be considered.

The existence of particular solutions of this kind is due to the presence in the λ-plane

of points at which the determinant of χ has simple poles. Thus it is representable as

a rational function of the parameter λ with a finite number of simple poles, in such a

way that it goes to the unit matrix when λ → ∞ , as required by (1.24). The reality

condition (1.21) for g implies that the poles can lie either on the real axis of the complex

λ-plane or come in complex conjugate pairs. From the symmetry condition (1.23), which

implies that det χ(λ) det χ(α2/λ) = 1 , it follows that for each pole λ = µ, there is a

corresponding point λ = α2/µ where det χ = 0 . The inverse matrix χ−1 has the same

properties, as can be seen from (1.21) and (1.23). Therefore the matrix χ has the form

χ = I +n∑

r=1

Rk

λ− µk

, (1.30)

where the matrices Rk and the functions µk do not depend on λ . With respect to

the expression (1.30), the reality conditions for χ , implies that to each real pole µk, it

correspond a real matrix Rk , while to each complex µk, there must be another function

µk+1 = µk , to which it corresponds Rk+1 = Rk . Thus from (1.30) and (1.25), the

solution of the equation (1.4) assumes the form

g(ζ, η) =

(I −

n∑

k=1

µ−1k Rk

)g0 . (1.31)

14

To determine the functions µk and the matrices Rk , it is necessary to substitute (1.30)

into (1.20) and impose that these equations be satisfied at the poles λ = µk(ζ, η) . Since

the right hand sides of (1.20) have in λ = µk first order poles, whereas the left hand sides

have second order ones, the requirement that the coefficients of the powers (λ − µk)−2

vanish yields the following equations for the pole trajectories µk(ζ, η) :

µk,ζ =2 α,ζ µk

α− µk

, µk,η =2 α,η µk

α + µk

. (1.32)

These equations have the following invariance: if µk is a solution of (1.32), then α2/µk

is also a solution. The solutions of (1.32) are the roots of the quadratic equation

µ2k + 2 (β − wk) µk + α2 = 0 , (1.33)

where β is the other independent solution (1.11) of the wave equation (1.9) for α , and

wk are arbitrary complex constants. For each given wk , (1.33) yields two roots, µk and

α2/µk . Hence these solutions can be written in the form

µink = (wk − β)

1− [1− α2(β − wk)

−2]1/2

(1.34)

µoutk = (wk − β)

1 + [1− α2(β − wk)

−2]1/2

(1.35)

since, in the λ-plane, respectively, they are never located outside or inside the circle of

radius λ = |α| .Rewriting (1.20) in the form

A

λ− α= (D1 χ) χ−1 + χ

A0

λ− αχ−1 , (1.36)

B

λ + α= (D2 χ) χ−1 + χ

B0

λ + αχ−1 , (1.37)

since the left hand sides of these equations are regular at the polee of χ , λ = µk , the

residues of these poles on the right hand sides must vanish. This leads to the equations

for the matrices Rk :

Rk,ζ χ−1(µk) + RkA0

µk − αχ−1(µk) = 0 , (1.38)

Rk,η χ−1(µk) + RkB0

µk + αχ−1(µk) = 0 . (1.39)

To obtain these equations use has been made of the relation

Rk χ−1(µk) = 0 , (1.40)

15

which follows from the identity χχ−1 = I , at the poles λ = µk . The relation (1.40)

implies that Rk and χ−1(µk) are degenerate matrices, therefore their matrix elements

can be written in the form

(Rk)ab = n(k)a m

(k)b , [χ−1(µk)]ab = q(k)

a p(k)b . (1.41)

According with these representations, we have that9

m(k)a q(k)

a = 0 . (1.42)

As a consequence of (1.38) and (1.39), we obtain

(n(k)a m(k)

c ),ζg(k)c p

(k)b +

1

µk − αn(k)

a m(k)d (A0)dc q(k)

c q(k)b = 0 , (1.43)

which, combined with (1.42), yields the differential system for the vector functions m(k)a :

[m

(k)a,ζ + m

(k)b

(A0)ba

µk − α

]q(k)a = 0 ,

[m(k)

a,η + m(k)b

(B0)ba

µk + α

]q(k)a = 0 . (1.44)

Anyway, these equations present four ( m(k)a and q

(k)a , for a = 0, 1 ) unknown quantities,

hence we need other relations. To find these, let us define the matrices

Mk = (ψ−10 )λ=µk

= ψ−10 (µk, ζ, η) . (1.45)

Being ψ0 a particular solution of (1.15), from these equations we obtain:

Mk,ζ + MkA0

µk − α= 0 , Mk,η + Mk

B0

µk + α= 0 . (1.46)

By comparing (1.46) with (1.44), it is found that the matrices Mk are proportional to

the vectors m(k)a ; that is, a solution of (1.44) will be

m(k)a = m

(k)0b (Mk)ba = m

(k)0b [ ψ−1

0 (µk, ζ, η) ]ba , (1.47)

where the m(k)0b are arbitrary complex constant vectors. The expression (1.47) for m

(k)a

could also present an arbitrary factor depending on k , and on the coordinates; since it

disappears in the final expressions of the residues Rk , it can be set equal to 1 without

loss of generality.

To complete the determination of Rk we need to find the vectors n(k)a , as prescribed by

the representation (1.41) . Substituting (1.30) into (1.23), and considering the relation

9Here and in the following, summation will be understood to be over repeated vector and tensorindices a , b , c , d , which, we recall, takes the values 0 and 1 .

16

obtained in such a way at the points λ = µk , since g does not depend on λ , we have

the following system of n algebraic matrix equations for the matrices Rk :

Rk g0

[I +

n∑

l=1

(α2 − µk µl)−1µk Rl

]= 0 . (1.48)

Substituting in this the representation (1.41) of Rk , we obtain a system of linear algebraic

equations for the vectors n(k)a :

n∑

l=1

Γkl n(l)a = µ−1

k m(k)c (g0)ca , (1.49)

where the n× n matrix Γkl is symmetric and its elements are

Γkl = −m(k)c m

(l)b (g0)cb(α

2 − µkµl)−1 . (1.50)

Introducing the symmetric matrix Πkl inverse to Γkl , and the vectors

L(l)a = m(l)

c (g0)ca , (1.51)

we obtain

n(k)a =

n∑

l=1

µ−1l ΠklL

(l)a . (1.52)

Now, using (1.41), (1.52) and (1.51) we get, from (1.31), the metric componets gab :

gab = (g0)ab −n∑

k,l=1

µ−1k µ−1

l Πkl L(k)a L

(l)b . (1.53)

This expression is obviously symmetric. To ensure that the matrix g is also real, as

deducible from the second of the conditions (1.21) and from (1.47), it is necessary to

choose the arbitrary constants m(k)0b so that the vectors m

(k)a corresponding to real poles

λ = µk are real. It results that all the complex poles have to appear only as conjugate

pairs: for each complex pole λ = µ , its complex conjugate λ = µ must also appear. We

can therefore denote, for each complex pole λ = µk , µk+1 = µk . In this case, the choice

of the constants m(k)0b , is constrained by the request that the vectors m

(k)a and m

(k+1)a

corresponding to such pairs of complex conjugate poles are complex conjugate to each

other.

1.3.1 The physical metric components gab

The matrix g of (1.53) does not satisfy (1.3), therefore, as explained at the end of para-

graph 1.2, it is necessary to perform a renormalization to obtain the physical solution.

17

Here, we will not describe all the steps to deduce this expression. We will limit ourselves

to describe the scheme which bring to the final formulas, remanding to [9] for the details.

The form (1.53) is not convenient for this calculation. It is better to dispose of a formula

for χ given by a product among factors, each one being specific of each single soliton.

This is feasible since the n-soliton solution can be obtained either as a simultaneous

dressing or as an iteration of n steps introducing one soliton at each step, in succession:

g(n) = χ(n) g0 =(χ(1)

n χ(1)n−1 ... χ

(1)1

)g0 = χ(1)

n

(χ

(1)n−1

(...

(χ

(1)1 g0

))). (1.54)

Therefore, it is better to start with one-soliton dressing. In this case the matrix χ1 can

be expressed in the following form

χ1 = I + µ−11 (λ− µ1)

−1(µ21 − α2) P1 , (1.55)

where the matrix P1 , given by

(P1)ab =m

(1)c (g0)ca m

(1)b

m(1)d (g0)df m

(1)f

, (1.56)

satisfies the properties

P 21 = P1 , Tr P1 = 1 , det P1 = 0 . (1.57)

Denoting with g1 the one-soliton solution, we have from (1.25) and (1.55)

g1 = χ1(0) g0 =[I − µ−2

1 (µ21 − α2) P1

]g0 , (1.58)

the determinant of which, using the general relation holding for any 2 × 2 matrix F ,

det(I + F ) = 1 + Tr F + det F , is:

det g1 = µ−21 α2 det g0 . (1.59)

This result is obviously independent by the soliton indices. That is, it can automatically

be generalized as

det gk+1 = µ−2k+1 α2 det gk . (1.60)

Therefore, the determinant of the final n-soliton matrix g will be

det g = α2n

(n∏

k=1

µ−2k

)det g0 = α2n+2

n∏

k=1

µ−2k . (1.61)

We can now write explicitly the equation (1.27)10:

g(ph) = α

(α2n+2

n∏

k=1

µ−2k

)−1/2

g = α−n

(n∏

k=1

µk

)g . (1.62)

10It is worth noting that both signs are allowed in front of the matrix g(ph) due the invariance ofthe Einstein Equations (1.3)-(1.7) with respect to the reflection g(ph) → −g(ph) . This sign should bechosen separately for each case in order to ensure the correct signature of the metric.

18

1.3.2 The physical metric components f

To complete the construction of the n-soliton solutions for the metric (1.2), we also need

to calculate the metric coefficient f from (1.5)-(1.6) using the matrix g already found.

It is possible to see that the coefficient f , in the general n-soliton case, can also be

calculated explicitly by algebraic operations only like the metric components gab . Here

we trace the general outline to get the final result. The first step consists in substituting

into (1.5)-(1.6) the nonphysical solution g given by (1.53), to obtain the nonphysical

factor f . It is convenient to proceed one soliton a time. Thus, the one-soliton case yields

f1 = C1 f0

(µ2

1 − α2)−1

α µ21 Γ11 , (1.63)

where C1 is an arbitrary constant, f0 is the particular background solution for f , which

corresponds to the solution g0 and Γ11 is the single component of the matrix (1.50) . It

is, in this case for which k = 1 and l = 1 , a 1× 1 matrix:

Γ11 =(µ2

1 − α2)−1

m(1)a m

(1)b (g0)ab . (1.64)

The repetition of this operation n-times yields:

f = Cn f0 αn

(n∏

k=1

µ2k

)[n∏

k=1

(µ2

k − α2)]−1

det Γkl , (1.65)

where Cn is an arbitrary constant. Substituting into (1.5)-(1.6) g(ph) , A(ph) and B(ph)

(calculated by means of (1.28) and (1.28)), we find that the physical coefficient f (ph) is

given by the formula

f (ph) = f α1/2F , (1.66)

where the function F is11:

F = CF α−(n2+2n+1)/2

(n∏

k=1

µk

)n−1 [n∏

k=1

(µ2

k − α2)] [

n∏

k>l=1

(µk − µl)−2

]. (1.67)

CF is an arbitrary constant, and∏n

k>l=1 (µk − µl)−2 is equal to 1 for n = 1. From

(1.65)-(1.66) we get the final expression for the physical value of the coefficient f :

f (ph) = Cf f0 α−n2/2

(n∏

k=1

µk

)n+1 [n∏

k>l=1

(µk − µl)−2

]det Γkl , (1.68)

where Cf is an arbitrary constant which should be taken with the appropriate sign in

order to ensure the correct sign for f (ph) .

11We remand to the book [9] for further details about the deduction of this function.

19

1.4 Some remarks

It is worth noting that the number of solitons is constrained in the case of stationary

solutions. In fact if e = −1 , using canonical coordinates12, (1.3) becomes:

det g = −α2 = −ρ2 , (1.69)

consequently, the expression (1.61) gives:

det g = (−1)nρ2n

(n∏

k=1

µ−2k

)det g0 . (1.70)

If we take the particular solution g0, which by definition satisfies det g0 = −ρ2 , it follows

from (1.70) that the number of solitons , n , must always be even, since an odd number

would change the sign of det g and lead to an unphysical metric signature. Therefore, in

contrast to the nonstationary case, on a physical background all stationary axisymmetric

solitons (even those which correspond to real poles λ = µk ) can only appear in pairs

forming bound two-soliton states. Nevertheless, we can obtain physical solutions with

an odd number of solitons, but for this it is necessary to take a background solution with

a nonphysical signature, det g0 = ρ2 . The first examples of solutions of this kind were

obtained and investigated in [86].

To end this chapter, it is important to recall that, from the physical point of view, the

metrics of the kind (1.2) have many applications in gravitational theory. A lot of well

known solutions belong to such a class as the Robinson-Bondi plane waves, the Einstein-

Rosen cylindrical wave solutions, the homogeneous cosmological models of Bianchi types

I-VII, the Schwarzschild and Kerr solutions, Weyl’s axisymmetric solutions, etc.. The

reader can find a wide review of the solutions belonging to this class in [83]. In particular,

for the solutions obtained by means of the Belinski-Zakharov technique, we indicate the

table on page 546 of this book.

12See footnote number 1.2 at page 10.

Chapter 2

Solitonic solutions ofEinstein-Maxwell equations:Alekseev technique

As we have seen in the previous chapter, the Belinski-Zakharov technique produces soli-

ton solutions for the equations (1.1), that is the Einstein equations in vacuum. The

generalization represented by the inclusion of matter, i.e. the appearance of a nonzero

right hand side in the Einstein equations, generally destroys the applicability of the

Inverse Spectral methods. This is because the stress-energy tensor produces a non-

vanishing right hand side in the basic equation (1.4) that prevents the application of

the technique. However, there are some exceptions. This chapter describes the case of

the coupled Einstein-Maxwell field equations. Some examples of the application of the

Inverse Spectral methods to other non-vacuum cases can be found in the book [9].

The main new step for the solution of the Einstein-Maxwell equations was made by G.

A. Alekseev in 1980 [1]; the most detailed presentation of his approach was given in

his 1988 paper [3]. In this chapter we will follow very near the exposition given in [9],

adopting the notation used therein.

It is worth to be mentioned that the integrability ansatz of the Einstein-Maxwell equa-

tions was also given by other authors using different approaches; we remand to the books

[9, 83] for the relevant references.

2.1 The Einstein-Maxwell field equations

The problem we are dealing with, is represented by the coupled differential equation

of Einstein and Maxwell; in the general case (with the usual notation and with an

20

21

appropriate choice of units), they read as

Rij − 1

2g4

ij R = Tij

(√−g4 F ik

),k

=√−g4 j i

,

where g4ij is the unknown four-dimensional metric tensor1, and g4 is its determinant; Rij

is the Ricci tensor, R is the Ricci scalar and Tij is the energy-momentum tensor. For

the electromagnetic fields it is:

Tij =1

2

(−FikF

kj +

1

4FlmF lm g4

ij

),

where, being Ai the covariant components of the electromagnetic vector potential,

Fij = Aj,i − Ai,j .

Anyway, since the solitonic technique works with metric tensors of the form (1.2),

ds2 = gab(xρ) dxadxb + f(xρ) ηµν dxµdxν , (2.1)

it is convenient to adapt the Einstein-Maxwell equations according with this symmetry

reduction. The notation will be the same as that adopted in the previous chapter.

Hereafter, for convenience of the reader, we will sometimes rewrite some of the formulas

already used. The two-dimensional matrix ηµν is2

ηµν =

(−e 0

0 1,

)(2.2)

where e = 1 and e = −1 for the non-stationary and stationary solutions, respectively.

The determinant of the two-dimensional matrix gab is

det g = e α2. (2.3)

In order to make the integrable ansatz compatible with the metric (2.1) one should

assume the following structure for the electromagnetic potentials:

Aµ = 0 , Aa = Aa(xρ) . (2.4)

1We put the index 4 only to distinguish the total metric tensor from the 2×2 block gab , a, b = 0, 1 ,as made in the previous chapter.

2In this chapter we will often use a matrix notation. Thus, for definiteness in any matrix(Mik, M ik, M i

k or M ki ) the first index, independent of its up or down position, will always enumerate

the rows, and the second index will enumerate the columns.

22

Then, the only nonvanishing components for the covariant and controvariant electromag-

netic tensor field are

Fµa = Aa,µ , F µa =1

fηµνgacAc,ν . (2.5)

As usual, gab are the components of the inverse matrix of gab and thus defined by

gacgcb = δab , being δa

b the Kronecker symbol.

The Einstein-Maxwell equations can be written (with an appropriate choice of units) in

the form

Rba = 2

(FλaF

λb − 1

2δbaFλcF

λc

), (2.6)

Rµν = 2

(FνcF

µc − 1

2δµν FλcF

λc

), (2.7)

(fαF µa),µ = 0 . (2.8)

Since the two-dimensional trace in the Greek indices on the right hand side of 2.7 vanishes

identically, these equations can be written in the following equivalent form:

Rµµ = 0 , Rµ

ν −1

2δµν Rλ

λ = 2

(FνcF

µc − 1

2δµν FλcF

λc

). (2.9)

Direct calculation shows that the first of these equations is

ηµν(ln f),µν + ηµν(ln α),µν +1

4gabgcdηµνgbc,µ gda,ν = 0 , (2.10)

and that the second does not contain the second derivatives of the metric coefficient f :

ηµρ

[1

2(ln f),µ(ln α),ρ +

1

2(ln f),ρ(ln α),µ

]− 1

2δνµ ηρσ(ln f),ρ(ln α),σ

− ηνρ(ln α),µρ +1

2δνµ ηρσ(ln α),ρσ

− 1

4gabgcd

[ηνρgbc,µgda,ρ − 1

2δνµ ηρσgbc,ρgda,σ

]

= gcdAd,ρ

[2 ηνρAc,µ − δν

µ ηρσAc,σ

].

(2.11)

The trace of this equation vanishes identically. Consequently it gives only two indepen-

dent relations from which the metric coefficient f can be found by quadratures if the

matrix gab and the potential Aa are known. As in the vacuum case, (2.10) will then

be satisfied due to the Bianchi identities, and we can forget about this equation from

now on. Also, as in the vacuum case, the integration of the coefficient f from (2.11)

does not present a major difficulty and will be carried out at the end of the procedure.

Thus we turn our attention to the problem of the matrix gab and the potentials Aa

from the system (2.6) and (2.8). It is easy to see that this system does not contain the

23

coefficient f and that it forms a closed self-consistent system of equation for gab and

Aa . Calculating Rab and using the definitions (2.5) we can write these equation in the

form:

ηµν 1

α(α gbcgac,µ),ν = −4 ηµνgbcAa,µAc,ν + 2 δb

a ηµνgcdAc,µAd,ν , (2.12)

ηµν(α gacAc,µ),ν = 0 . (2.13)

It is noteworthy that the trace of right hand side of (2.12) vanishes identically and that

the function α(xµ) , in accordance with its definition (2.3), should satisfy the vacuum

’wave’ equation,

ηµνα,µν = 0 . (2.14)

Let us introduce the two-dimensional antisymmetric matrices,

εµν = εµν =

(0 1

−1 0

), (2.15)

and the same for Latin indexes,

εab = εab =

(0 1

−1 0

). (2.16)

Now we are in the position to start the integration scheme. Following Alekseev’s sugges-

tion of exploiting the duality property of the electromagnetic field, we introduce some

auxiliary potentials Ba , which will only play an intermediary role and will not be present

in the final results. In terms of the original potentials Aa , these are defined by,

Ba,µ = − 1

αηµν ενλ gab εbcAc,λ . (2.17)

It is easy to verify that the integrability condition for this equation, εµνBa,µν = 0 ,

coincides with the Maxwell equation (2.13). Relation (2.17) can also be written in the

inverse form:

Aa,µ =1

αηµν ενλ gab εbcBc,λ . (2.18)

Let us now combine Aa and Ba into a single complex potential Φa , defined by

Φa = Aa + i Ba . (2.19)

Then (2.17)-(2.18) are, respectively, the imaginary ad real part of the equation for Φa :

Φa,µ = − i

αηµνε

νλ gabεbc Φc,λ , (2.20)

24

from which the Maxwell equations for Φa trivially follow:

ηµν(α gac Φc,µ),ν = 0 . (2.21)

By direct calculation one can show that Einstein equations (2.12) can be written as

ηµν 1

α

(α gbc gac,µ

),ν

= −2 gbcηµνΦa,µΦc,ν . (2.22)

The imaginary part of the right hand side of this equation vanishes because the left hand

side is real. This is indeed a consequence of (2.17). Also due to this relation and the

identity

εadεbc = δab δ

dc − δa

c δdb , (2.23)

the real part of the right hand side of (2.22) coincides exactly with the right hand side

of (2.12). It is thus clear that any solution of (2.19)-(2.22), gab and Aa = Re Φa , is also

a solution of the Einstein-Maxwell equations (2.12)-(2.13).

2.2 The spectral problem for Einstein-Maxwell fields

Now we want to represent the Einstein-Maxwell equations (2.19)-(2.21) as self-consistency

condition of a linear spectral problem. The five-dimensional generalization3 of the

Belinski-Zakahrov technique, suggests that it is reasonable to look for the solution in

the framework of the same spectral problem (1.15), but for three-dimensional matrices

A , B and ψ . This three-dimensional generalization is straightforward4 and can be

written as:

Πµψ =

(e λ

λ2 − e α2ηµρ ερσKσ − e α

λ2 − e α2Kµ

)ψ , (2.24)

where the operators Πµ are

Πµ = ∂µ +2 e (λ2 ηµρ ερσα,σ − λα α,µ)

λ2 − e α2∂λ , (2.25)

and ηµρ and ερσ are given in (2.2) and (2.15). The matrices Kµ and ψ are now three-

dimensional and the function α(xµ) is the same as before, i.e. it satisfies (2.14).

Let us note that the equations (2.24) and (2.25) are written in terms of the coordinates xµ

introduced at the beginning of the paragraph 1.1, while the Lax-Pair (1.15) is written in

terms of the null coordinates ζ, η . This is made in order to develop a universal approach

3See paragraph 1.5 in [9].4Namely, even if it is written in a different form, at least of the dimension of the matrices, it

is formally equal to Lax Pair of Belinski-Zakharov for Einstein equations in vacuum. See below theexpressions of this Lax Pair for the non-stationary and stationary cases.

25

both to the stationary and the nonstationary cases. Hereafter we open a short aside to

give the link between the expressions of the Lax Pair (2.24) with its expression as given

by the (1.15) in null coordinates, and that one given in chapter 8 of the book [9] for the

stationary case. In these transformations the dimension of the matrices is not important.

———————–

Written for each component the system (2.24) becomes:

Π2ψ = − 1

λ2 − e α2(λK3 + e αK2) ψ , Π3ψ = − e

λ2 − e α2(λK2 + αK3) ψ ,

where

Π2 = ∂2 − 2 λ (λα,3 + e α,2 α)

λ2 − e α2∂λ , Π3 = ∂3 − 2 e λ (λ α,2 + α,3 α)

λ2 − e α2∂λ .

We recall that the null coordinates ζ, η are defined in terms of the coordinates xµ by:

ζ = (√

e x2 + x3)/2 , η = −(√

e x2 − x3)/2 .

• For nonstationary case we have: x2 = t , x3 = z , e = 1 , thus, the system (2.24),

expressed in terms of the null coordinates, reduces to the system (1.15)

D1 ψ =A

λ− αψ

D2 ψ =B

λ + αψ

, where D1 = ∂ζ − 2 α,ζ λ

λ− α∂λ , D2 = ∂η +

2 α,η λ

λ + α∂λ ,

if Π3 + Π2 = D1 , Π3 − Π2 = D2 , and A = −(K3 + K2) , B = K3 −K2 .

• For stationary case it results: x2 = ρ , x3 = z , e = −1 and α = ρ . Now

the system (2.24) becomes (see chapter 8 of the book [9]):

D1ψ =ρ V − λU

λ2 + ρ2ψ

D2ψ =ρU + λV

λ2 + ρ2ψ

, where D1 = ∂z − 2λ2

λ2 + ρ2∂λ , D2 = ∂ρ +

2λρ

λ2 + ρ2∂λ ,

if Π2 = D1 , Π3 = D2 , and U = K3 , V = K2 . In this case, U = ρ g,ρ g−1 and

V = ρ g,z g−1 .

———————–

26

Due to the ’wave’ equation (2.14) for the function α the operators Πµ commute:

ΠµΠν − ΠνΠµ = 0 . (2.26)

The self-consistency conditions for (2.24) are:

ηµν Kµ,ν = 0 , (2.27)

εµν

(Kµ,ν +

1

αKµ Kν − 1

αα,νKµ

)= 0 . (2.28)

The second of these equations implies that the two matrices Kµ can be written in terms

of a single matrix X in the form

Kµ = αX,µX−1 . (2.29)

Note that the matrix X plays the same role as g in the two-dimensional vacuum case.

Then the (2.28) is just the integrability condition of (2.29) for X , and (2.27) gives

a really nontrivial condition in the form of the following differential equation for the

matrix X

ηµν(αX,µ X−1),ν = 0 . (2.30)

In general this equation does not reproduce the Einstein-Maxwell system. We should

perform a reduction on X introducing some additional constraints that are not a con-

sequence of the self-consistency conditions of the spectral problem (2.24), but that are

compatible with them. To formulate these constraints, let us first introduce the following

matrix Ω :

Ω =

0 1 0

1 0 0

0 0 0

. (2.31)

In what follows we shall use the small Latin indices from the first part of the alphabet

for the first and the second rows and columns of the three-dimensional matrices and, we

recall, that they can take the values 0 and 1; for the third rows and columns we shall use

the star symbol. Capital letters of the first part of the Latin alphabet will be used to

enumerate the matrix components, thus A = (a, ∗) , B = (b, ∗) , ... . With this convention

the matrix Ω , for example, can be written in the following form:

Ω = ΩAB =

(Ωab Ωa∗

Ω∗b Ω∗∗

)=

(εab 0

0 0

). (2.32)

It is now convenient to introduce two special combinations made up of the matrix X and

its derivatives. Thus, we define Uµ by

Uµ = i e α ηµρ ερσX−1X,σ + 4 e (α2X−1),µ Ω . (2.33)

27

The additional constraints we need to impose on the matrix X can now be written as

X = X† , (2.34)

XUµ = −4 i e α ηµν ενσ Ω Uσ , (2.35)

where † means Hermitian conjugation. These are the two fundamental constraints.

However, we can still impose three new constraints on the matrix X . These new con-

straint are weaker, are easily imposed and do not represent a loss of generality. It is easy

to see from (2.31)-(2.35) that X33,µ = 0 , so that X33 = constant . Due to the invari-

ance of equations with respect to the rescaling X → cX , α → c α ( c is an arbitrary

constant), the value of X33 can be chosen at will. We chose to put

X33 = 2 . (2.36)

The second of these new constraints is the requirement that the determinants of the

two-dimensional blocks, constructed from the first two rows and columns of the matrices

Uµ , are not zero. As follows from definition (2.33), the upper and left blocks of matrices

Uµ are

(Uµ)ab = i e α ηµν ενσ

[(X−1)a3 X3b

,σ + (X−1)ac Xcb,σ

]+ 4 e

[α2(X−1)ac

],µ

εcb . (2.37)

Because det X 6= 0 and det Ω = 0 , it follows from (2.35) that det Uµ = 0 . Thus our

second new constraint can be formulated as

rank(Uµ) = 2 , det[(Uµ)a

b] 6= 0 . (2.38)

These properties will be satisfied automatically by the construction of the solutions and,

in practice, do not mean a loss of generality.

The third of these new constraints is that at least one of the diagonal elements of the

two-dimensional matrix

Xab − 1

2Xa3X3b (2.39)

does not vanish. This condition can also be easily imposed and does not represent a loss

of generality for the Einstein-Maxwell fields. It is easy to show that the unique structure

for the matrix X that follows from the constraints (2.34)-(2.39) is

X =

−4εacgcdε

db + 8εacΦcεbdΦd 4εacΦc

4εacΦc 2

,

X−1 =

−1

4εacgcdεdb −1

2εacg

cdΦd

−14εbcgcdΦd −1

2+ gcdΦcΦd

.

(2.40)

28

It is useful to notice that the matrix X is similar to the matrix X ′ ,

X ′ = 4

(gab + 2 ΦaΦb Φa

Φb12

).

That is X ′ = R−1XR if R = diag (εab, 1) . Moreover, it is worth underlining that

det X = 32 det g = 32 e α2 5.

Equation (2.20), which is satisfied by the complex electromagnetic potentials Φa , is now

a consequence of of the (a∗)-components of (2.35). Now we can see that the substitution

of this form of the matrix X into the self-consistency equations (2.30) exactly reproduces

the Einstein-Maxwell equations (2.21)-(2.22) and nothing else.

It is now worth to emphasizing that it is not possible to construct a dressing method

based on the spectral equation (2.24) and producing dressed solutions satisfying the con-

straints (2.34)-(2.35), at least following the standard approach described in chapter 16.

For this reason, Alekseev introduces, mapping the λ-spectral plane on a new w-spectral

plane, a different Lax Pair.

We introduce a new generating matrix φ ,which is related to the generating matrix ψ of

the vacuum case by

ψ = (X − 4iΩ) ϕ . (2.41)

The substitution of this expression into (2.24) shows that, due to the additional con-

straints (2.35), this new generating matrix ϕ satisfies the following spectral equation:

Πµϕ =i λ

(λ2 − e α2)2

[ (λ2 + e α2

)Uµ − 2 e α λ ηµρ ερσUσ

]ϕ , (2.42)

where Uµ are the matrices (2.33). The advantage of this representation of our spectral

problem is that it consists of rational functions not only with respect to the original

parameter λ , but also with respect to a new parameter w defined by

w = −1

2

(λ + 2β +

e α2

λ

). (2.43)

5The factor 32 is just a consequence of the choice X33 = 2 .6To preserve the structure (2.40) of the X matrices under the dressing Darboux transformation

(1.25), it is necessary to generalize the condition (1.23) substituting the transposition with the Hermitianconjugation: X = χ (λ)X0 χ†

(α2/λ

). As a consequence the 3 × 3 generalization of (1.53) becomes

XAB = (X0)AB − ∑k,l µ

−1k µ−1

l ΠklL(l)A L

(k)B , where Πkl =

[− (

α2 − µkµl

)−1m

(k)A m

(k)B

(X0

)AB

]−1

and

L(l)A = m

(l)B (X0)BA . A direct calculus shows that these formulas do not allow to generate new electro-

magnetic fields; namely the dressed components of the electromagnetic potential result to be triviallynull.

29

Here β is the second independent solution of the ‘wave’ equation (2.14), which has the

following connection to the function α :

β,µ = −e ηµρ ερσ α,σ . (2.44)

Due to this connection, the parameter w(xµ, λ) satisfies the identity

Πµw = 0 . (2.45)

The relation (2.43) can be understood as a transformation λ = λ(α, β, w) from the

parameter λ to new spectral parameter w . After this transformation is applied to any

generating matrix ϕ(xµ, λ) it must be understood that such a matrix becomes a function

of xµ and w only (more precisely as ϕ [xµ, λ(α, β, w)] ). In this sense and due to identity

(2.45), for any matrix φ we have

Πµφ = (∂µφ)w , (2.46)

where the right hand side is the usual partial derivatives with respect to xµ performed

under the assumption that w is some free parameter independent from xµ . The key

point now is that the application of this transformation to (2.42) shows its rational

dependence on w together with a simple structure of differential operators:

∂ ϕ

∂xµ=

1

2 i [(w + β)2 − eα2][(w + β) Uµ + e α ηµρ ερσUσ] ϕ. (2.47)

The analyticity of this equation with respect to the spectral parameter w is important,

because it allow us to apply to the construction of its solitonic solutions the dressing

procedure used in the vacuum case, but with the meromorphic structure of the dressing

matrices in the complex w-plane. At the same time the simplicity of the differential

operators allow us to impose the additional constraints (2.34)-(2.35) in a simple way.

2.3 The component gab and the potentials Aa

The construction for the metric components gab and the electromagnetic potentials Aa

needs, as first thing, the building of the n-soliton solution of the spectral problem (2.47);

it will be resolved at first in general, i.e. without assuming any additional structure for

the matrices Uµ . After that, we will impose all the necessary additional constraints (i.e.

the conditions which follow from (2.33)-(2.39).

2.3.1 The n-soliton solution of the spectral problem

Let us start this first stage with the introduction of a new matrix Λ νµ :

Λ νµ =

1

2 i

(w + β)δνµ + e α ηµρ ερν

(w + β)2 − e α2, (2.48)

30

and then the spectral equation (2.49) takes the form

ϕ,µ = Λ νµ Uνϕ . (2.49)

Let ϕ(0) and U(0)µ be some background solution of (2.49) with some given function

α and β . Then we search for the new ’dressed’ solution, corresponding to the same

functions α and β , of the form

ϕ = χϕ(0) . (2.50)

Because ϕ(0) is a solution, from (2.49) we obtain the following equation for the dressing

matrix χ :

χ,µ = Λ νµ ( Uν χ− χU (0)

ν ) . (2.51)

Now we will use the Latin indices of the last part of the alphabet (i.e. letters i, j, k... )

to enumerate quantities related to the poles of matrix χ . We assume that χ and χ−1

have n simple poles,

χ = I +n∑

k=1

Rk

w − wk

, χ−1 = I +n∑

k=1

Sk

w − wk

. (2.52)

Here and in the following we do not assume summation on indices i, j, k, ... ; such a

summation will be always indicated by the symbol∑

. At this stage wk and wk can

be arbitrary functions of the coordinates xµ . Also, in what follow, we consider that all

the 2n functions wk and wk are different. From the identity χχ−1 = I we have the

following condition for the matrices Rk(xµ) and Sk(x

µ) :

Rk χ−1(wk) = 0 , χ(wk) Sk = 0 , (2.53)

where the expression of the type F (wr) means the value of the function F (w, xµ) at

w = wk . The dependence of the coordinates is omitted for simplicity. Equation (2.53)

imply that we can look for matrices Rk and Sk of the form

(Rk)AB = n

(k)A m(r)B , (Sk)A

B = p(k)A q(k)B . (2.54)

It is worth noting that the construction of the solution of the spectral problem (2.47)

that we are carrying out is valid for matrices of any dimension.

The substitution of (2.54) into (2.53) gives two systems of algebraic equations from which

one can express all vectors n(k)A and q(k)A in terms of vectors m(k)A and p

(k)A as

n∑

l=1

p(k)B m(l)B

wl − wk

n(l)A = p

(k)A , (2.55)

n∑

l=1

m(k)Bp(l)B

wk − wl

q(l)A = −m(k)A . (2.56)

31

If we now introduce the n× n matrix Tkl and its inverse (Tkl)−1 ,

Tkl =p

(k)B m(l)B

wl − wk

,

n∑

l=1

Til(T−1)lk = δik , (2.57)

we obtain

q(k)A = −n∑

l=1

(T−1)lk m(l)A , n(k)A =

n∑

l=1

(T−1)kl p(k)A , (2.58)

for the vectors q(k)A and n(k)A .

To obtain the vectors m(k)A and p(k)A we use (2.51). It can be written in the form

Λ νµ Uν = χ,µ χ−1 + Λ ν

µ χU (0)µ χ−1 , (2.59)

or, equivalently, as

Λ νµ Uν = −χ(χ−1),µ + Λ ν

µ χU (0)µ χ−1 . (2.60)

All the terms in that functions are meromorphic functions of w that vanish at w →∞ .

Thus, to satisfy these equations it suffices to eliminate the residues of all their poles.

The first terms on the right hand side generate the second order poles at these points if

wk and wk depend on the coordinates xµ . Consequently, the first result we have from

(2.59) and (2.60) is that

wk = constant , wk = constant . (2.61)

Note that, due to the simplicity of the differential operators in the spectral equation

(2.49), the poles and the zeros of matrices χ and χ−1 in the w-plane are stationary

points and not trajectory as in the vacuum case.

Now the right hand side of (2.59) contain only simple poles at the points w = wk and

w = wk . The elimination of their residues gives the following equations for the matrices

Rk and Sk :

Rk,µ χ−1(wk) + Λ νµ (wk) Rk U (0)

ν χ−1(wk) = 0 , (2.62)

χ−1(wk) Sk,µ − Λ νµ (wk) χ(wk) U (0)

ν Sk = 0 . (2.63)

The solution of these equations can be expressed in terms of the background matrix

ϕ(0) . It is easy to check that if we substitute the matrices Rk and Sk into (2.62)-(2.63),

taking into account the conditions (2.53) and the fact that ϕ(0) and U (0) are solutions

of (2.49), then the system (2.62)-(2.63) is a set of differential equations for the vectors

m(k)A and p(k)A . The general solution of which is:

m(k)A = k(k)B [ (ϕ)(0)−1(wk) ]BA , (2.64)

p(k)A = l(k)B [ ϕ(0)(wk) ]AB . (2.65)

32

where k(k)B and l(k)B are 2n arbitrary constant vectors.

The structure of the coefficients Λ νµ , see (2.48), shows that in (2.59) we still have poles

with nonzero residues at the two points where (w + β)2 − e α2 = 0 . If we define√

e as

√e = 1 if e = 1 ,

√e = i if e = −1 , (2.66)

these poles can be written as w = w+ and w− , where

w+ = −β + e√

e α , w− = −β − e√

e α . (2.67)

The elimination of the residues of (2.59), at these poles does not produce any new

constraints on the matrices Rk and Sk , but gives the value of the matrices Uµ in terms

of Rk , Sk and the background matrices U(0)µ :

Uµ =1

2

[χ(w+) U (0)

µ χ−1(w+) + χ(w−) U (0)µ χ−1(w−)

]

+1

2e√

e ηµρ ερσ[χ(w+) U (0)

σ χ−1(w+)− χ(w−) U (0)σ χ−1(w−)

].

(2.68)

With this formula we have finished the construction, in general, of the n-soliton solution

of the spectral equation (2.49). This means that we can now express the matrices Uµ ,

ϕ and χ in terms of the background solution U(0)µ , ϕ(0) up to the freedom of choosing

arbitrary constants wk , wk and the arbitrary constants k(k)B and l(k)B in the vectors

m(k)A and p(k)A .

It is clear that one can use such a freedom to further specify the solution when necessary.

This is indeed necessary because the solution we have constructed for the matrices Uµ

does not guarantee that these are the same matrices that can be expressed in terms of X

matrix in (2.33), and that such a matrix X satisfies (2.30) and the additional constraints

(2.34)-(2.39).

It is remarkable and nontrivial that all these additional requirements can be satisfied

due to the freedom of the parameters. This is a consequence of the fact that our spectral

equations have ‘conserved integrals’ (some authors call them ‘involutions’), i.e. some

expressions quadratic in the generating matrix that give zero under the action of the

operators Πµ .

2.3.2 The matrix X

Let us now return to the Einstein-Maxwell three-dimensional problem. Here the analogue

of the two-dimensional metric tensor is the three-dimensional matrix X . However this

matrix needs to be Hermitian, not symmetric. Furthermore the dressing matrix χ(w) is

rational on w , which means that the replacement λ → e α2/λ is irrelevant in this case.

33

This suggests that we impose the basic additional constraints (2.34)-(2.35) assuming the

existence of a ‘conserved integral’ of the following form:

∂µ

[ϕ†(w, xµ) W (w, xµ) ϕ(w, xµ)

]= 0 (2.69)

with some and, as yet unknown, matrix W 7.

The existence of the integral (2.69) means that ϕ† W ϕ = Q(w) , where Q does not

depend on xµ ; we impose that matrix Q(w) be hermitian: Q† = Q . In this case the

freedom of the transformation ϕ(w, xµ) → ϕ(w, xµ)γ(w) , which obviously exist for the

spectral equation (2.49), allows to normalize each solution in such a way that the matrix

Q transforms as Q → γ†Qγ . Since Q is hermitian its transformed form can be made

universal, i.e. the same for all solutions, by choosing an appropriate transformation

matrix γ(w) for each solution. Moreover, this universal form can be made diagonal,

real and independent of w . Thus, without loss of generality, and within the class of

Hermitian matrices Q , the integral (2.69) can be written as

ϕ†(w, xµ) W (w, xµ) ϕ(w, xµ) = C , (2.70)

where

C = diag (C1, C2, C3) , C1, C2, C3 = constant , (2.71)

and where the three constants, C1, C2 and C3 , are real. This implies that

C = C† . (2.72)

Even if the constants C1, C2, C3 can be eliminated from the solutions by making their

modulus equal to 1 , we will keep the matrix C in the more general form (2.71) in order

to leave open the possibility for more convenient choices of arbitrary parameters in the

final form of the solution.

Since (2.70) is universal, it is also valid for the background solution

ϕ(0)† W (0) ϕ(0) = C , (2.73)

where W (0) is the matrix W calculated for the background solution. From (2.70) and

(2.73) we have

W−1 = χ(W (0))−1 χ†. (2.74)

Now let us assume that the matrix W−1 has no singularities at the points where the

matrices χ and χ−1 have poles. Then in order to satisfy (2.74) one needs first to

7Note: The definition of the Hermitian conjugation of matrix functions that depend on the complexparameter w is the following: to obtain the hermitian conjugation M†(w, xµ) of any matrix M(w, xµ)as function of w, one should first calculate the value of the matrix M at the complex conjugate point w,i.e. the value M(w, xµ), and then take the usual Hermitian conjugate of this value.

34

eliminate the residues on the right hand side of this relation, i.e. at w = wk and

w = wk . Let us consider first the set of points w = wk . The residues at these points

vanish if (I +

n∑

k=1

Rk

wl − wk

)(W (0))−1(wl)R

†l = 0 . (2.75)

or in components

n∑

k=1

m(k)D[(W (0))−1(wl)

]DB

m(l)B

wk − wl

n(k)A =

[(W (0))−1(wl)

]AD

m(l)D . (2.76)

It follows from (2.72)-(2.73) that we should construct any background solution in such

a way that the matrix W (0) is hermitian. Now it is easy to check that the equation

eliminating the residues on the right hand side of (2.74) at the second set of poles, i.e.

at w = wk , coincides exactly with (2.76). Therefore this is the only equation we need in

order to have regularity of (2.74) at the points where matrices χ and χ−1 are singular.

Equation (2.76) is an algebraic system where vectors n(k)A can be expressed in terms of

the vectors m(k)A . If we substitute into such a system (2.65) for the vectors p

(k)A , (2.55)

takes the form:

n∑

k=1

m(k)D[ϕ(0)(wl)

]DB

l(l)B

wk − wl

n(k)A =

[ϕ(0)(wl)

]AD

l(l)D . (2.77)

Of course, this equation should coincide with (2.75). The coincidence takes place when

wk = wk , (2.78)

and [(W (0))−1(wl)

]DB

m(l)B = [ ϕ(0)(wl) ]DB l(l)B . (2.79)

The first condition shows that the pole of the inverse matrix χ−1 should be located at

the points which are complex conjugate to the pole of the matrix χ . To discover the

second condition we should substitute (2.64) into (2.79) for the vectors m(k)A and the

expression for (W (0))−1 in terms of the ϕ(0) and C which follows from (2.73):

(W (0))−1 = ϕ(0)C−1ϕ(0)† . (2.80)

After this substitution we have to take in account the conditions (2.71)-(2.72) for matrix

C and the fact that now wk = wk . Then the resulting from 2.79 is very simple:

k(k)A = CAB l(k)B , (2.81)

where CAB are the components of the diagonal matrix C . This allows us to write all

the constants k(k)A in terms of l(k)A , or viceversa.

35

The same results can be obtained if we start our analysis from the ’conserved integral’

(2.74) written in its inverse form:

W = (χ−1)†W (0)χ−1 . (2.82)

2.3.3 Verifications of the constraints

Up to this point we do not need to know the explicit structure of the matrix W , apart

from their regularity at points w = wk and w = wk and the hermiticity of their

background values. Under these conditions the relations (2.78) and (2.80) between the

free constant parameters ensure the absence of poles at the points w = wk and w = wk

on the right hand side of (2.74) or (2.82). Now, we have to fix the exact structure of

matrix W in such a way that (2.82) is satisfied not only at the poles but everywhere in

the complex w-plane, and that it also satisfies the constraints (2.33)-(2.35). This goal

can be achieved if we choose the matrix W to be a linear function of w of the following

form:

W = X − 1

4XEX + 4 i (w + β) Ω , (2.83)

where

E =

0 0 0

0 0 0

0 0 1

. (2.84)

With this choice the matrix

W − (χ−1)†W (0)χ−1 (2.85)

clearly has no singularities at finite values in the w-plane. It has also no singularities at

infinity because the matrix χ−1 tends to unity as w → ∞ and W → W (0) since the

fixed constant matrix Ω has the same values for the background and dressed solutions.

This eliminates the poles in (2.85) at infinity in the w-plane. However, this expression

still has non-zero finite values at w → ∞ , which should vanish if we wish to satisfy

(2.82). Using (2.52) and (2.83) it is easy to calculate the first nonvanishing term of the

matrix (2.85) at w →∞ . Equating this term to zero we get

X − 1

4XEX = X(0) − 1

4X(0)EX(0) + 4 i (S†Ω + ΩS) , (2.86)

where X(0) is the background value of the matrix X and

S =n∑

k=1

Sk . (2.87)

36

The components of S follow from (2.54) and (2.58),

S BA = −

n∑

k,l=1

(T−1)kl p(l)A m(k)B . (2.88)

Now (2.82) is completely satisfied because (2.85) represents an analytic function at each

point on the w-plane which vanishes at infinity. Such a function is everywhere zero in

the w-plane.

Due to the special structure of the matrix E , it is easy to prove by direct computation

the hermiticity of the matrix X from the hermiticity of the matrix X− 14XEX . Then it

is also easy to see from (2.86) that the hermiticity of the dressed matrix X(0) implies the

hermiticity of the dressed matrix X . Another important property follows from (2.86),

namely, that X∗∗ = 2 if X(0)∗∗ = 2 . Because the background solution X(0) satisfies, by

definition, all the additional constraints (including (2.34) and (2.36) ), (2.86) guarantees

that all these constraints are satisfied for the dressed solution X .

Since (2.86) gives the matrix X− 14XEX , we need to know how to calculate the matrix

X from this. Let us introduce a new matrix

G = X − 1

4XEX . (2.89)

This expression can be easily inverted:

X = G + GEG , if X∗∗ = 2 . (2.90)

Due to this property and the trivial identity EΩ = 0 , it is easy to prove that we have

an equivalent form of condition (2.35), which is obtained by just replacing X on the left

hand side by G

GUµ = −4 i e α ηµρ ερσΩ Uσ . (2.91)

Another equivalent equation can be obtained by multiplying (2.91) by e√

e ηνλ ελµ , and

taking the sum and the difference of the new equation and the original one. The result

is

(G± 4 i e√

e α Ω)(Uµ ± e√

e ηµρ ερσ Uσ) = 0 . (2.92)

From (2.83), (2.89) and (2.67) we have,

G± 4 i e√

e α Ω = W (w±) . (2.93)

Using (2.82) we see that the dressing formulae for the first factors in (2.92) are

G± e√

e α Ω = (χ−1)†(w±)(G(0) ± e

√e α Ω

)χ−1(w±) . (2.94)

37

The dressing formulae for the second factor in (2.92) can be obtained by multiplying

(2.68) by e√

e ηνλ ελµ , and taking the sum and the difference of this new equation with

(2.68) itself. We thus obtain

Uµ ± e√

e ηµρ ερσUσ = χ(w±(U (0)

µ ± e√

e ηµρ ερσ U (0)σ

)χ−1(w±) . (2.95)

The product of these last two equations shows that if the left hand side of (2.92) is zero

for the background solution, it is also zero for the dressed solution. Thus we conclude

that condition (2.35) is valid because the background solution verifies it and because we

have already ensured that (2.49) and (2.69) are satisfied.

It follows from (2.95) that the traces of the matrices Uµ e√

e ηµρ ερσUσ are also conserved

under the dressing procedure. However, it is more convenient to deal directly with the

traces of the matrices Uµ , by taking trace of (2.68). From this equation we have simply

that Tr(Uµ) = Tr(U(0)µ ) . Since (2.33) is trivially valid for the background solution, we

have that Re[ Tr(U(0)µ ) ] = 0 , which one can easily verify using (2.40) for the background

matrix (X(0))−1 and the fact that the two-dimensional background matrix is real and

symmetric. As a consequence we have that for the dressed matrices Uµ ,

Re [ Tr(Uµ) ] = 0 . (2.96)

Finally we need to prove that our matrices Uµ and X are connected by (2.33). Again,

for the background solution such a relation is trivially satisfied because we started with

a given matrix X(0) and the new matrices U(0)µ were defined just using (2.33). In this

case (2.33) represents an additional constraint connecting Uµ and X .

To prove the validity of the constraint (2.33) one can start from the conserved integral,

[ϕ†(G + 4 i (w + β) Ω) ϕ

],µ

= 0 . (2.97)

After differentiation and by substitution into this formula of the expression (2.47) for

ϕ,µ (and for is hermitian conjugated ϕ†,µ ), we multiply the result by (w + β)2−e α2 and

obtain on the left hand side of (2.97) a quadratic polynomial in the spectral parameter

w , or more precisely, in w + β . Since our solution already ensures that condition (2.97)

is satisfied, all the coefficients in this polynomial vanish. The zero value of the coefficient

of the quadratic term gives the identity

(G + 4 i β Ω),µ = 2 (U †µ Ω− Ω Uµ) . (2.98)

The remaining coefficients of the polynomial give nothing new: the linear coefficient just

lead again to (2.98), and the free coefficient leads to (2.91).

It would be nice to prove the validity of (2.33) by the method we used before, i.e. by

38

proving that the dressing procedure preserves this relation. However, we have no suitable

dressing formula for the right hand side of (2.33). Instead, we can calculate the exact

structure of the matrices Uµ from those equation for which we have already proved the

validity. Then we can check the correctness of (2.33) by direct substitution. At this

stage we know that (2.34)-(2.39), (2.96) and (2.98) are valid and that matrix X has the

structure of (2.40). Detailed analysis in which a key rule is played by (2.35) in the form

(2.91) and by (2.96) and (2.98), shows that this system leads to the following unique

structure for the matrices Uµ :

(Uµ) ba = gac,µε

cb − i

αηµρ ερσ gac εcd gdf,σ εfb + 2 Φa,µΦc εcb

(Uµ) ∗a = −Φa,µ

(Uµ) b∗ = 2 Φc εcd (Uµ) b

d

(Uµ) ∗∗ = 2 εab Φa,µ Φb

. (2.99)

Direct substitution of this result together with the matrix X , see (2.40), into (2.33)

shows that this equation is identically verified. This is the final step in the proof that the

matrix X which first appeared in (2.29), has the structure (2.40) and satisfies (2.30). As

a consequence, the functions gab and Re(Φa) , which can be extracted from (2.86) using

(2.89), (2.90) and (2.40), indeed represent a solution of the Einstein-Maxwell equations.

2.4 The metric component f

To complete the construction of the n-soliton solution for the Einstein-Maxwell equa-

tions, we need to compute the metric coefficient f from (2.11). Without giving the

details (the reader can find them in [9]), we limit here to tell that the equation (2.11)

can be written as:

(ln f),µ = (ln |D|),µ − (ln α),µ + iD−1ελνα,ν Tr[(G−1 + E) U †

λ Ω Uµ

], (2.100)

where

D = ηµνα,µα,ν . (2.101)

The integration of (2.100) is rather long and will not be done here. It was performed by

Alekseev [3], and yields the very simple result:

f = C0f(0)T T , (2.102)

where, C0 = constant , f (0) is the background value of the metric coefficient f and T

is the determinant of the n× n matrix Tkl ,

T = det Tkl . (2.103)

39

2.5 Summary of prescriptions

Let us summarize now, step by step, the set of practical prescriptions for constructing

n-soliton solution of the Einstein-Maxwell equations starting with a given background

solution.

1. Take some background solution g(0)ab and A

(0)a of the Einstein-Maxwell equations

(2.12)-(2.13). Calculate the determinant of the matrix g(0)ab and find the function

α(xµ) from the relation α2 = e det g(0)ab , after choosing some definite root of this

quadratic equation, for example α > 0 .

2. Take the previous g(0)ab , A

(0)a and α , and find, using (2.17), the auxiliary potentials

B(0)a (up to two arbitrary real additive constants), and write the background value

of the complex electromagnetic potentials Φ(0)a = A

(0)a + i B

(0)a .

3. Substitute the values Φ(0)a and g

(0)ab into (2.40). This gives the background value

X(0) of the matrix X .

4. Calculate the background matrices U(0)µ by substituting into (2.33) the previous

values of X(0) and α .

5. Use (2.44) to find the function β(xµ) , up to some arbitrary real additive constant.

6. From (2.83) compute the background matrix W (0) in terms of X(0) and β .

7. Substitute α and β and U (0) into the spectral equation (2.47) and find the nor-

malized solution for the background generating matrix ϕ(0)(w, xµ) , i.e. the solution

that satisfies (2.73) with the matrix C defined in by (2.71)-(2.72).

8. Using the previous ϕ(0) , construct the vectors m(k)A and p(k)A according to (2.64)-

(2.65), where wk = wk , and where the constants k(k)A and l(k)A are related by

(2.81).

9. With these values for m(k)A and p(k)A construct the matrix Tkl using (2.57), and

again taking wk = wk .

10. Substitute the matrix Tkl and the vectors m(k)A and p(k)A into (2.88) to obtain

the matrix S .

11. Finally, from (2.86), with the help of (2.89) and (2.90) calculate the components of

the matrix X in terms of X(0) and S . The matrix X , thus obtained when written

40

in the form (2.40), gives the dressed solution gab and Aa of the Einstein-Maxwell

equations in terms of the Xab and Xa∗ components of X as

gab =1

4εca

(Xcd − 1

2Xc∗Xd∗

)εdb , (2.104)

Aa =1

4εcaReXc∗ . (2.105)

12. Takeing the coefficient f (0) from the background solution (2.1) and, after calculat-

ing the determinant T of the Tkl matrix defined by (2.57), use (2.102) to obtain

the coefficient f .

2.6 Some remarks

It is worth making some remarks on the relation between soliton solutions described

here for the particular case when Φa = 0 (vacuum) and the vacuum soliton solutions

which can be constructed using the Belinski-Zakharov technique described in the first

chapter. There is not a comprehensive analysis of this relation yet. However the results

obtained in [29, 30, 31, 37] show that to all appearances the n-soliton vacuum solution

corresponding to n complex coordinate-independent poles in the complex w-plane in

Alekseev’s approach is equivalent to the 2n-soliton solution corresponding to n pairs of

complex conjugate (coordinate-dependent) poles in the complex λ-plane in the framework

described in chapter 1. By equivalent we mean two solutions can be transformed into each

other by a coordinate transformation. Nevertheless, in the vacuum case, the Inverse Scat-

tering Method described in chapter 1 using the complex structure in the λ-plane, gives a

richer set of soliton solutions, since it also includes solutions which correspond to an odd

number of poles in the λ-plane. In the Alekseev approach for each single complex pole a

distinct complex conjugate pole must appear in the inverse of the generating matrix.

Hence, if a single pole of the Belinski-Zakharov approach must be real, such kind of poles

has no place in the Alekseev framework since, in this case, the generating matrix and its

inverse would have the same pole in such a way that the procedure becomes singular.

Thus there are no analogues of such solutions in the framework that uses the complex

structure in the w-plane, at least following all the prescriptions of Alekseev procedure.

At this regard, it is worth mentioning the work of Micciche and Griffiths [67], in which,

following a different prescription, they obtained such 1-soliton solutions in w-plane by

introducing distinct real poles in the inverse of the generating matrix. Anyway, they

found this possibility only for solutions in vacuum, that is for null electromagnetic fields.

Chapter 3

Exact stationary axially symmetricone-soliton solution on Minkowskybackground

In this chapter, we show how to apply the Alekseev solitonic technique to generate a

stationary axially symmetric one-soliton solution on a Minkowsky background. The aim

of this chapter is to illustrate practically the application of the procedure, since both the

solution that will be found [66] and the reparameterization to get it [3] are well known.

It is worth reminding here that a generating technique provides exact solutions of the

field equations in terms of a certain number of mathematical constants; after this primer

product is obtained, further work is necessary on it to find, when possible, its physical

meaning. This work generally consists, on the one side, in a reparameterization that

links the mathematical constants to the physical constants; on the other side, in finding

further conditions that have to be imposed on the parameters to eliminate some possible

spurious behaviour, or to characterize, or select, a particular solution of interest among

a more general family, naturally obtained by means of the generating technique.

This chapter is organized as follows: in the first part the generating procedure will

be applied, step by step, until the determination of the T matrix defined in (2.57).

In the second part a known solution will be introduced, possessing the same degree of

freedom (that is the number of independent constants) of the one-soliton solution; after

an appropriate comparison, the transformations to express the mathematical constants

in terms of the physical ones will be found1. The final part consists in the application

of the last steps of the procedure to calculate those components that are useful to the

1The same result can be obtained by an analysis of the asimptotic behaviour of the fields far fromtheir source.

41

42

determination of the one-soliton solution in the form of the introduced known solution.

The result, of interest here, is not the exact solution representing a one-body source

characterized by mass, angular momentum, electric and magnetic charge and NUT pa-

rameter. Rather, it is principally the reparameterization to get the one-soliton solution

in a physically readable form.

Index notation and form of the line elements.

Hereafter, the coordinates will be enumerated as (x0, x1, x2, x3) = ( t, ϕ, ρ, z) , the small

Latin letters take the two values 0, 1 while, Greek small indexes take the two values

2, 3 . The background line element and the dressed one-soliton solution are therefore of

the form:

ds2 = gab(xρ)dxadxb + f(xρ)ηµνdxµdxν , (3.1)

where, since we are dealing with an axially symmetric stationary metric, the sign indi-

cator e is set equal to −1, hence the ηµν matrix will be:

ηµν=

(1 0

0 1

),

while, since det g(0) = −ρ2 and α2= e det g(0) , then α = ρ .

3.1 Application of the first nine steps of the gener-

ating procedure.

3.1.1 Step-1: Background Einstein-Maxwell solution.

As mentioned above, we start by taking a flat background solution, that is, automatically

null values for A(0)a electromagnetic background potentials. In cylindrical coordinate,

the Minkowsky line element is:

ds2 = −dt2 + ρ2dϕ2 + dρ2 + dz2 . (3.2)

Therefore it results that, respectively for the background metric components and for the

background electromagnetic potentials, we have:

g(0)ab =

(−1 0

0 ρ2

), f (0) = g(0)

µµ = 1 (3.3)

and

A(0)a = 0 . (3.4)

43

3.1.2 Step-2: Background value of the complex electromagneticpotential Φ

(0)a .

Recalling that:

εµν = εµν=

(0 1

−1 0

), εab = εab=

(0 1

−1 0

),

the immaginary components B(0)a of the complex background electromagnetic fields Φ

(0)a

follows as the solution of the differential system (2.17)

Ba,µ = − 1

αηµν ενλ gab εbcAc,λ .

Being the imaginary part of the complex electromagnetic potential depending only on ρ

and z variable and being a constant potential inessential for the field value, then we put

B(0)a = 0

and therefore:

Φ(0)a = A(0)

a + iB(0)a = 0 .

3.1.3 Step-3: Calculus of X(0) and X(0)−1.

From the general structure of the X and X−1 matrices given in (2.40),

X =

−4 εacgcdε

db + 8 εacΦcεbdΦd 4 εacΦc

4 εacΦc 2

X−1 =

−1

4εacgcdεdb −1

2εacg

cdΦd

−14εbc gcdΦd −1

2+ gcdΦcΦd

.

Substituting the values found above for g(0)ab and Φ

(0)a , and since

εacg(0)cd εdb =

(−ρ2 0

0 1

),

we obtain:

X(0) =

4ρ2 0 0

0 −4 0

0 0 2

X(0)−1 =

1

4ρ20 0

0 −1

40

0 01

2

.

44

3.1.4 Step-4: Calculus of U(0)µ .

From the equation (2.33)

Uµ = i e α ηµρ ερσX−1X,σ + 4 e(α2X−1

),µ

Ω

putting

εµσ .

= ηµρερσ =

(0 1

−1 0

),

and recalling the definition (2.31)

Ω=

(εab 0

0 0

)=

0 1 0

−1 0 0

0 0 0

,

substituting the background values just found above, for the stationary case we have:

U (0)µ = −

[i ρ εµ

σ X(0)−1X(0),σ + 4

(ρ2X(0)−1

),µ

Ω]

.

Since

X(0),σ =

8 ρ δ2σ 0 0

0 0 0

0 0 0

,

where we have put ρ,σ = δ2σ , being δ the Kronecker symbol,

X(0)−1X(0),σ =

2

ρδ2σ

1 0 0

0 0 0

0 0 0

and

(ρ2X(0)−1

),µ

= 2 ρ δ2µ X(0)−1 + ρ2

(X(0)−1

),µ

= −1

2ρ δ2

µ

0 0 0

0 1 0

0 0 −2

,

than we have for each component of U(0)µ matrices that:

U(0)2 = −

[i ρ

(6ε2

2X(0)−1X(0),2 + ε2

3X(0)−1X(0),3

)+ 4

(ρ2X(0)−1

),2

Ω]

=

= −[i ρ X(0)−1X

(0),3 + 4

(ρ2X(0)−1

),2

Ω],

U(0)3 = −

[i ρ

(ε3

2X(0)−1X(0),2 + 6ε3

3X(0)−1X(0),3

)+ 4

(ρ2X(0)−1

),3

Ω]

=

= −[−i ρX(0)−1X

(0),2 + 4

(ρ2X(0)−1

),3

Ω].

45

Therefore, the final expressions are:

U(0)2 =

0 0 0

−2 ρ 0 0

0 0 0

U

(0)3 =

2i 0 0

0 0 0

0 0 0

.

3.1.5 Step-5: Deduction of β(xµ).

From the equation (2.44) β,µ = −e ηµρ ερσ α,σ , which now reduces to

β,2 = ρ,3

β,3 = −ρ,2

,

it follows immediately that

β = −z + z0 ,

where z0 is an arbitrary constant that gives information on the location of the source

on the axis of symmetry.

3.1.6 Step-6: Calculus of W (0).

From the equation (2.83) we have that:

W (0) = X(0) − 1

4X(0)EX(0) + 4 i (w + β) Ω ,

where w ∈ C and E = diag( 0, 0, 1 ) .

Since

X(0)EX(0) =

4ρ2 0 0

0 −4 0

0 0 2

0 0 0

0 0 0

0 0 1

4ρ2 0 0

0 −4 0

0 0 2

= 4E ,

then

W (0) =

4ρ2 0 0

0 −4 0

0 0 2

− 1

64 64 E + 4 i (w + β)

0 1 0

−1 0 0

0 0 0

.

Introducing2 λ = − (β + w) = z − w , where it has be choosen Re(w) = z0 , the W (0)

matrix assumes the form:

W (0) =

4ρ2 −4iλ 0

4iλ −4 0

0 0 1

.

2Here, the λ symbol has nothing to do with the same symbol used to indicate the spectral parameterin the context of the Belinski-Zakharov technique.

46

3.1.7 Step-7: Deduction of background generating matrixϕ(0) (w, xµ) and its normalization.

This is the only step in which we have to solve a differential problem; in fact, the great

advantage of the soliton method is that, now, we only have to integrate a linear system

instead of the nonlinear one of Einstein-Maxwell.

Integration of Lax Pair. To get the background generating matrix ϕ(0) we have to

integrate the spectral equation (2.47)

ϕ(0),µ =

1

2i

(w + β

(w + β)2 + ρ2Uµ − ρ

(w + β)2 + ρ2εµ

σUσ

)ϕ(0) .

Introducing the symbol Γ =√

λ2 + ρ2 , the Lax Pair assumes the form:

ϕ(0),µ =

1

2iΓ2(−λUµ − ρ εµ

σUσ) ϕ(0) ,

that is:

ϕ(0),2 =

1

2iΓ2(−λU2 − ρU3) ϕ(0)

ϕ(0),3 =

1

2iΓ2(−λU3 + ρU2) ϕ(0)

.

Since

(−λU2 − ρU3)=

−2iρ 0 0

2λρ 0 0

0 0 0

and (−λU3 + ρU2)=

−2iλ 0 0

−2ρ2 0 0

0 0 0

,

the spectral system is reduced to:

∂ρϕ(0) =

1

Γ2

−ρ 0 0

−iλρ 0 0

0 0 0

ϕ(0)

∂zϕ(0) =

1

Γ2

−λ 0 0

iρ2 0 0

0 0 0

ϕ(0)

;

or, introducing explicit value of Γ , and recalling that the capital Latin letters of the first

part of the alphabet ( i.e. from A to H ) can assume the values ( 0, 1, ∗ ) and since

47

∂z ≡ ∂λ ,

∂ρϕ(0)0A = − ρ

λ2 + ρ2ϕ

(0)0A

∂ρϕ(0)1A = − iλρ

λ2 + ρ2ϕ

(0)1A

∂ρϕ(0)∗A = 0

∂λϕ(0)0A = − λ

λ2 + ρ2ϕ

(0)0A

∂λϕ(0)1A = − iρ2

λ2 + ρ2ϕ

(0)1A

∂λϕ(0)∗A = 0

.

The integration of this system yields the following solution:

ϕ(0)0A =

aA

Γ

ϕ(0)1A =

iaAλ

Γ+ bA

ϕ(0)∗A = cA

where aA, bA, cA are nine real integration constants.

Normalization of generating matrix. Using the above general solution for the gen-

erating matrix, its hermitian conjugate and the matrix W (0) :

ϕ(0) =

1Γ

aA

bA + iΓ

λ aA

cA

, ϕ(0)† =

(1Γ

aA, bA − iΓ

λ aA, cA

), W (0) =

4ρ2 −4iλ 0

4iλ −4 0

0 0 1

,

from the conserved integral (2.70) ϕ(0)†W (0)ϕ(0) = C where the 3× 3 C matrix can be

choosen to be real and diagonal C = diag (C1, C2, C3) we have that:

(ϕ(0)†W (0)ϕ(0)

)= 4aAaB − 4bAbB + cAcB = C ,

that is

4 a 20 − 4 b 2

0 + c 20 = C1

4 a 21 − 4 b 2

1 + c 21 = C2

4 a 2∗ − 4 b 2

∗ + c 2∗ = C3

,

4 a0 a1 − 4 b0 b1 + c0 c1 = 0

4 a1 a∗ − 4 b1 b∗ + c1 c∗ = 0

4 a∗ a0 − 4 b∗ b0 + c∗ c0 = 0

.

Finally, choosing aA = (1, 0, 0) , bA = (0, 1, 0) , cA = (0, 0, 1) , it follows that C1 = 4 ,

48

C2 = −4 , C3 = 1 and:

[ϕ(0)

]AB

=

1

Γ0 0

iλ

Γ1 0

0 0 1

,[ϕ(0)−1

]AB=

Γ 0 0

−iλ 1 0

0 0 1

.

Rationalization of Γ function. It is convenient to replace the cylindrical coordinates

(ρ, z) with a system of ellipsoidal coordinates (r, θ) expressed by:

ρ =√

R2 − w2 s θ z = R c θ (3.5)

where s.= sin , c

.= cos and R = R (r) is at the moment an unknown function of the

radial coordinate r . In fact, it is easy to see that, in these new coordinates, Γ depends

on them in the following rational way

Γ = R− w c θ .

3.1.8 Step-8: Costruction of m(k)A and p(k)A vectors.

From the matrix ϕ(0) , and its inverse, it is now possible to evaluate the vectors m(k)A

and p(k)A through the formulas (2.64)-(2.65):

m(k)A= k(k)B

[(ϕ(0)

)−1(wk)

]BA

, p(k)A = l(k)B

[ϕ(0) (wk)

]AB

,

where we have replaced directly wk with wk in virtue of (2.78), and where wk are

the values of the fixed poles. For one-soliton, the values of the soliton index k is 1 .

Hence, since we are dealing with w1 , henceforth in this chapter, we replace it with w ,

remembering that, from now on, it will be no more a variable but a constant. The soliton

index will be removed from all the other symbols too. The vectors k(k)A depend on l(k)A

vectors through the equation (2.81), k(k)A = CAB l(k)B , where CAB = diag (C1, C2, C3) .

Therefore, as Γ(w) = Γ and λ(w) = λ , it follows that:

m0 = C1 Γ l0 − i C2 λ l1 p0 =1

Γl0

m1 = C2 l1 p1 = i1

Γλ l0 + l1

m∗ = C3 l∗ p∗ = l∗

;

49

or, substituting the values chosen for the diagonal components of constant C matrix:

m0 = 4 Γ l0 + i 4 λ l1 p0 =1

Γl0

m1 = −4 l1 p1 = i1

Γλ l0 + l1

m∗ = l∗ p∗ = l∗

. (3.6)

3.1.9 Step-9: Costruction of Tkl matrix.

Having at our disposal the vectors mA and pA , it is now possible to construct the unique

component (since the indexes of the matrix T enumerate the solitons) T11 of the matrix

Tkl . We denote it simply with T through the equation (2.57). We rewrite this equation

here, substituting directly w with w

T =pA mA

w − w. (3.7)

It is worth noting that z and the complex pole w appear in T , in the S BA matrix

and, therefore, in the final solution, always through the combination λ = z−w . Then ,

for the arbitrariness of the z0 constant, and since Re(w) is an arbitrary fixed constant

too, it follows that we have only one independent constant that we have already used

above when we put z0 = Re(w) . As the most convenient choice, we now take the pole

to be purely imaginary, that is w = i σ , where σ ∈ R ; this position is not a reduction

of the generality of the result since the arbitrariness in the choice of z0 reflects, in this

one-soliton case, the invariance under translation along the z-axis of the Minkowsky

background line element (3.2). From this position it follows that:

Γ = R− i σ c θ , (3.8)

λ = R c θ − i σ , (3.9)

T =1

2 i σΓ

[4 Γ | l0|2 + Γ

(−4 | l1|2 + | l∗|2 )+ 8 σ l0 l1

]. (3.10)

3.2 Determination of the mathematical parameters

in terms of the physical one.

In the previous paragraphs, we have constructed the m(1)A and p(1)A vectors and the

unique component of the T matrix. They all will be the ingredients by which the S

matrix is defined through the (2.88) equation. The mathematical constants present in

50

these objects are the six real constants of the three complex constant vectors lA and

the imaginary part σ of the pole w . Looking at the formulas (2.64)-(2.65) which give

the m(1)A and p(1)A vectors, it results that there may also be arbitrary complex factors

which can depend on the index k and the coordinates xµ , and that such factors are not

present in final expressions for the matrices Rk and Sk . This means that we dispose of

a rescaling freedom lA → ι lA for an arbitrary complex constant ι , which lets us fix one

of the three constant vectors lA according to our convenience, without loss of generality.

Therefore, the six real constants given by lA are reduced to four; adding σ , we have a

total of five arbitrary independent constants.

It is necessary to remind the unknown function R(r) present in the definition (3.5) of

the ellipsoidal coordinates (r, θ) , has not been defined yet.

Now, we are going to introduce the one-body solution of McGuire-Ruffini [66], which

depends on five free parameters representing: mass, angular momentum for unit mass3,

electric and magnetic charge4 and NUT parameter of one-body source. A direct compar-

ison between the respective f conformal factors and the g00 components will enable us

to express the five mathematical constants in terms of the above mentioned physical ones.

It is worth to recall that here the qualification of ”physical”, extended to the mag-

netic charge and the NUT parameter, is improper; we have used it for them just to

distinguish the five parameters of the McGuire-Ruffini from the purely mathematical

parameters present in the one-soliton solution. The magnetic charge parameter appears

in the McGuire-Ruffini as a consequence of the invariance of the Einstein-Maxwell equa-

tions respect to a duality rotation e = q c θ g = q s θ , where q is a constant and the

electric charge e and the magnetic charge g are described in terms of θ . The NUT

parameter5 appeared in the literature as a new constant present in the solution of New-

man, Unti and Tamburino [70]. It reduces to the Schwarzschild solution if this parameter

is equated to zero otherwise it presents singularities along the symmetry axis at θ = 0

and θ = π . If one of them can be removed by a coordinate transformation, this does

not get rid of the other one. Some literature was devoted to look for an interpretation

of this parameter (see among the most recent works for example [65] and the literature

cited therein). Anyway it carries some singularities. Because of this, it should be better

to refer to the magnetic charge and NUT terms as unphysical parameters and hence to

3Dealing with axially symmetric solution the angular momentum is parallel to the axis of symmetryand oriented along the θ = 0 direction.

4We call McGuire-Ruffini solution the generalization of the Kerr-Newmann-NUT solution [33] givenby the presence of this additional magnetic charge parameter.

5It can also be referred to as the dual mass, the magnetic mass, the gravitomagnetic monopolemomentum and so on.

51

remove them from the final result if a physical meaningful solution is desired.

3.2.1 The McGuire-Ruffini one-body solution.

The line element of McGuire-Ruffini one-body solution in ellipsoidal coordinates

(t, ϕ, r, θ) is:

ds2 = − 1

Σ

(∆− a2s2θ

)dt2 + 2

1

Σ

(∆ χ− a% s2θ

)dt dϕ +

1

Σ

(%2s2θ −∆ χ2

)dϕ2+

+Σ

(1

∆dr2 + dθ2

) (3.11)

where:

B = b + a c θ , Σ = r2 + B2 , ∆ = r2 − 2 mr − b2 + a2 + q2 ,

χ = a s2θ − 2 b c θ , % = r2 + b2 + a2 , q2 = e2 + g2

and m , b , a , e and g are respectively the Schwarzchild mass, the NUT parameter, the

angular momentum per unit mass, the electric and magnetic charge.

To perform a first comparison, we transform the conformally flat bedimensional block of

the line element (3.1) in ellipsoidal coordinates (3.5),

ρ =

√R2 − w2s θ

z = R c θ, where R = R (r) ,

and equate it to the corresponding block of (3.11). Under this transformation of coordi-

nates we have:

dρ2 + d z2 = R2,r

(R2 − w2c2θ

R2 − w2

)dr2 +

(R2 − w2c2θ

)d θ2 ,

where R,r= ∂rR . From the identification

f

[R2

,r

(R2 − w2c2θ

R2 − w2

)dr2 +

(R2 − w2c2θ

)d θ2

]= Σ

(1

∆dr2 + d θ2

),

we get the system

f R2r

(R2 − w2c2θ

R2 − w2

)=

r2 + B2

r2 − 2 mr − b2 + a2 + q2

f (R2 − w2c2θ) = r2 + B2

; (3.12)

52

from the second equation of (3.12), we get that

f =r2 + B2

(R2 − w2c2θ).

and, substituting this in the first of (3.12),

R2r

R2 − w2=

1

r2 − 2 mr − b2 + a2 + q2.

The solution (with positive determinations of the radicals) of these last relation gives:

R +√

R2 − w2 = (r −m) +

√(r −m)2 −m2 − b2 + a2 + q2 .

Since w = i σ, it follows that:

R = r −m, (3.13)

σ2 = −m2 − b2 + a2 + q2 . (3.14)

The line element of (3.11) is now rewritable in the form

ds2 = gab(r, θ)dxadxb + f(r, θ)(dρ2 + dz2

),

where the components, expanded and ordered in accordance with decreasing power of r

and c θ to facilitate future comparison, are:

g00 = − 1

Σ

[r2 − 2mr + a2c2θ +

(q2 − b2

)](3.15)

g01 =1

Σ

[(−2b c θ) r2 + 2m

(a c2θ + 2b c θ − a

)r

+ a(2b2 − q2

)c2θ + 2b

(b2 − a2 − q2

)c θ + a

(q2 − 2b2

)] (3.16)

g11 =1

Σ

s2θ r4 +

(−a2c4θ − 4ab c3θ − 6b2c2θ + 4ab c θ + a2 + 2b2)r2

+ 2m(a c2θ + 2b c θ − a

)2r

+ a2(b2 − q2 − a2

)c4θ + 4ab

(b2 − q2 − a2

)c3θ

+[2(a2 − 2b2

)q2 +

(a4 + 3b4 − 8a2b2

)]c2θ

− 4ab(b2 − a2 − q2

)c θ

+(b4 + 3a2b2 − a2q2

)

(3.17)

f =r2 + (b + a c θ)2

(r −m)2 + σ2c2θ(3.18)

It is easy to see that, equating some of the five parameters to zero, this solution contains,

as particular cases, well known one-body solutions as showed below in the table.

53

m b a e g

Schwarzschild 6= 0 = 0 = 0 = 0 = 0

Reissner-Nordstrom 6= 0 = 0 = 0 6= 0 = 0

NUT 6= 0 6= 0 = 0 = 0 = 0

Kerr 6= 0 = 0 6= 0 = 0 = 0

Kerr-NUT 6= 0 6= 0 6= 0 = 0 = 0

Kerr-Newman 6= 0 = 0 6= 0 6= 0 = 0

Kerr-Newman-NUT 6= 0 6= 0 6= 0 6= 0 = 0

3.2.2 Calculus and comparison of the f conformal factors.

Introducing the following terms:

p.= 4|l0|2

q.= −4|l1|2 + |l∗|2

,

ξ.= 8 σRe

(l0l1

)

η.= 8 σIm

(l0l1

) , ζ.= ξ + iη ,

the (3.10) expression for T becomes

T =1

2iσΓ

(Γ p + Γ q + ζ

).

Using this expression in the (2.102) for the f factor, and since f (0) = 0, we have that:

f = C(0)T T = C(0)1

4σ2|Γ|2(Γ p + Γ q + ζ

) (Γ p + Γ q + ζ

).

If F.= 4σ2|Γ|2T T , then from the relations

|Γ|2 = R2 + σ2c θ

Γ2 + Γ2 = 2 (R2 − σ2c2θ)(p ζ + q ζ

)γ +

(p ζ + q ζ

)Γ = 2 [(p + q) ξR− (p− q) η σc θ] ,

it follows that

F =[P 2R2 + 2PξR + Q2σ2c2θ − 2Qησ c θ + |ζ|2] ,

where P = p + q and Q = p − q . Using the expression for R , given by (3.13), we

have

f = C(0)1

4σ2|Γ|2[P 2r2 + 2P (ξ − Pm) r + Q2σ2c2θ − 2Qησc θ + |ζ|2 + Pm (Pm− 2ξ)

],

54

which, compared with (3.18), gives the system

C(0)P 2

4σ2= 1

C(0)2P (ξ − Pm)

4σ2= 0

C(0)Q2σ2

4σ2= a2

−2C(0)Qησ

4σ2= 2ab

C(0)|ζ|2 + Pm (Pm− 2ξ)

4σ2= b2

.

Therefore P =ε1 2 σ√

C(0)

, Q =ε2 2 a√

C(0)

, ξ =ε1 2 mσ√

C(0)

, η =−ε2 2 b σ√

C(0)

, where ε1 and ε2

are two undetermined indicators of sign. This leads to the following relations for the lA

vectors:

4|l0|2 − 4|l1|2 + |l∗|2 = ε12σ√C(0)

4|l0|2 + 4|l1|2 − |l∗|2 = ε22a√C(0)

Re(l0l1

)= ε1

m

4√

C(0)

Im(l0l1

)= −ε2

b

4√

C(0)

, (3.19)

from which

4|l0|2 =ε1σ + ε2a√

C(0)

(3.20)

−4|l1|2 + |l∗|2 =ε1σ − ε2a√

C(0)

. (3.21)

Hence we have:

T =1

i√

C(0)Γ[ ε1r − ε2i (b + a c θ) ] . (3.22)

Now, we can proceed with the final steps of the generating procedure to get all the

remaining components of the metric tensor and of the electromagnetic potentials. From

the comparison with the g00 component (3.15), we will be able to define, in a complete

manner, both the undetermined terms C(0) , ε1 , ε2 , and the lA vectors in terms of the

five physical constants.

55

3.2.3 Step-10 & 11: Determination of S matrix and calculusof gab components.

Before we calculate all the components of the S matrix, it is convenient to have the final

formulas for the dressed solution expressed only in terms of the background solution and

of the S matrix. In fact, not all its components appear in these formulas. We get this

explicit version of the (2.104) and (2.105) formulas from the (2.86) through the (2.89)

and (2.90). In the particular case, adapted to the Minkowsky background, we have:

g00 = −1 + 2 Im (S 10 )− 4|S ∗

0 |2

g01 = i[S 0

0 + S11

]− 4S ∗0 S∗1

g10 = −i[S 1

1 + S00

]− 4S ∗1 S∗0

g11 = ρ2 − 2 Im (S 01 )− 4|S ∗

1 |2

, (3.23)

A0 = −2 Im (S ∗0 )

A1 = −2 Im (S ∗1 )

. (3.24)

A direct calculation shows that the condition valid to assure, at the same time, the

reality and simmetry of gab matrix

Re(S 00 + S 1

1 )− 4 Im(S ∗0 S ∗

1 ) = 0 ,

reduces to

Im(8 σ p0 p1 − p0 m0 + p1 m1

)= 0 .

This can be verified to be identically satisfied if we substitute the expressions (3.6) into

it.

Comparison of g00 components.

To get the g00 component we have to calculate the S 10 and S ∗

0 .

———————–

• Calculus of S 10 component:

S 10 = − 1

Tp0m

1 = − 1

T

(− 4

Γl0 l1

)= i

(ε1m− ε2i b

ε1r − ε2iB

)=

1

Σ(ε2b + ε1im) (ε1r + ε2iB) =

=1

Σ[ε1ε2 (br −mB) + i (bB + mr)]

56

and therefore:

Im(S 1

0

)=

1

Σ(bB + m r) .

———————–

• Calculus of S ∗0 component:

S ∗0 = − 1

Tp0m

∗ = − 1

T

(− 1

Γl0l∗

),

therefore

|S ∗0 |2 =

1

|T |21

|Γ|2 |l0|2|l∗|2 ;

and since |T |2|Γ|2 =Σ

C(0)

and |l0|2 =1

4(ε1σ + ε2a) then:

|S ∗0 |2 =

√C(0)

1

4Σ(ε1σ + ε2a) |l∗|2 .

———————–

Now, using the first one of the (3.23) equations, we have

g00 = −1 +1

Σ

[2 (bB + mr)− 64 1

64√

C(0) (ε1σ + ε2a) |l∗|2]

,

which, compared with the corresponding component (3.15), leads to the relation:

|l∗|2 =q2

√C(0) (ε1σ + ε2a)

. (3.25)

This last expression, together with (3.20) and (3.21), gives

|l0|2 =1

4√

C(0)

(ε1σ + ε2a)

|l1|2 =1

4√

C(0)

(a2 + q2 − σ2

ε1σ + ε2a

)

|l∗|2 =q2

√C(0) (ε1σ + ε2a)

.

The first two of these last formulas, from (3.19) through the relation[Re

(l0l1

)]2+[

Im(l0l1

)]2= |l0|2|l1|2 , give the same expression for the square of σ as in (3.14). Hence

57

we have:

|l0|2 =1

4√

C(0)

(ε1σ + ε2a)

|l1|2 =1

4√

C(0)

(m2 + b2

ε1σ + ε2a

)

|l∗|2 =q2

√C(0) (ε1σ + ε2a)

.

Therefore, it is possible to write the lA vectors as:

l0 =1

2 4√

C(0)

Aeiψ

l1 =1

2 4√

C(0)A(ε1m + ε2ib) eiψ

l∗ =1

4√

C(0)Aqeiγ

,

where A.=√

ε1 σ + ε2 a is constant. Using the rescaling freedom mentioned at the be-

ginning of this section, and choosing l0 = 0 , then ψ = 0 , and C(0) = A4/16 . About γ

phase, recalling that q2 = e2 + g2 , it is naturally defined by e = ε3q cγ and g = ε4q sγ .

It results that all sign indicators ε# are inessential regarding the final result, hence they

are set in the following way: ε1 = ε3 = +1 , ε2 = ε4 = −1 , from which it follows:

C(0) =(σ − a)2

16.

In this way, we have obtained the expressions of the lA vectors in terms of physical

parameters:

l0 = 1

l1 =m− i b

σ − a

l∗ = 2e− i g

σ − a

(3.26)

Comparison of ga1 components.

The ensuing comparisons of ga1 with the corresponding components of McGuire-Ruffini

solution are just a way to verify completely the identification hypothesis between the

58

two solutions, since all the freedom available have been fixed. It is worth noting that

this identification could be valid at least of a linear coordinate transformations, involving

just the t and ϕ coordinates, and preserving the values of g00 component that we have

used to set the (3.26) relations. Calculi for these components are much more laborious

respect those already performed and will not be reported here. The calculus for g01

component gives a results which is a linear combination with constant coefficients of g00

and g01 components of McGuire-Ruffini solutions. This suggests the following coordinate

transformations (which leaves g00 component unchanged)6:

dt = dt′ + E dφ′

dφ = dφ′, where E =

q2

σ − a− 2a . (3.27)

It is easy to check that values of the E factor coincides, for the one-soliton case, with

the general formula, given at page 239 in [3], valid for a generic n-soliton solution. We

rewrite it here, according to our notation and bearing in mind that it depends on the

choice l0 = 1 ,

En = −i

n∑

k,l

γ−1kl

(1 + l(k)1 l(l)1

), where γkl =

4− 4 l(l)1 l(k)1 + l(l)∗ l(k)∗

4 (wl − wk). (3.28)

The transformation (3.27) changes the components gab and Aa in:

g′00 = g00

g′01 = E g00 + g01

g′11 = E 2g00 + 2 E g01 + g11

(3.29)

A′0 = A0

A′1 = E A0 + A1 .

(3.30)

As a final verification, the calculus of g11 inserted together with g00 and g01 into the

third of (3.23) yields g′11 . It results to be just equal to the corresponding component

given by (3.17).

3.2.4 Components of the electromagnetic potential.

To complete the construction of the one-soliton solution, it remains to calculate, from the

equations (3.24), the Aa components and then, through the (3.30), the electromagnetic

6This coordinate transformation can be obtained, independently from the comparison with theMcGuire-Ruffini solution, also by an asymptotical analysis of the behaviour of g01 .

59

potential A′a . Again, we skip the description of calculus giving directly the final result:

A′0 =

e r + g (b + a c θ)

Σ

A′1 =

me + b g

σ − a− 1

Σ g r2 c θ + e r (a s2θ − 2 b c θ) + g [ (a2 − b2) c θ + a b s2θ ]

(3.31)

where we recall that Σ = r2 + (b + a c θ)2 .

Chapter 4

A perturbative approach forstationary axially symmetric solitonsolutions

4.1 Some preliminary remarks.

It is important to note that in the one-soliton solution, given by (3.15)-(3.17) and (3.31),

σ appears only inside a constant addendum of A′1 which is physically irrelevant for the

determination of the electromagnetic fields; being always present in its second power

elsewhere, using the relation (3.14) it disappears. Because of this, we can ignore the

constrains to be imposed on the physical parameters to assure the reality of σ , looking

just at the final result. In any case this is possible just for one-soliton solutions since the

multi-soliton solutions will contain mixed products σkσl with k 6= l . The difficulties

in performing an analytical continuation to remove the constrains among the physical

parameters is a limitation of the soliton technique.

To illustrate the physical meaning of this limitation, we can consider, as simple example,

the solution of Reissner-Nordstrom. It is obtainable, as a particular case, from the

general one-soliton solution, leaving different from zero only the mass and electric charge

parameters. We thus have:

g00 = −1 +2m

r− e2

r2σ =

√−m2 + e2 .

Since, to assure σ to be real and different from zero then −m2 + e2 > 0 , it follows that

there are no values of coordinate r for which g00 changes its sign. This implies the

absence of an horizon and that the solution does not describe a blackhole but a naked

singularity or, in other words, as called conventionally in literature, a superextremal part

60

61

of Reissner-Nordstrom solution1.

Therefore a multi-soliton dressing will be able to produce only a superposition of super-

extremal sources which represents a weakness of this generating technique.

Apart from the reduction of the generality of the solutions that can be obtained, a

deeper meaning of this limitation appears clearer under the following additional physical

considerations.

Let us consider a two-soliton solution, representing the superposition of two Reissner-

Nordstrom like sources, that is, of two charged masses. The resulting metric will depend

on five independent parameters: the two masses m1 , m2 , the two charges e1 , e2 and

the distance d between the sources. It will obviously belong to the class of kind (3.1).

Moreover, because of the absence of any angular momentum, it will also be diagonal,

that is, static. In accordance with the thinking of classical physics, to have two sources in

static equilibrium, it is necessary to have a compensation between the attractive gravita-

tional force and the repulsive electrostatic force. In natural units, this condition reads as

the equality between the product of the two masses and the product of the two charges:

m1 m2 = e1 e2 . (4.1)

The relativistic generalization, given in [4], of the (4.1) is more complicate. Because of

the nonlinearity of the equations, it depends also on the distance parameter. At the

classical limit of weak fields, for great values of the parameter of distance2, the (4.1)

is recovered. This implies that the system has to be composed either by two extremal

sources or by a subextremal and a superextremal source, that is by a black hole and a

naked singularity.

The fact that the two-soliton dressing could yield fields generated by two superextremal

objects has as a consequence that, on the points of the segment of the axis between the

two sources, the elementary flatness of the solution is violated and not that the final

result is not a solution of Einstein-Maxwell equations. In other words the points of that

segment are conical singularities. That is, contracting a small circle, linked together with

the axis of symmetry ρ = 0 , to one of the points of the axis placed between the sources,

the limit of the ratio of the length of the circle to its radius times 2π does not go to the

unity. Different words can be found in literature to denote this kind of singular line as

strings, props, struts, rods, since to them is possible [54], [81] to associate a particular3

1The cases for −m2 + e2 = 0 and −m2 + e2 < 0 are called respectively extremal and subextremal,corresponding this last one to a black hole solution.

2It is worth noting that the equilibrium constrains give the possibility to have some particularconfiguration peculiar of the relativistic regime for which this limit does not exist.

3In General Relativity a stress generates gravitational fields.

62

kind of stress of topological nature, not realizable in nature4, which, even if does not

generate gravitational fields, balances the reciprocal actions between the sources which

are not able, by themselves, to remain in static equilibrium. This stress can be described

in terms of a force which shows, depending on the sign of the deficit of angle of the con-

ical singularity, a repulsive or attractive character. We will come back again on this to

illustrate in detail this connection in the chapter dedicated to the analysis of the stability

of the double Reissner-Nordstrom solution of Alekseev and Belinski.

It is now clear the weakness of the soliton dressing procedure. It would obviously be

preferable to dispose of a generating technique capable to yield regular electrovacuum

solutions and therefore the possibility to remove such kind of spurious singularity, giving,

in such a way, the balancing condition among the parameters of the solution.

It is important to take into account another important aspect. We have found the

reparameterization formulas (3.26) and the characterization of the imaginary part of the

poles (3.14), comparing directly the one-soliton generated solution with the known so-

lution of McGuire-Ruffini. To construct a two-soliton solution we could use the same

formulas for each soliton5. This will leads to an exact solution of the Einstein-Maxwell

equations in terms of the eleven parameters mk , bk , ak , ek , gk , plus the distance

between the two sources d . But they will loose their individual physical meaning due

to the non linearity of the interaction. Therefore, to obtain an interpretable final so-

lution either additionally considerations will be necessary or it needs to start from an

alternative reparameterization.

4.2 A proposal for a perturbative approach.

On the line of the considerations mentioned just above, hereafter we will describe a

proposal for a perturbative approach to look for hints to overcome some, or possibly

each one, of the difficulties presented by the solitonic technique. The motivation for this

attempt lies on considerations [10] concerning, for example, the experience coming from

the deduction of the Alekseev-Belinski solution [5]. The discrepancy between the rather

cumbersome way6 to obtain that solution and the rather surprising simplicity of the final

result suggests that the same result could be reached, in a simpler and direct way, by

means of the dressing procedure. Procedure that, then, could be applied to deal with

more general problems as that concerning the general balance conditions for two charged

4It does not correspond to any kind of gravitational mass.5Hence we have to restore in the notation the indexes to each parameter to indicate it refers to

which source or soliton.6We recall that Alekseev and Belinski did not use the dressing technique to find their solution.

63

and rotating sources.

We are going to construct a perturbative generating procedure based on the expan-

sion respect the Newton’s constant. We will denote it with γ . We recall that until

now we adopted geometrized units where the gravitational constant γ and the speed of

light c are set equal to one. Now, keeping again c = 1 , we have to reintroduce in its

right place γ . Anyway, for formal reasons we will explain soon after, we will work with

the square root of γ for which we will use the symbol κ . Relatively to the physical

(and unphysical) parameters with which we are dealing with, this can be done simply

performing the following substitution:

m → κ2 m, b → κ2 b , a → a , e → κ e , g → κ g , (4.2)

where

κ =√

γ .

We could perform the expansion in the κ parameter directly on the final product of

the generating technique7. Anyway we would find some hint to overcome the problems

carried by the solitonic technique. Resuming them here, they are: the reparameteriza-

tion between mathematical and physical constants, the characterization of the complex

poles and their analytical continuation. To solve these problems, it could be necessary

to modify the dressing procedure even at the level of the Lax Pair. We recall that the

proposal of Alekseev is not the only one. For this reason we will deal with the expansion

starting from a deeper level or the dressing procedure.

In this chapter, we will give an outline of the dressing procedure expanded in κ . We

can schematically separate it in two moments. One concerning with the expansion of the

generating matrix function ϕ , the other one regarding the expansion of the poles and of

the constants quantities. In the next chapter, we will apply this perturbative scheme to

dress a flat background. In this case, we can see that all the terms of the approximate

terms of the generating matrix will be formally equals.

7These expansions will present obviously only terms corresponding to even powers in κ ; that is,they will result to be expansions in γ .

64

4.3 Outline of the perturbative solitonic generating

technique.

Hereafter, we will adopt the following notation to denote the terms of each expansion:

f =0

f + κ1

f + κ22

f + ... .

4.3.1 Expansion of Einstein-Maxwell Equations.

Introducing the Newton’s constant γ in the Einstein-Maxwell equations, we have:

ηµν 1

α

(α gbcgac,µ

),ν

= −2 γ gbcηµνΦa,µΦc,ν

ηµν (α gacΦc,µ),ν = 0

. (4.3)

Remembering the expression (2.40) for the matrix X , and noticing that its out of di-

agonal block elements depend linearly by the electromagnetic potential, it is convenient

to introduce the control parameter κ . Hence we can read the (4.3) as the result of the

formal substitution Φ → κ Φ . This implies that, even if it should be natural to consider

just the expansions of the potentials in γ , from now on, with the introduction of the

control term κ , we will deal with even power expansions for the gravitational potentials

gab and odd power expansion for the electromagnetic potentials Φ . Hence, instead to

write

Φ → κ Φ = κ (0

Φ + γ2

Φ + ...)

changing the notation, we will directly put:

Φ → κ1

Φ + κ33

Φ + ... .

Noticing that α = ρ and hence α ≡ 0α , the equations (4.3) therefore splits, up to the

fifth order in κ , into the following systems:

ηµν 1

α

[α

0g bc 0

gac,µ

],ν

= 0

ηµν

(α

0g ac

1

Φc,µ

)

,ν

= 0 ,

(4.4)

65

ηµν 1

α

[α

(0g bc 2

gac,µ +2g bc 0

gac,µ

)],ν

= −2 ηµν 0g bc

1

Φa,µ

1

Φc,ν

ηµν

[α

(0g ac

3

Φc,µ +2g ac

1

Φc,µ

)]

,ν

= 0 ,

(4.5)

ηµν 1

α

[α

(0g bc 4

gac,µ +2g bc 2

gac,µ +4g bc 0

gac,µ

)],ν

=

= −2 ηµν

(0g bc

1

Φa,µ

3

Φc,ν +0g bc

3

Φa,µ

1

Φc,ν +2g bc

1

Φa,µ

1

Φc,ν

)

ηµν

[α

(0g ac

5

Φc,µ +2g ac

3

Φc,µ +4g ac

1

Φc,µ

)]

,ν

= 0 .

(4.6)

It is worth noting that, starting from the assumption of a decoupled problem between

the gravitational fields and electromagnetic fields, the0g potentials can be taken as a

generic solution of Einstein equations in vacuum given by the first of (4.4). Therefore the

equations of successive orders will give the first corrections for week interactions between

the two fields over this fixed generic background solution.

About the terms likekg ab , we recall that given a generical matrix M and its perturbative

representation M =0

M + κ1

M + κ22

M + ... , thenk

M −1k

M 6= I . That isk

M −1 6= (k

M)−1 .

The termsk

M −1 are instead defined by the condition:

M−1M = (0

M −1 + κ1

M −1 + κ22

M −1 + ...)(0

M + κ1

M + κ22

M + ...) = I . (4.7)

4.3.2 Lax Pair expansion.

Rewriting the Lax Pair (2.24) in the simplified manner

Πµψ = Aµψ , 8 (4.8)

and looking at the equations (2.27) and (2.31), it is clear that we have to expand ψ and

Aµ . Because of this expansion, the system (4.8) will split, at least of O(κ2) , into the

8Hence Aµ =e

λ2 − e α2

(λ ε σ

µ Kσ − α Kµ

).

66

following one:

Πµ

0

ψ =0

Aµ

0

ψ

Πµ

1

ψ =0

Aµ

1

ψ +1

Aµ

0

ψ .

(4.9)

It can be rewritten in the more compact notation as:

Πµψ = Aµ ψ , (4.10)

if

ψ.=

0

ψ 0

1

ψ0

ψ

and Aµ

.=

0

Aµ 0

1

Aµ

0

Aµ

. (4.11)

Notice that these are 6 × 6 matrixes. Hereafter, with the hat put on a symbol, we

will mean its expanded version without any specification about the order of expansion.

Hence, for example, with ψ we mean the 3(N + 1)× 3(N + 1) lower triangular matrix

ψ =

0

ψ1

ψ0

ψ 02

ψ1

ψ0

ψ...

......

. . .N

ψN−1

ψ . . . . . .0

ψ

,

if N is the order of expansion. When the structure of the covered9 symbols will have

some different structure from the above one, it will be specified explicitly. Therefore the

Kµ matrices present inside the Aµ coefficients of the (4.8) will have to be obviously

taken in their “hat” version inside Aµ .

It is now easy to represent the expanded compatibility conditions for the system (4.10)

as

ηµν(α X,µX

−1)

,ν= 0 . (4.12)

At the fifth order, they will reproduce the systems (4.4)-(4.6) if X has the structure

given by (2.40). The block components of X−1 are determined as described in the

example (4.7).

We recall that the dressing procedure work in general not only for 3 × 3 matrix but it

can be naturally extended to matrices of any dimension. This is the reason for which the

“hat” notation will straightforwardly lead to the expanded version of the exact generating

9I.e. dressing the hat.

67

procedure.

Hence, to pass from the expansion of the Lax Pair (2.24) to that of the (2.47), it is

sufficient to use “hat” version of the definition (2.41):

ψ =(X − 4 i Ω

)ϕ ,

where Ω = diag(Ω, Ω, ...) . Being also

Uµ = i e α ηµρερσX−1X,σ + 4 e (α2 X−1),µΩ ,

then we have:

∂ ϕ

∂xµ=

1

2 i

[w + β

(w + β)2 − e α2Uµ +

e α

(w + β)2 − e α2ηµρ ερσUσ

]ϕ . (4.13)

4.3.3 Expanded dressing procedure.

Now, taking a background solution ϕ(0) of the system (4.13), we can obtain the perturbed

dressed generating matrix through:

ϕ = χ ϕ(0) .

Here it is useful to specify the structure of the χ and χ−1 matrices. Their are given by:

χ = I +n∑

k=l

(1

w − wk

)Rk ,

χ−1 = I +n∑

k=l

(1

w − wk

)Sk .

The scalar factors are given by

(1

w − wk

)=

1

w − 0wk

+ κ

1wk

(w − 0wk)2

+ ... ,

(1

w − wk

)=

1

w − 0wk

+ κ

1wk

(w − 0wk)2

+ ... ,

since we have to expand in κ the poles too.

If we have two proportional matrices A and B through a scalar k such as A = kB ,

then for the expansion we have A = kB , where just up to the first order as an example:

k =

0

k 01

k0

k

and A =

0

k0

B 00

k1

B +1

k0

B0

k0

B

.

68

For what concerns the R and S matrices, we have that:

(Rk)BA = n

(k)A m(k)B , (Sk)

BA = p

(k)A q(k)B .

To have the right structure of these matrices, n(k)A , p

(k)A and and m(k)A , q(k)A are respec-

tively 3N ×N and N × 3N matrices, since the corresponding exact ones n(k)A and p

(k)A

are three dimensional column vectors and m(k)A and q(k)A are three dimensional row

vectors.

Therefore, taking as an example n(k)A and m(k)A we have for them, up to the first order,

the following structure:

n(k)A =

0n

(k)A 0

1n

(k)A

0n

(k)A

, n

(k)A =

0n

(k)A + κ

1n

(k)A + ... ,

m(k)A =

0m(k)A 0

1m(k)A 0

m(k)A

, m(k)A =

0m(k)A + κ

1m(0)A + ... .

Now, we dispose of the scheme which will enable us to construct the approximate terms

of soliton solutions.

Chapter 5

Generation of approximate solitonsolutions over a flat background.

In this chapter, we apply the perturbative methods, described in the previous chapter,

to generate the lower order solitonic corrections on a Minkowsky background space-time

together with a null background electromagnetic field. Note that the perturbative scheme

permits us to separate the dressing of the two fields. Hence the choice of null back-

ground electromagnetic field is not a consequence of the flatness of the background

space-time. That is, we could also take a non null electromagnetic fields decoupled with

the Minkowsky background as starting point. In the same manner, this freedom makes

the perturbative approach interesting for more general applications, as the generation of

pure electromagnetic soliton perturbed solutions on a curved seed space-time.

After an introductive section, where we will present the perturbed version of the for-

mulas relative to the final steps of the dressing procedure, we will apply the perturbed

procedure to one-soliton case. What we expect is to generate exactly the same perturbed

terms we could find by taking the exact one-soliton solution, presented in the third chap-

ter, and performing directly on it the expansion in the γ parameter; therefore it will

only play the role of an useful verification. Then, in the last section, we will start to

generate the approximate terms of the two-soliton solution. About them, we will only

give the results relative to the lowest order expansion, since the work about the first (in

γ ) or second and third (in κ ) order corrections is still in progress. The results of these

preliminary calculi, even if obviously yield no new physically relevant results, consist in

the reparameterization to obtain the usual complex representation of week electromag-

netic fields of two rotating charged sources on Minkowsky background.

For what we specified in section 4.2 at the end of page 62, the perturbative procedure

can be separated in two phases. The first one regarding the expansion of the generating

matrix, i.e. the determination of the components of the ϕ(0) matrix, the second one the

69

70

expansions of the poles and, as a consequences, the expansions of all that intermediate

quantities depending on them, by which the final fields are constructed. The choice of

a flat background, joined with the fact that the two-soliton solution will be constructed

by a simultaneous double dressing and not by an iterated dressing, will reduce our work

only to the second phase regarding the expansions of the poles. Therefore, using, as a

frame, the same enumeration of the perturbative procedure as that used for the exact

one, we can start directly from the step number 8 . All the results relatives to the previ-

ous ones (together with the choices of all the arbitrary constants), are exactly equal to

those found in the second chapter for the one-soliton exact solution.

5.1 Perturbative building block quantities

Here, we give the basic formulas to generate the approximate terms of the soliton solu-

tions at least up to the third order in κ . Since our main task is to construct then the

approximate terms of two-soliton solution, we leave the soliton indexes, intending them

running from 1 to 2. Hence, recalling the index notation we have:

i, j, h, k, ... = 1, 2 ; a, b, c, ... = 0, 1 ; A, B, C, ... = 0, 1, ∗ .

In the following formulas the parameterizations (3.14) and (3.26), extended to each

soliton, will not be assumed as in the n-soliton solution presented in [3]. The only

generally valid assumption that we can keep is to take l0k = 1 . Otherwise, we use them

just for dimensional considerations. In fact, their expansions suggests that both those of

the imaginary parts of the poles σk and those of the vectors l1 present only even order

terms, while l∗ only odd ones.

The set of expressions for all the approximate terms, listed hereafter, will be useful as a

formulary for algebraic computer computations.

Bipolar coordinates expansion

In general, for multi-soliton solution is still convenient to adopt multipolar coordinates

(rk, θk)1, of the kind (3.5). Thus, we have for each soliton:

ρ =√

R2k + σ2

k

√1− y2

k

z = Rk yk + ζk

, where

Rk = rk −mk

yk = cos θk

. (5.1)

1These systems of coordinates are obviously redundant.

71

In particular, for two solitons we can choose:

ζ1 = −d

2, ζ2 = +

d

2. (5.2)

Expansions of the poles, of the λk and Γk functions and of the lA vectors

Recalling the definitions (3.8) and (3.9), that is

λ(w) = z − w , Γ(w) =√

ρ2 + λ(w)2 ,

we denote their values on the poles wk = ζk + i σk , with

λk.= λ(w)|w=wk

, Γk.= Γ(w)|w=wk

,

therefore:

λk = Rk yk − i σk , Γk = Rk − i σk yk .

For the imaginary parts of the poles expansions we have, as metioned above, only even

order terms, i.e.

σk =0σk +

2σkκ

2 + O(κ4) ,

Therefore the expansions of the poles gives:

0wk = ζk + i

0σk ,

1wk = 0 ,

2wk = i

2σk ,

3w = 0 . (5.3)

The perturbed terms of the λk and Γk functions are respectively:

0

λk = rk yk − i0σk ,

1

λk = 0 ,2

λk = −mk yk − i2σk ,

3

λk = 0 ; (5.4)

0

Γk = rk − i0σk yk ,

1

Γk = 0 ,2

Γk = −mk − i2σk yk ,

3

Γk = 0 . (5.5)

For the lAk complex vectors we have:

l0k = 1 , l1k =2

l1k κ2 + O(κ4) , l∗k =1

l∗k κ +3

l∗k κ3 + O(κ4) ,

and therefore:

1

l 0 = 0 ,2

l 0 = 0 ,3

l 0 = 0 ;0

l 1 = 0 ,1

l 1 = 0 ,3

l 1 = 0 ;0

l ∗ = 0 ,2

l ∗ = 0 .

72

Expansions of mA(k) and p(k)A vectors

As prescribed by the step number 8, now we have to expand the mA(k) and p(k)A com-

plex vector functions as given by their respective expressions (2.64) and (2.65). In the

following formulas we will omit the soliton index k .

0mA :

0m0 = 4

0

Γ ,0m1 = 0 ,

0m∗ = 0 ; (5.6)

0pA :

0p0 =

10

Γ

,0p1 = i

0

λ0

Γ

,0p∗ = 0 . (5.7)

1mA :

1m0 = 0 ,

1m1 = 0 ,

1m∗ =

1

l∗ ; (5.8)

1pA :

1p0 = 0 ,

1p1 = 0 ,

1p∗ =

1

l∗ . (5.9)

2mA :

2m0 = 4

2

Γ + 4 i0

λ2

l1 ,2m1 = −4

2

l1 ,2m∗ = 0 ; (5.10)

2pA :

2p0 = −

2

Γ0

Γ 2

,2p1 = i

2

λ0

Γ

−0

λ2

Γ0

Γ 2

+

2

l1 ,2p∗ = 0 . (5.11)

3mA :

3m0 = 0 ,

3m1 = 0 ,

3m∗ =

3

l∗ ; (5.12)

3pA :

3p0 = 0 ,

3p1 = 0 ,

3p∗ =

3

l∗ . (5.13)

Expansions of Tkl and (T−1)kl matrices

Recalling the formula (2.57), we had that:

Tkl =p

(k)A m(l)A

wl − wk

Putting Nkl = p(k)A m(l)A and Wlk = wl− wk then: Tkl =

Nkl

Wlk

, or, omitting the indexes,

simply T =N

W. Being the expansions of N and W given by:

0

Nkl =0pA

(k) 0m(l)A ,

0

W lk =0wl −

0wk ,

1

Nkl =0pA

(k) 1m(l)A +

1pA

(k) 0m(l)A ,

1

W lk =1wl −

1wk ,

2

Nkl =0pA

(k) 2m(l)A +

1pA

(k) 1m(l)A +

2pA

(k) 0m(l)A ,

2

W lk =2wl −

2wk ,

3

Nkl =0pA

(k) 3m(l)A +

1pA

(k) 2m(l)A +

2pA

(k) 1m(l)A +

3pA

(k) 0m(l)A ,

3

W lk =2wl −

3wk ,

,

73

it is easy to see that both1

N ≡3

N ≡ 0 and1

W ≡3

W ≡ 0 , thus we have only even order

terms of the T matrix expansion:

0

T =

0

N0

W

,2

T =

2

N0

W

−2

W0

W

0

T , while1

T = 0 ,3

T = 0 .

Therefore, the terms of the expansion of the inverse matrix T−1 are, up to the fourth

order in κ , only0

T−1 and2

T−1 . This last one reduces to:

2

T−1 = −0

T−12

T0

T−1 .

Expansion of S BA matrix

Again, to lighten the notation, we rewrite the formula (2.88)

S BA = −

n∑

k,l=1

(T−1)kl p(l)A m(k)B ,

omitting all indexes, but leaving unchanged the order of the factors to remember then,

easily, where to reinsert them in the right place; hence we write simply:

S = −T−1p m .

The eliminations of all null terms leads to the following reduced expressions:

0

S = −0

T−1 0p

0m,

1

S = −0

T−1(

0p

1m +

1p

0m

),

2

S =

(0

T−12

T0

T−1

)0p

0m−

0

T−1(

0p

2m +

1p

1m +

2p

0m

),

3

S =

(0

T−12

T0

T−1

) (0p

1m +

1p

0m

)−

0

T−1(

0p

3m +

1p

2m +

2p

1m +

3p

0m

).

Expansions of metric tensor and electromagnetic potential

Inserting the terms of the expansions of mA and pA vectors into the formulas for the S BA

matrix, it results that some of its components are null. The resulting general expressions

for the corrections of the metric tensor components, as given by the dressing and hence

74

before the final coordinates transformation (3.27), are

0g00 = −1 + 2 Im(

0

S01)

0g01 = i

[0

S00 +

0

S11

]

0g10 = −i

[0

S11 +

0

S00

]

0g11 = ρ2 − 2 Im(

0

S10)

,

2g00 = 2 Im(

2

S01)− 4

1

S0∗

1

S∗0

2g01 = i

[2

S00 +

2

S11

]− 4

1

S0∗

1

S∗1

2g10 = −i

[2

S11 +

2

S00

]− 4

1

S1∗

1

S∗0

2g11 = −2 Im(

2

S10)− 4

1

S1∗

1

S∗1

.

It is worth noting that only the even order terms are present in the above final set of

formulas. It is, in fact, easy to verify that the righthand sides of the relations for the

odd terms, are all identically zero. Hereafter we write only those about the first order

corrections:1g00 = Im(

1

S01) ≡ 0

1g01 = i

[1

S00 +

1

S11

]− 4

1

S0∗

0

S∗1 ≡ 0

1g10 = −i

[1

S11 +

1

S00

]− 4

1

S1∗

0

S∗0 ≡ 0

1g11 = −2 Im(

1

S10) ≡ 0

.

Recalling that the dressing technique yields directly a real and symmetric metric tensor,

we can use the above formulas for the corrections of the g01 and g10 components as a

good test for the calculi. In particular, in our simple 0-order case, it is very easy to see

that such conditions are verified. For the second order, the relation equivalent to both

such conditions (equating the righthand side of2g01 and

2g10 ) is:

Re (2

S00 +

2

S11)− 4 Im (

1

S0∗ 1

S1∗) = 0 .

Anyway, it is to say that the usage of this equation to perform this test in terms of thek

lA vectors is not convenient since, already to the second order, it reduces to a very not

trivial expression. Instead, it is much more convenient to equate directly the final results

of the corrections of the out of diagonal components of the metric tensor and perform

some numerical test, after having chosen some random numerical values for the constants.

75

The formulas for the corrections of the electromagnetic potentials components are:

1

A0 = −2 Im (1

S0∗) ,

1

A1 = −2 Im (1

S1∗) ;

3

A0 = −2 Im (3

S0∗) ,

3

A1 = −2 Im (3

S1∗) .

From these components, as a final step, we will have to perform the coordinate transfor-

mation defined by (3.27). We have obviously to expand such transformations too. Hence,

from the defining relations (3.28), for the constant En appearing in such transformation,

we will obtain the correctionsk

En . Thus, from the expansions of the righthand side of

the (3.23) and (3.30), constructed by means of the combinations of these constant terms,

we will obtain the corrections for the fields components for each desired order.

The corrections for the bidimensional conformal factor f are:

0

f =0

C(0)

0

f (0)0

T0

T ,2

f =0

C(0)

[2

f (0)0

T0

T + 20

f (0) Re

(0

T2

T

)]+

2

C(0)

0

f (0)0

T0

T ,

Where now, we recall, T is the determinant of the Tkl matrix. For flat background,

being f (0) = 1 , we simply have:

0

f =0

C(0)

0

T0

T ,2

f = 20

C(0) Re

(0

T2

T

)+

2

C(0)

0

T0

T .

5.2 One-soliton approximate solution.

For this first deduction2 (which, we recall, is here performed only as a verification), we

keep the assumptions given by the (3.14) and (3.26) formulas. Hence, to determine the

perturbative terms of the m(k)A and p(k)A vectors, it is before necessary to expand σ

and the lA vectors, according to their determinations (3.14) and (3.26). Moreover, we

equate to zero the unphysical constants b and g . For this case of one-soliton solution,

we will preform the expansion up to O(κ4) . From (3.14) we have:

σ = ±(|a|+ 1

2

e2

|a| κ2

)+ O(κ4) .

Since, from the expansion of the lA vectors, we can see that their perturbative terms are

proportional to (±|a| − a)−1 , then we have to replace ±|a| with −a . The first terms

2Dealing with only one soliton, we will omit, in this section, to write the soliton index.

76

of the expansion of σ are then:

0σ = −a ,

2σ = −1

2

e2

a.

Therefore, since for one-soliton solution we put again ζ ≡ Re (w1) = 0 , the perturbative

terms of the unique pole are:

0w = −i a ,

2w = −i e2

2 a.

For the lA vectors we have:

0

l 0 = 1 ,1

l 0 = 0 ,2

l 0 = 0 ,3

l 0 = 0 ,

0

l 1 = 0 ,1

l 1 = 0 ,2

l 1 = −1

2

m

a,

3

l 1 = 0 ,

0

l ∗ = 0 ,1

l ∗ = −e

a,

2

l ∗ = 0 ,3

l ∗ =e3

4 a3.

While, for the expansions of λ and Γ functions, we obtain:

0

λ = r c θ + i a ,2

λ = −mc θ +i e2

2 a,

0

Γ = r + i a c θ ,2

Γ = −m +i e2

2 ac θ .

Hereafter we will list in a schematic manner the principal expressions and the results for

each order. We will denote withkg′ab and

k

A′a , the corrections of the dressed components,

withkgab and

k

Aa , the final components obtained by the transformation of coordinates

(3.27), defined by the constant (3.28) denoted with E1 =0

E1 +2

E1 κ2 + O(κ4) .

———————–

• 0-order0

T and0

S matrices

0

T =2

0

Γ

i0σ

0

Γ

0

SAB =

−2 i0σ 0 0

20σ

0

λ 0 0

0 0 0

77

Metric tensor components0gab

0g′00 = −1

0g′01 = 2

0σ

0g′11 = ρ2 − 4

0σ2

and, with0

E1 = 20σ , then

0g00 = −1

0g01 = 0

0g11 = ρ2

0

f =0

C(0)

0

f (0)0

T0

T =0

C(0)

0

f (0) 40σ2

Choosing0

C(0) =0σ2/4 = a2/4 , then

0

f = 1 .

Electromagnetic potential components0

Aa

Since0

Sa∗ = 0 then

0

Aa = 0 .

———————–

• 1-order1

T and1

S matrices

1

T = 01

SAB =

0 0 − i e

20

Γ

0 0e

0

λ

20

Γ

−2 i e0

Γ 0 0

Metric tensor components1gab

Since1

Sab = 0 and

1

T 11 = 0 , then1gab = 0 and

1

f = 0 .

78

Electromagnetic potential components1

Aa

1

A′0 =

e r

r2 + a2 c2θ

1

A′1 =

e a r (1 + c2θ)

r2 + a2 c2θ

1

A0 =1

A′0

1

A1 =0

E1

1

A′0 +

1

A′1

1

A0 =e r

r2 + a2 c2θ

1

A1 = − e a r s2θ

r2 + a2 c2θ

———————–

• 2-order0

T and0

S matrices

2

T =

0

Γ

2 i0σ

0

Γ2

(4 m− e2 r

a2+

3 i e2 c θ

a

)

In the following expressions for the components of2

SAB we put y = c θ .

2

S00 =

1

2 (r2 + a2y2)2 [m (r2 − a2) y − e2ry] + i [e2r2 + 3e2a2y2 − 2 m a2r(1 + y2)]

2

S01 =

m (a y + i r)

r2 + a2y2

2

S10 =

1

2 (r2 + a2y2) a

[−e2r3 + 2 ma2r2(3 + y2)− e2a2r (2 + 3y2)− 2 ma4(1− y2)]y+

+i a[−2 mr3(1− y2) + e2r2(3− 2y2)− 3 ma2r (2 + 3y2) + 5 e2a2y2

]

2

S11 =

1

r2 + a2y2[−m (r2 − a2) y + im a (1 + y2)]

2

Sa∗ = 0

79

Metric tensor components2gab

2g′00 =

2 mr − e2

r2 + a2y2

2g′01 =

−e2r2 + 4 ma2r (1 + y2)− e2a2(2 + 3y2)

2 (r2 + a2y2) a

2g′11 =

−2 e2r2 + 2 ma2r (1 + y2)2 − e2a2(1 + 4y2 + y4)

r2 + a2y2

Thus, since the terms of the expansion of the constant E1 for the transformation of

coordinate are0

E1 = −2 a ,1

E1 = 0 ,2

E1 =−e2

2 a,

then:2g00 =

2 mr − e2

r2 + a2c2θ

2g01 = −(2 mr − e2) a (1− c2θ)

r2 + a2c2θ

2g11 =

(2 mr − e2) a2(1− c2θ)2

r2 + a2c2θ

Choosing2

C(0) = e2/8 , then

2

f =2 mr − e2c2θ

r2 + a2c2θ

Electromagnetic potential components2

Aa

Since2

Sa∗ = 0 then

2

Aa = 0 .

———————–

80

• 3-order3

T and3

S matrices

1

T = 0 ,

3

Sab = 0

3

S0∗ = 0

3

Sa∗ =

e

4 (r2 + a2y2)a−e2y + i [m (r2 + a2y2)− e2r]

Metric tensor components1gab

Since3

Sab = 0 and

3

T 11 = 0 , then3gab = 0 and

3

f = 0 .

Electromagnetic potential components3

Aa

3

A′0 = 0

3

A′1 = − e

2 (r2 + a2y2) a[m (r2 + a2y2)− e2r]

Since3

E1 = 0 and thus

3

A1 =3

A′1 +

0

E1

3

A′0 +

2

E1

1

A′0 then

3

A0 = 0

3

A1 = −em

2 a

———————–

It results that all the boxed formulas found above are just equal to the corresponding

perturbative terms obtainable through the expansion of the exact one-soliton solution

described in chapter 3.

It is worth noting that the lowest order electromagnetic potential can be expressed as

the real part of the Lynden-Bell [60] potential3

Ψ = e/0

Γ =e

r + i a c θ. (5.14)

3It is the same of the Ernst electormagnetic complex potential for the Kerr-Newman solution [83].

81

It is easy to see that Ψ =0

Φ0 and hence that, since

0

A0 = Re Ψ and0

B0 = Im Ψ ,

the component0

A1 is deducible from Ψ by means of the integration of the system (2.17).

5.3 Two-soliton lowest order approximate solution

At the two-soliton case regard, we wish to underline the following fundamental aspect

of our strategy. Since the problems of it concerns the reparameterization and the char-

acterization of the poles, we will avoid to assume the same choices as in one-solitonic

case. Therefore, we reject the assumptions (made in [3]) to take the indexed two-soliton

versions of the (3.26) and (3.14). That is, namely, we will not use the definitions:

σ2k = −m2

k − b2k + a2

k + q2k , l1k =

mk − i bk

σk − ak

, l∗k = 2ek − i gk

σk − ak

, k = 1, 2 .

We will only preserve l0k = 1 .

The analysis of the two-soliton perturbed solution, limited to the lowest order, needs

only of starting formulas (5.3) for the poles, (5.4) (5.5) for the functions0

Γk ,1

Γk ,0

λk ,1

λk

and (5.6)-(5.9) for the vectors0m(k)A ,

0p

(k)A and

1m(k)A ,

1p

(k)A . The definitions (5.1) and

(5.2), concerning the coordinates and the positions of the poles on the axis, are obviously

assumed. Hereafter we will use the notation ξk = Re1

l∗k , ηk = Im1

l∗k . Therefore, the

only task of this lowest order analysis is to find the expressions to represent0σk and

ξk , ηk in terms of ak , ek4. These expressions will be deduced imposing the linear

superposition of the electromagnetic potentials as expressed by the (5.14), namely:

Ψ2−sol = Ψ1 + Ψ2 ,

that is:

Ψ2−sol =e1

r1 + i a1 c θ1

+e2

r2 + i a2 c θ2

. (5.15)

Since1σk = 0 and

1

Γk = 0 , hereafter we will simply denote with σk and Γk respectively0σk and

0

Γk , thus

Γk = rk − i σk yk , where yk = c θk .

Besides, we introduce the symbols:

σ+ = σ1 + σ2

σ− = σ1 − σ2

, and

Σ+ = d + i σ+

Σ− = d + i σ−

.

4We recall that the mk mass parameters can enter only at least with the second order of expansion.

82

These notations simplify the building block expressions in such a way that we have:

0

T =

−2 i Γ1

σ1 Γ1

4 Γ2

Σ+ Γ1

− 4 Γ1

Σ+ Γ2

−2 i Γ2

σ2 Γ2

, thus det0

T = − 4

σ1 σ2

|Σ− |2|Σ+ |2

Γ1 Γ2

Γ1 Γ2

and hence

0

T−1 =i

2 |Σ− |2

σ1 |Σ+ |2 Γ1

Γ1

−2 i σ1σ2 Σ+Γ2

Γ1

2 i σ1σ2 Σ+Γ1

Γ2

σ2 |Σ+ |2 Γ2

Γ2

.

For the0

SAB matrix, which, we recall, is defined by

0

SAB = −

2∑

k, l =1

(0

T −1)k l0pA

(l) 0m(k)B ,

since it results that

0p

(l)

A

0m

(k)B

=

4Γk

Γl

0 0

4 i0

λlΓk

Γl

0 0

0 0 0

,

it follows, after some simplification, that:

0

S00 = −2 i σ+ ,

0

S10 = 2 z σ+ + d σ− + 2 i σ2

+ .

These values yields the following components of the metric tensor, which, obviously,

results to be flat:

0g′00 = −1

0g′01 = 2 σ+

0g′11 = ρ2 − 4

0σ+

2

, and with0

E2 = 20σ+ , then

0g00 = −1

0g01 = 0

0g11 = ρ2

.

The calculi for the1

Sa∗ components, since

1

S0∗ = −(

0

T −1)k l0p0

(l) 1m(k)∗ ,

1

S1∗ = −(

0

T −1)k l0p1

(l) 1m(k)∗ ,

83

gives

1

S0∗ =− i

2 |Σ− |2[ ( |Σ+ |2 − 2 i σ2 Σ+

) σ1 l∗1Γ1

+( |Σ+ |2 + 2 i σ1 Σ+

) σ2 l∗2Γ2

], (5.16)

1

S1∗ =

1

2 |Σ− |2[(|Σ+ |2

0

λ1 − 2 i σ2 Σ+

0

λ2

)σ1 l∗1Γ1

+

(|Σ+ |2

0

λ2 + 2 i σ1 Σ+

0

λ1

)σ2 l∗2Γ2

]. (5.17)

To impose the condition (5.15), we can use only the component1

S0∗ . In fact

Re(Ψ2−sol) =1

A0 = −2 Im(1

S0∗) , Im(Ψ2−sol) =

1

B0 = +2 Re(1

S0∗) .

Thus, from the conditions

Re

(e1

Γ1

+e2

Γ2

)= −2 Im(

1

S0∗) = Re (2 i

1

S0∗) ,

Im

(e1

Γ1

+e2

Γ2

)= +2 Re(

1

S0∗) = Im (2 i

1

S0∗) ,

that is

Ψ2−sol =e1

Γ1

+e2

Γ2

= 2 i1

S0∗ ,

it follows that:

0σk = −ak ,

ξ1 = − d2 + a+a−d2 + a2

+

e1

a1

η1 = − d (a+ − a−)

d2 + a2+

e1

a1

,

ξ2 = − d2 − a+a−d2 + a2

+

e2

a2

η2 = +d (a+ + a−)

d2 + a2+

e2

a2

. (5.18)

It is possible to check that these results are coherent with the1

S1∗ component. In fact

substituting (5.18) in (5.17) it follows that

1

S1∗ =

1

2

[(0

λ1 − 2 i a2

)e1

Γ1

+

(0

λ2 − 2 i a1

)e2

Γ2

].

Moreover, we have1

A1 =0

E2

1

A0′ +

1

A1′ = −2 Im(

0

E2

1

S0∗ +

1

S1∗)

and finally1

A1 = −Im

(0

λ1e1

Γ1

+0

λ2e2

Γ2

).

84

It is now easy to verify that this expression is the same as that deducible by means of

the integration of the system (2.18) which, for this component, becomes

1

A1,ρ = −ρ1

B0,z

1

A1,z = ρ1

B0,ρ

.

Equilibrium conditions

The equilibrium condition can now be found looking for the distance for which the

interaction potential assumes extremal values. It can be expressed as the opposite of the

total Lagrangian of the system

Vint = −L =1

16 π

∫Fij F ij dV . (5.19)

The scalar invariant in the above volume integral is constructed with the non null co-

variant and controvariant components of electromagnetic tensor field Fµa = Aa,µ and

F µa =1

fηµνgac Ac,ν . The integral (5.19) gives5

Vint =e1e2 d

d 2 + a 2+

.

Therefore, the distances between the two sources for which this potential assumes ex-

tremal values is

d = ±a+ . (5.20)

These distances will correspond to stable or unstable configurations depending by the

signs of the charges and the magnitudes of the angular momentums.

Substituting the values given by (5.20) into the expressions (5.18) it results:

1

l∗1 = −e1

a1

(a1 ± i a2

a+

),

1

l∗2 = −e2

a2

(a2 ∓ i a1

a+

).

5This result is present in [79]; as pointed by its author, it is worth to mention, about this calculation,L. Samuelsson.

Chapter 6

Electric force lines of the doubleReissner-Nordstrom exact solution

6.1 Introduction

The new solution which has been recently found by Alekseev and Belinski [4] (in the

following denoted with AB) has solved the long standing problem of the static equilibrium

of two charged masses in the context of General Relativity (GR).

While in the Newtonian theory the equilibrium condition is simply m1m2 = e1e2 , in the

relativistic regime the problem is much more complicate because one has to solve the

full system of the Einstein-Maxwell equations:

Rij − 12R gij = 2

(FikFj

k + 12FlmF lmgij

)

(√−gF ik),k =

√−gji

(6.1)

and find a static solution with two sources. Furthermore, in general this solution will

present conic singularities at the symmetry axis1; to find the equilibrium condition is

equivalent to require the absence of any conic singularity, i.e. the axis has to be el-

ementary flat. This means that there must be neither “struts” nor “strings”(see [81]

for the rigorous relation between the value of the angle deficit and the effective energy-

momentum tensor of these struts and strings) which prevent the two bodies to fall or

run away each other.

The key point to understand the main differences between classic and relativistic regime

is the repulsive nature of gravity in GR near a naked singularity. This can be seen just

by looking at the Reissner-Nordstrom (RN) metric

gtt = 1− 2M

r+

Q2

r2, (6.2)

1It is called “conic singularity” because the ratio between a small circumference around the axis andits radius is not 2π (as for a circle painted on a cone around its vertex).

85

86

where gravity is repulsive for r < Q2

M: it is for that reason that the equilibrium is allowed

only at certain distances. Indeed, e.g. if one considers the geodesic of a neutral particle

on that background, it is easy to find a (stable) equilibrium precisely at

rc =Q2

M. (6.3)

For charged particles an equilibrium is also possible at a fixed distance [21]; in these

cases it can be both stable or unstable, according to the choice of the parameters. In the

AB-solution, the Newtonian equilibrium condition is restored taking the limit of large

distance between the two singularities.

Although in principle such exact solution could be found already many years ago - by

using the Inverse Scattering Method (ISM) or the Integral Equation Method (IEM) -

practically nobody was able to eliminate the conic singularity in a reasonable explicit

way. Indeed, the important achievement of the AB-solution is the extreme compactness

of all the formulas, despite of complexity of calculations by which it was found [5]. They

get the wanted task using the IEM which presents some advantages with respect to the

ISM2.

As they showed, the equilibrium is possible, apart from the well-known Majumdar-

Papapetrou case where the charge of each source is equal to its mass, only for a naked

singularity near a black hole (b.h.). We excluded from our analysis the b.h.-b.h. and

naked-naked configurations since they do not exist at all in the equilibrium state.

This chapter is organized as follows: we give a brief historical review of the works in

literature (Sec. 6.2) (this section can be skipped by the ones interested only to the

physical contents); to make easier the reading, we also add the reproduction of the

Alekseev-Belinski solution in Sec. 6.3; we give some details clarifying the use of the

coordinates systems involved (Sec. 6.4); then we recall the definition of the electric field

in GR (Sec. 6.5) and finally we graph the plots of the electric force lines in the various

qualitatively different cases (Sec. 6.6). More precisely, in this last section, we consider

at the beginning the general case with two charges, firstly with e1e2 > 0 and then

e1e2 < 0 ; and finally that in which only one object (the naked singularity) is charged.

This last case was presented in different form in [6]. Of each configuration, we present

also the limit in which one source has a much smaller mass and charge than the other.

In particular we consider the limit case of a small charged particle near a Schwarzschild

black hole, finding electric force lines plots congruent with the Hanni-Ruffini [43] ones.

2In the ISM there are also some unphysical parameters (NUT parameter, magnetic charge) and therotation which are not easy to be eliminated.

87

6.2 Some Historical Remarks

The problem of the equilibrium of two charged masses and their resulting gravitational

and electric fields has a long history in GR literature (see table 6.1). It is possible to

distinguish two different kind of results: approximate results, and exact solutions.

Table 6.1: Some historical remarks.

Perturbation Methods Exact Solutions

(1927) Copson: Electric field of a test chargenear a Schwarzschild b.h.

(1947) Majumdar-Papapetrou: mi = ei

(1973) Hanni-Ruffini: Electric force lines of atest charge near a Schwarzschild b.h.

(1976) Linet: A correction of Copson solu-tion

(1978) Belinski-Zakharov: Vacuum solitons(1979) Hauser-Ernst: Integral equation

method for rational axis data(1980) Alekseev: Electrovacuum solitons(1984) Sibgatulling: Integral equation

method for rational axis data(1985) Alekseev: Integral equation method

for rational monodromy data(1993) Bonnor: Equilibrium of a test particle

on RN background(1997) Perry-Cooperstock: Equilibrium is

possible (three numerical example)(2007) Bini-Geralico-Ruffini: Equilibrium of

a test charge on RN with back-reaction until first order

Alekseev-Belinski: Exact solution forequilibrium (without strut) of two RNsources

In the contest of the approximate results, the first to be mentioned is the one of Copson

[28], who gave in 1927 the electric potential of a test charge on the Schwarzschild back-

ground (therefore it was neglected the backreaction of the particle on the metric tensor).

That work was important because it gave the potential in a closed analytic form, however

that result was not completely correct because it implicated that the black hole would

have an induced charge: the correct potential was given by Linet [59] only in 1976 —the

electric potential of the AB-solution indeed reduces to that form in the limit in which

the naked singularity source can be considered as a test particle.

88

In 1973 Hanni and Ruffini [43] gave for the first time the plots of the electric force lines3,

again for a test particle near a Schwarzschild black hole (but they used a multipole ex-

pansion of the electric potential).

Later a certain number of papers have been published in which different authors (us-

ing both exact generating techniques and approximate one, like Post-Newtonian (PN)

and Parameterized-Post-Newtonian (PPN) approaches) arrived to different conclusions

about the possibility/impossibility of an equilibrium configuration. However no final

statements were achieved because of the use of supplementary hypothesis or for the in-

completeness of the analysis.

In 1993 the already mentioned article of Bonnor [21] gave an important hint to clarify

the problem: studying the equilibrium configurations in the test particle limit, namely a

test charge on the RN background, he pointed out that equilibrium configurations were

possible when the ratio e/m was less than unity for the background and more than unity

for the particle, or viceversa; he showed also that equilibrium was possible for charges

of opposite signs too. It is worth noting that the Alekseev-Belinski solution confirms

practically word-by-word (from a qualitative point of view) that picture.

Then in 1997 Perry and Cooperstock [73] found three numerical example showing that

the equilibrium was possible for naked-b.h. configurations using an exact solution.

Finally it is to mention the Bini-Geralico-Ruffini articles [18, 17], in which the authors

found, using the Zerilli perturbative approach, the correction to the test particle ap-

proximation, considering the back-reaction of the particle to the background until the

first order. Surprisingly they found that the Bonnor condition remain unchanged also

considering these corrections.

For what concerns the exact solutions history, the first two important articles were the

ones of Majumdar and Papapetrou [63, 72], which exhibited the fields of an arbitrary

number of sources in reciprocal equilibrium, each one with mi = ei .

For many years that was the only exact result known, the next step was made by Belinski

and Zakharov [14, 15] in 1978 with the foundation of the Inverse Scattering Method in

General Relativity (purely gravitational), which was then extended also to the Einstein-

Maxwell equations by Alekseev [1] (see [9] for a self-consistent review). This method al-

lows to find stationary, axially symmetric solutions with an arbitrary number of sources.

From this time in principle the solution of our problem was available. However, practi-

cally, the constraints necessary to eliminate the rotation, the conic singularity and the

unphysical parameters (NUT parameter, magnetic charge) were too complicate to be

handled analytically.

The next step was made by Ernst and Hauser [46, 47], Sibgatullin [80] and Alekseev

3We follow this work for the construction of the plots of the present solution.

89

[2], who developed different integral equation methods for constructing of solutions of

Einstein-Maxwell equations. (The first method of such kind for pure gravity was already

formulated in [14, 15]). The method of [2] was used by Alekseev and Belinski to find

the present solution [4] (see also [5]), the important achievement of which is the extreme

simplicity of the formulas and of the equilibrium condition.

6.3 Summary of the Alekseev-Belinski formulas

The following (6.4)-(6.13) formulas are the reproduction of formulas (1)-(10) of [4].

The solution, which can be interpreted as the non-linear superposition of two RN source

at a fixed distance on the z-axis, is of the form

ds2 = Hdt2 − ρ2

Hdϕ2 − f(dρ2 + dz2) (6.4)

(6.5)

At = Φ, Aϕ = Aρ = Az = 0 (6.6)

where H , f and Φ are real function of ρ and z only. In what follows m1, m2 and

e1, e2 are the physical masses and charges of each source respectively4; the masses

include also the interaction energy therefore Mtot = m1 + m2 , and Qtot = e1 + e2 . It

is convenient to use the spheroidal coordinates (r1, θ1) and (r2, θ2) which are linked to

the Weyl coordinates (ρ, z) by:

ρ =√

(r1 −m1)2 − σ21 sin θ1

z = z1 + (r1 −m1) cos θ1

ρ =

√(r2 −m2)2 − σ2

2 sin θ2

z = z2 + (r2 −m2) cos θ2

(6.7)

By definition l ≡ z2 − z1 is the distance, expressed in the Weyl coordinate z , between

the two objects. Then, the explicit solution is:

H =[(r1 −m1)

2 − σ21 + γ2 sin2 θ2][(r2 −m2)

2 − σ22 + γ2 sin2 θ1]

D2(6.8)

Φ =[(e1 − γ)(r2 −m2) + (e2 + γ)(r1 −m1) + γ(m1 cos θ1 + m2 cos θ2)]

D(6.9)

f =D2

[(r1 −m1)2 − σ21 cos2 θ1][(r2 −m2)2 − σ2

2 cos2 θ2](6.10)

4The expressions were found with the help of the Gauss theorem.

90

where

D = r1r2 − (e1 − γ − γ cos θ2)(e2 + γ − γ cos θ1) , (6.11)

while γ , σ1 and σ2 are defined by:

γ = (m2e1 −m1e2)(l + m1 + m2)−1 ,

σ21 = m2

1 − e21 + 2e1γ , σ2

2 = m22 − e2

2 − 2e2γ .

(6.12)

It is easy to see that (fH)ρ=0 = 1 on the whole axis, i.e. automatically there is no conic

singularity. The above formulas give the solution satisfying the Einstein-Maxwell system

only under the equilibrium condition

m1m2 = (e1 − γ)(e2 + γ). (6.13)

Each of the parameters σ1 and σ2 can be either real (in the case of a black hole) or

imaginary (for a naked singularity); however in the following it will be always

σ21 > 0 , σ2

2 < 0, and σ1 > 0 (6.14)

i.e. the first source is “dressed” and the second is “naked”. Since we want to deal only

with separable objects, we require also the non-overcrossing condition

l − σ1 > 0 (6.15)

(it means that the naked singularity must be outside the horizon). Using (6.13), the

distance l can be written as a function of the other parameters by the very simple

formula:

l = −m1 −m2 +m1e2 −m2e1

2(m1m2 − e1e2)

[(e2 − e1)±

√(e1 + e2)2 − 4 m1m2

]; (6.16)

we always choose the sign in front of the root in (6.16) in order to satisfy the non-

overcrossing condition (6.15). From (6.16) it is clear that the parameters must satisfy

the restriction

(e2 + e1)2 > 4 m1m2 . (6.17)

6.4 Some further details of the solution

The solution has a very simple form, the only price to pay is just the simultaneous use

of two pairs of coordinates. Obviously for practical purposes, as for the electric lines

plot, one needs the use of only one system. We choose (r1, θ1) , the one related to the

91

black hole (which is centered on the origin, since we took z1 = 0 for simplicity, and

consequently z2 = l ). The linking relations are:

r2 −m2 = 1√2

√b2 +

√b4 − 4σ2

2(z − z2)2

cos θ2 = (z − z2)(r2 −m2)−1

(6.18)

where b2 ≡ ρ2 +σ22 +(z−z2)

2 , while ρ and z have to be expressed using the first couple

of (6.7). We take the plus sign of the roots in the first of (6.18) since r1 and r2 must

coincide at infinity.

The peculiarity of the coordinates used needs a clarification in order to understand the

physical property of the solution, first of all where the “true” divergences are and what

happens on the horizons.

Using (r1, θ1) : These coordinates are centered on the black hole and can be considered

as the natural generalization of the Schwarzschild ones. For the peculiar choice of the

(r1, θ1)− coordinates, the horizon remains a perfect circle (it can be seen also analytically

that H vanishes at rh = m1±σ1 as for the single RN black hole). However the spherical

symmetry is only apparent, indeed the invariants have a θ1-dependence and vary on the

horizon. In this frame is not possible to reach the inside of the spheroid r2 < m2 (we

called the surface r2 = m2 the ‘critical spheroid’, as in [4]), therefore the second source

(the naked RN centered in z = z2 ), appears squeezed “inside” a horizontal segment that

cuts the vertical axis: this happens because the naked singularity lies inside the region

not covered by (r1, θ1) .

Table 6.2: The two peculiar regions in Weyl and in the spheroidal coordinates.

Physical description Location

Horizonρ = 0, z1 − σ1 ≤ z ≤ z1 + σ1

or equivalentlyr1 = m1 + σ1 , ∀θ1

Critical spheroidof the naked singularity

0 ≤ ρ ≤ Im(σ2) , z = z2

or equivalentlyr2 = m2 ,∀θ2

92

Using (r2, θ2) : Conversely, using (r2, θ2) , the ‘critical spheroid’ of the naked RN will

appear as a sphere of coordinates r2 = m2 , while the black hole horizon as a segment

squeezed on the axis: in this case it is the ‘critical spheroid’ of the first source, i.e.

r1 < m1 , that cannot be reached. Again, that has nothing to do with physics but

just with the choice of the coordinate system). In table 6.2 we localize the two peculiar

Table 6.3: Characteristic points of the first source.

Note the degeneracy of the Weyl coordinates.For the numerical evaluation we used m1 = 1, e1 = 0.7, m2 = 0.3, e2 = 0.44,l = 5 (the same used for fig.1).The central singularity of the b.h. is split in two points:

r(I)1 =

m1+m2+l−√

(m1+m2+l)2−4e1e2

2, and r

(II)1 =

m1−m2−l+√

(m1−m2−l)2−4e1e2

2.

Description (r1, θ1) (ρ, z) H Φ f F ijFij

Two branch points

r1 = m1

θ1 = 0, π(0, 0) finite fin. fin. fin.

Equatorial pointof the ext. hor.

r1 = m1+ σ1

θ1 = π/2(0, 0) 0 fin. fin. fin.

Equatorial pointof the int. hor.

r1 = m1− σ1

θ1 = π/2(0, 0) 0 fin. fin. fin.

B.h. singularity I

r(CI)1 = 0.0497

θ1 = 0(0, r1−m1) +∞ −∞ 0 −∞

B.h. singularity II

r(CII)1 = 0.0594

θ1 = π(0,−r1+ m1) +∞ −∞ 0 −∞

North poleof the ext. horizon

r1 = m1+ σ1

θ1 = 0(0, σ1) 0 fin. +∞ fin.

South poleof the int. horizon

r1 = m1− σ1

θ1 = π(0, σ1) 0 fin. +∞ 0

North poleof the int. horizon

r1 = m1− σ1

θ1 = 0(0,−σ1) 0 fin. +∞ 0

South poleof the ext. horizon

r1 = m1+ σ1

θ1 = π(0,−σ1) 0 fin. +∞ fin.

regions (the horizon and the critic spheroid), using Weyl coordinates, with the respective

translations in (r1, θ1) or (r2, θ2) ; while in tables 6.3-6.4 we give a detailed description

of the relevant physical quantities in the notable points of these two zones. It is also

93

to note the “degeneracy” of the Weyl coordinates: to the same point in (ρ, z) it can

corresponds different values of the spheroidal coordinates.

Table 6.4: Characteristic points of the naked source:

The first three points correspond all to (ρ = 0, z = z2 = l) .The same numerical values of table 6.3 are used.

Description (r2, θ2) (ρ, z) H Φ f F ijFij

Naked sing.

r2 =

l+Mtot−√

(l+Mtot)2−4e1e2

2

θ2 = π(0, l) +∞ −∞ 0 −∞

Crossing of thecut with the axis(up border)

r2 = m2

θ2 = 0(0, l) fin. fin. fin. fin.

Crossing of thecut with the axis(down border)

r2 = m2

θ2 = π(0, l) fin. fin. fin. fin.

Extremesof the cut

r2 = m2

θ2 = π/2(Im σ2, l) fin. fin. +∞ fin

The electromagnetic invariant

In order to understand where the charges are located it is useful to consider the electro-

magnetic invariant F = F ijFij/2 . For the solution (6.4) it has the form:

F = − [(r1 −m1)2 − σ2

1](∂r1Φ)2 + (∂θ1Φ)2

f H [(r1 −m1)2 − σ21 cos2 θ1]

. (6.19)

It can be seen numerically (see tables 6.3,6.4) that it diverges inside the horizon and

inside the critical spheroid of the naked RN 5.

It is also worth noting that on the critical spheroid, although in the (r1, θ1) representa-

tion it is a line, the up- and down-limit of F do not coincide, since they correspond to

different points of the physical space-time.

Looking at F it is possible to see that no real discontinuity exists on the horizon, indeed

it diverges only on the central singularities.

The other invariant, εijklFijFkl = E ·B , is identically zero.

5The spheroid, i.e. the line 0 < ρ < Im(σ1) , z = z2, seems apparently regular in (r1, θ1) coordinatesjust because its interior can be reached only using (r2, θ2)

94

6.5 Electric force lines definition

Just to understand better the meaning of the plots, we want to recall the definition of

the electrical vector (which is not a trivial choice in GR). Following [43], we define the

electric field as the three non-diagonal time-like components of the controvariant tensor

F ij :

Eα = Fα0 , α = 1, 2, 3 . (6.20)

That identification is geometrically justified by the Gauss theorem generalized to the

curved manifolds [88]:

4πQ =

∫

C

*F =

∫

C

∗Fij dxi ∧ dxj , (6.21)

where ∗Fij = 1/2εijklFkl√−g is the dual tensor of F ij . Then it is natural to define the

force lines in the usual way as the trajectories of the dynamical system:

d

dλr1 = Er1

d

dλθ1 = Eθ1

(6.22)

or equivalently by

dr1

dθ1

=Er1

Eθ1,

Er1

Eθ1=

[(r1 −m1)

2 − σ21

] ∂r1Φ

∂θ1Φ. (6.23)

Then, from the equation of motion for this problem, restricting to our case, we have:

F r1t u

t dθ1 − F θ1t u

t dr1 = 0 , (6.24)

having used the coordinates xi = (t, ϕ, r1, θ1) .

The physical interpretation (Christodoulou-Ruffini, quoted in [43]) is the following: a

force line is a line tangent to the direction of the electric force measured by a free-falling

test charge momentarily at rest, with initial 4-velocity

ut = (√

gtt, 0, 0, 0). (6.25)

Note that such interpretation is valid only for gtt > 0 , for this reason we have not plotted

the lines inside the horizon.

In the (t, ϕ, r1, θ1) coordinates the metric (6.4) becomes

ds2 = H dt2 − ρ2

Hdϕ2 − f [(r1 −m1)

2 − σ21 cos2 θ1]

[dr1

2

(r1 −m1)2 − σ21

+ dθ12

](6.26)

95

while the electric potential remains unchanged. Then for the electric field we have:

Eϕ = 0

Er1 = gttgr1r1 ∂At

∂r1

Eθ1 = gttgθ1θ1 ∂At

∂θ1

. (6.27)

Therefore the force lines are given by the solution of

dr1

dθ1

=[(r1 −m1)

2 − σ21

] ∂r1At

∂θ1At

. (6.28)

It is worth noting that the force lines depend only on the two ratios ∂r1At/∂θ1At and

gr1r1/gθ1θ1 (indeed the conformal factor f and neither gtt nor gϕϕ do not appear in

(6.28).

6.6 Plots of the force lines

In the plots, what we called “second source” (i.e. the naked RN) is always up, while the

“first source” (i.e. the black hole) is always down and centered on the origin. The lines

are plotted in (x, y) Cartesian coordinates defined as

x = r1 sin θ1

y = r1 cos θ1

(6.29)

(they coincide with (ρ, z) defined in (6.7) when r1 →∞ ).

In the plots we have used geometrical units ( G = c = 1 ), in which the unitary length is

given by the Schwarzschild mass m1 = 1 .

The graphical Faraday criterium is used, namely we plotted the electric force lines such

thatNumber of lines from the first source

Number of lines from the second source∼= e1

e2

.

The separatrix

In general, when there are two charges, the electric force diagram will present a separa-

trix, which is a force line which reach asymptotically a saddle point of the potential and

separates the lines of the two charges in the case they have the same sign. In the case of

opposite sign charge, it delimits the region in which the lines flow from one to the other

source. We marked these separatrix lines in bold; may be it is worth to mention that on

the saddle point they have an invariant definition since on that point F = 0 .

96

Inside the horizon

In the following plots the force lines are graphed only outside the horizon since there

it is no more possible to consider a static observer; the physical interpretation given in

Sec. 6.5 does not hold because (6.25) becomes imaginary. However, when the separatrix

starts from the inside of the horizon, the study of that region is important to understand

the difference between cases with the same or opposite charges. Therefore, in the case of

fig. 6.3, in which the saddle point is inside the horizon, we calculated the point where the

separatrix touches the horizon, and we plotted the diagram just from there. (This was

possible because mathematically the eqn. (6.28) is well defined also inside the horizon).

In the following three sub-sections we analyze the three qualitatively different sub-cases:

e1e2 > 0 (6.6.1), e1e2 < 0 (6.6.2), and finally e1 = 0 (6.6.3).

6.6.1 Two charges of equal sign ( e1e2 > 0 )

General case: two comparable RN sources

Let us consider the case in which the two RN sources have charges and masses of com-

parable dimensions

m1 ≈ m2 e1 ≈ e2

m21 > e2

1 m22 < e2

2

e1e2 > 0 .

(6.30)

This is the closest case to the classical picture, indeed here the equilibrium is mainly due

to the classical balance of the electrostatic force and gravitational field. The resulting

plot is given in fig.(6.1).

The qualitative behavior of the force lines does not change with the changing of the

distance l .

Small charge (naked) near a RN black hole

Here and in the following we say ”small” charge and not ‘test’ charge because the exact

nature of the solution automatically takes in account all the back-reaction terms even if

they can be very small (while the ‘test’ limit is in general referred as the one in which all

those terms are completely neglected). The equilibrium configurations of this case (see

97

10

8

6

6

4

4

2

0

2

-2

-4

0-2-4-6

Figure 6.1: Force lines in the general case (6.30), when the two RN have charges of thesame sign. Note that the critical spheroid in that coordinate representation (6.29) is anhorizontal segment. The bold line is the separatrix. The circle on the bottom is theexternal horizon of the first source. Parameters used: m1 = 1 , e1 = 0.7 , m2 = 0.3 ,e2 = 0.44 , l = 5 .

fig. 6.2), with

m1 >> m2 , |e1| >> |e2|

m21 > e2

1 m22 < e2

2

e1e2 > 0 ,

(6.31)

have been studied in the test particle approximation first in [21], and recently in [18, 17],

where they took in account also the back-reaction of the test particle.

98

6

4

4

2

2

0

-2

0-2-4

Figure 6.2: Force lines of a small charge near a RN with horizon, case (6.31). Parametersused: m1 = 1 , e1 = 0.1 , m2 = 10−3 , e2 = 1.3 · 10−2 , l = 2.5 ). The bold line is theseparatrix.

Small charge (with horizon) near a naked RN

This case does not exist for e1e2 > 0 .

6.6.2 Two charges of opposite sign ( e1e2 < 0 )

Although it is easy to show that in the previous cases with e1e2 > 0 the implications

m21 > e2

1 ⇒ σ21 > 0

m22 < e2

2 ⇒ σ22 < 0 ,

(6.32)

are always true, it is not so if e1e2 < 0 . However in the following we considered two

cases in which (6.32) holds.

99

Two comparable RN sources

This case, with

m1 ≈ m2 e1 ≈ −e2

m21 > e2

1 m22 < e2

2

e1e2 < 0 ,

(6.33)

is the case in which the relativistic effects are much evident since here also the electric

force is attractive (see fig. (6.3)): in this case the equilibrium is due to the repulsive

nature of the naked singularity.

10

8

6

6

4

4

2

0

2

-2

-4

0-2-4-6

Figure 6.3: Force lines in the general case (6.33), with charges of the opposite sign.Parameters used: m1 = 1 , e1 = 0.05 , m2 = 0.3 , e2 = −1.66 , l = 5 . The bold line isthe separatrix, which now encircles also the central singularity of the b.h.: inside thatregion the lines go from one charge to the other. Outside that region the lines go frome2 to infinity (some of them pass also through the horizon).

100

Small charge near a RN

It is also possible to find values that corresponds to a small charge with horizon near a

naked RN:m1 << m2 , |e1| << |e2| ,

m21 > e2

1 , m22 < e2

2 ,

e1e2 < 0 .

(6.34)

However in this case it would be useless to plot the force lines because the electric field

is trivially Coulombian (the first source is weakly interacting both gravitationally and

electrically).

The inverse case, namely a small charge naked near a RN with horizon, does not exist

for particles lying outside the horizon (i.e. requiring l > σ1 ), as noted by Bonnor [21].

6.6.3 Cases with only one charge

In the following we will consider the cases with a naked singularity near a neutral black

hole; they are qualitatively different from the previous ones since now there is no separa-

trix and the electric flux over the horizon surface is zero. In the particular case in which

the first source is neutral (i.e. e1 = 0 ), the equilibrium distance is even simpler,

l = −m1 −m2 +e22

2m2

1±

√1− 2 m1

(e22

2m2

)−1 , (6.35)

which can be always satisfied for sufficiently large values of the charge parameter e2 .

101

RN near a Schwarzschild black hole (comparable masses)

10

8

6

6

4

4

2

0

2

-2

-4

0-2-4-6

Figure 6.4: Force lines for the values (6.36). The blank circle of radius 2m1 is theSchwarzschild horizon. Parameters used: m1 = 1 , m2 = 0.3 , e2 = 1.5 , l = 5 .

Thanks to the exact nature of the solution, it is very interesting also the case in which

the RN source has comparable mass with the Schwarzschild black hole, say

m1 ≈ m2 , e1 = 0 ,

σ1 = m1 , m22 < e2

2 ,

(6.36)

indeed this case cannot be achieved by a perturbative approach, see fig. 4. It is possible

to see that the electric lines are just slightly deformed by the gravitational field.

Small charge near a Schwarzschild black hole

We can also consider the small-charge limit,

m1 >> m2 , e2 e1 = 0 ,

σ1 = m1 , m22 < e2

2 ,

(6.37)

i.e. the second source is a small RN naked singularity. That is the only case in which we

have a good comparing in literature, since it is the only case already studied (as we know)

102

by using the force lines plots [43], although by a perturbative approach. Strictly speaking

the Hanni-Ruffini case refers to a slightly different situation, since they considered a

particle momentarily at rest in the Schwarzschild metric, while the AB solution is exactly

static6. However the present solution confirms very nearly their multipole expansion,

since we find that the plots are in practice coincident. In order to have the best possible

comparing we considered the same distances between the charge and the horizon (figg. 5-

7). Since now l is not an independent parameter we fixed the masses values m1 = 1 and

m2 = 10−4 , then varying the distance we found (using (6.16)) the respetive parameter

e2 . The test particle is at z = l , or equivalently at r1 = l + m1 . (Just to clarify the

link with [43]’s notations: their r is our r1 , and their M is our m1 ).

10

10

5

5

0

-5

0-10

-5-10

Figure 6.5: Force lines for the values (6.37), with l = 3 m1 , i.e. in the spheroidalcoordinates the particle is in r1 = 4 m1 . The circle of radius 2m1 is the Schwarzschildhorizon. The plots are practically identical to the ones found by Hanni and Ruffini.

6From another point of view, Hanni-Ruffini do not use (6.35) to determine the fourth parameter(because in their approximation the fourth parameter, say m2 , is considered arbitrarily small, thereforeit is not present at all)

103

10

10

5

5

0

-5

0-10

-5-10

Figure 6.6: Now the distance is l = 2 m1 , or equivalently the charge is in r1 = 3 m1 .

10

10

5

5

0

-5

0-10

-5-10

Figure 6.7: Now the distance is l = 1.2 m1 , or equivalently the charge is in r1 = 2.2 m1 .

From eq. (6.23), considering that now σ1 = m1 , it is easy to see that the corrections

104

to the Hanni-Ruffini approximation are limited only on the exact form of the At poten-

tial, since to use the Schwarzschild metric or the functions H and f given in (7.115)

and (6.10) does not change the force lines.

6.7 Final remarks

The main result of our analysis is that the exact solution seems to confirm quite strictly

the test-charge approximation on the RN background (see. e.g. [21]), which seems to

give a good test of the exact picture.

Size of the naked singularity Sometimes in literature has been guessed ([36], [68])

that e2/2m should be considered as a ‘critical radius’ of the naked singularity inside of

which the RN solution has no physical meaning since it should be matched with a more

realistic matter field tensor, in order to avoid the well known problems of a point-like

source, as the divergence of the electric energy.

If the quantity e22/2m2 can be roughly considered as the physical size of the RN charge,

then from formula (6.35) it is easy to see that the equilibrium configurations exist only

for e22/2m2 larger than the Schwarzschild radius ( 2m1 ). This seems to suggest that a

real ‘small’ charge limit cannot be achieved, in the sense that the particle can be ‘small’

only gravitationally (and electrically), but not geometrically because it would have a size

larger than the Schwarzschild horizon.

However this is just a speculation since further investigations should be done to model

the interior of a realistic RN source and find its radius.

Coordinate dependence of the plots Any plot of the force lines change drastically

for different choices of the coordinates. However, what is interesting is to compare

different situation by using the same coordinate representation, e.g. as we did for the

Hanni-Ruffini case.

Stability If the solution would be unstable that would mean that it is a completely

academic problem, since the equilibrium will be physically not allowed. However in the

geodesic/test particle approximation, which gives the essential features of the problem,

the equilibrium is stable, therefore at least in some range of values it should be the same

also in the exact case (indeed the exact solution smoothly converge to the test particle

approximation in the limit ei , mi → 0 , with ei/mi finite, i = 1 or 2 ).

Anyway a systematic, even if not complete, analysis of stable configurations it is reported

in the next chapter. There we will see that, with respect to the different cases examined

above, a number of exact configurations are indeed stable.

Chapter 7

A stability analysis of the doubleReissner-NordstromAlekseev-Belinski exact solution

Introduction

In the static double Reissner-Nordstrom (RN) solution1of Alekseev-Belinski [4] (AB),

the equilibrium condition which ensure the absence of any conic singularity, implies that

the distance between the two sources has to be a function of the other four parameters

of mass and charge of the two sources: l0 = l0(m1,m2, e1, e2) (see below equation (7.7)).

In the following we will study the stability of this solution in a very restrictive sense, i.e.

with respect to spatial displacements of the two sources. Indeed there can be a lot of

different perturbations, as rotational ones, anyway we think that these are some of the

most physically significant. Our analysis make no use of the usual perturbative methods

(i.e. to put a perturbation in the Einstein-Maxwell equations and see how they evolve in

time), we use instead the dynamical properties of the conical singularity, following the

Sokolov-Starobinski definition [81].

If the two sources are placed in a distance different as regards the equilibrium one, say

l = l0 + x , then one can have still a static solution but in that case it appears a conic

singularity between the two bodies [5], namely on ρ = 0, z ∈ [z1, z2] (let us suppose

that source-1 is in z1 = 0 and source-2 in z2 = l ). This is interpreted as a strut or a

string to which it can be associated a force. It is called “strut” if it exerts a compensative

pressure, “string” if it exerts a tension; since the difference between these two situations

depends only by a sign, hereafter, we will use simply the word ”strut”.

Now, we assume that in the reality there are no struts if the two sources are displaced

1We recall that in this solution, the sources have to be a black hole (b.h.) and a naked singularity(n.s.).

105

106

from the equilibrium position, but that the two sources oscillate near the distance l0

(stable configuration), or go far away or collapse (unstable configurations). Then we

assume that the force exercised by the two bodies one to the other will be precisely the

opposite of FStrut , say

FBodies = −FStrut ; (7.1)

indeed the eventual presence of a strut with such a force would balance the re-

pulsion/attraction of the bodies, keeping the system exactly in “equilibrium”, with

Ftot = FBodies + FStrut = 0 . Therefore by means of the knowledge of FBodies , the

analysis of the equilibrium follows the usual procedure of classical mechanics.

What reported in this chapter is an extended version of part of [74].

7.1 Force of the strut.

To calculate the force of the strut we need to calculate the energy-momentum tensor

T ji on this segment2. Since we know the metric (from the AB solution in his general

5-parameters formulation [5]), we can define T ji , by means of the Einstein equations

(G = 1 , c = 1 )

8π T ji = Rj

i − 1/2 δji R , (7.2)

in terms of a Dirac δ-function, as described in [81].

If the parameter l 6= l0 , then the AB metric presents a conic singularity in the segmenta

of the axis between the two sources. This means that, if we expand the solution near

the axis, the line element corresponding to the two-dimensional space-like surface t, z =

const. , has the form:

ds2 = dρ2 + a2ρ2dϕ2 , ρ ' 0 , 0 < z < l , (7.3)

which is, for a < 1 , the line element of a conic surface. The coefficient a is a constant.

In terms of the δ parameter of the AB solution it is given by

a =1

(√

fH)|ρ=0

=1 + 2δ

1− 2δ. (7.4)

In general, when a 6= 1 between the two sources, it represents a deficit of angle specific

of the conical singularity.

2In this chapter we adopt the following notations: signat. = (−+ ++) ; i, j, ... = 0, 1, 2, 3 ,α, β, ... = 1, 2 ;

(x0, x1, x2, x3) = (t, ρ, ϕ, z

).

107

We recall, from [5], that:

δ =m1m2 − (e1 − γ)(e2 + γ)

(l0 + x)2 −m21 −m2

2 + (e1 − γ)2 + (e2 + γ)2, (7.5)

γ =m2e1 −m1e2

l0 + x + m1 + m2

, (7.6)

l0 = −m1 −m2 +m1e2 −m2e1

2(m1m2 − e1e2)

[(e2 − e1)±

√(e1 + e2)2 − 4 m1m2

]. (7.7)

The calculation of Rji for the line element

ds2 = −dt2 + dρ2 + a2ρ2dϕ2 + dz2 (7.8)

which approximates the AB solution near the axis, gives Rji = 0 , R = 0 , and thus

T ji = 0 . However , we can introduce a distribution-like source using the Gauss-Bonnet

theorem as in [81]:∫

S

Kdσ = 2π −∫

∂S

kg ds . (7.9)

In this formula K is the Gaussian curvature which results to be the half of the Ricci

scalar, K = R/2 ; kg is the geodesic curvature:

kg = εαβ

(d2xα

ds

dxβ

ds+ Γα

µν

dxµ

ds

dxν

ds

dxβ

ds

) (gαβ

dxα

ds

dxβ

ds

)−1/2

; (7.10)

S is a small disk centered on the vertex of the conic surface and ∂S is its border.

Since the right-hand side of (7.9) yields:

2π −∫

∂S

kg ds = 2 π (1− a) , (7.11)

introducing the Dirac delta-function δD(ρ) , normalized by∫ ∞

0

∫ 2π

0

δD(ρ) ρ dρ dϕ = 1 , (7.12)

we can define, by means of (7.9), the Gaussian curvature as:

K = 2π1− a

aδD(ρ) . (7.13)

It results that K = R/2 = R11 = R2

2 , hence Gji = −1/2 R diag (1, 0, 0, 1) . Thus, the

Einstein equations yiels

T 33 = −1− a

4 aδD(ρ) , (7.14)

108

using which, we find the expression of the force3:

FStrut =

∫ ε

0

∫ 2π

0

T 33 a ρ dρ dϕ =

1

4(a− 1). (7.15)

From (7.4) and (7.5), we obtain:

FStrut =δ

1− 2δ=

m1m2 − (e1 − γ)(e2 + γ)

l2 − (m1 + m2)2 + (e1 + e2)2, (7.16)

where we recall that l = l0+x and l0 is the equilibrium distance. For x = 0 than l = l0 ,

the value of which is obtained by the equilibrium condition: m1m2 = (e1 − γ)(e2 + γ) .

Therefore, in this case, it naturally follows that FStrut = 0 .

It is worth to mention the work of Manko [64], where this results is given, at least of the

notation used therein and of the sign4.

It is also useful to recall the work of Letelier and Oliveira [58], where a confusion in the

literature about different definitions of the force of the string is reported and discussed.

Anyway, these differences, since reduce to a positive rescaling of the expression of the

forces, do not affect our following analysis.

Classical limit of the force

Choosing an equilibrium configurations with all the parameters of mass and charge dif-

ferent from zero, and expanding the formulas (7.16) for l going to infinity, we find

−FStrut = FBodies = −m1 m2

l2+

e1 e2

l2. (7.17)

That is, we obtain the resultant between the Newtonian gravitational force law and

the Coulomb electrostatic one. The limit given by (7.17), together with the relations

Mtot = m1 + m2 , shows that the mk parameters, deduced by Alekseev and Belinski by

means of Komar integrals, have to be ever positive and that they coincide with Newtonian

masses.

3In the following formula ε is a small positive number for which the local approximation (7.3) holds.4In that work the force given by the formula (30) is declared to be ”the interaction force between

the costituents” and, then, for the uncharged case, the formula (31) gives ”the known expression for theinteraction force between two Schwarzschild black holes”. By means of this last formula it is easy to seethat, at large separation distance, we do not obtain Newtonian expression of the gravitational force butits opposite; hence, the formula (30), which coincide with our (7.16), is just the expression describingthe compensative forces exerted by the strut.

109

7.2 Analysis of equilibrium in the AB solution

The stability can now be deduced from the sign of the derivative of the force w.r.t. x ;

obviously we have to evaluate this quantity on the equilibrium point x = 0 :

(∂xFStrut)| x=0 =m2e1 −m1e2

(l0 + m1 + m2)2

[2γ0 − e1 + e2

l20 −m21 −m2

2 + (e1 − γ0)2 + (e2 + γ0)2

]; (7.18)

where γ0 is γ evaluated on x = 0 . The stability condition is:

(∂xFStrut)| x=0 < 0 . (7.19)

The previous formula can be simplified without loss of generality using the following

considerations5:

1. Using the arbitrariness of the electric charge’s sign definition:

e2 > 0 ; (7.20)

2. Since we are considering a black hole and a naked singularity:

|e1|m1

< 1 <e2

m2

; (7.21)

3. Separability requirement:

l0 > σ1 ; (7.22)

4. Finally, the existence of a real l0 needs [71]:

(e2 + e1)2 > 4 m1m2 . (7.23)

We recall that, as defined in [5] ,

σ12 = m1

2 − (e1 − γ)2 + γ2 , σ22 = m2

2 − (e2 + γ)2 + γ2 . (7.24)

From (7.20) and (7.21), we find that for all the configurations according with them, it

results that

γ0 < 0 . (7.25)

Then, from Eq.(7.22) it is easy to show that the denominator in Eq.(7.113) is always

positive, i.e.

l20 −m21 −m2

2 + (e1 − γ0)2 + (e2 + γ0)

2 = l20 − σ21 − σ2

2 + 2γ20 > 0 (7.26)

5Obviously a necessary condition for the stability is also that the equilibrium exists.

110

(indeed σ22 < 0 because source-two is naked). Thus, stability condition (7.113) can be

reduced to the following one:

2γ0 − e1 + e2 > 0 . (7.27)

If the previous inequality is not fulfilled, it means that the configuration is unstable.

Inequality (7.118) can be rewritten as:

(m2 −m1)(e1 + e2) + (e2 − e1) l0 > 0 ; (7.28)

for commodity in the following we define the quantity

X = (m2 −m1)(e1 + e2) + (e2 − e1) l0 ,

which is anyway an irrational 4-parameters quantity.

Resuming, the analysis of the stable configurations implies the discussion of the following

system of inequalities:

e2 > 0

|e1|m1

< 1 <e2

m2

l0 > σ1

(e2 + e1)2 > 4 m1m2

X > 0

. (7.29)

Being two of the inequalities present in the above system irrational, a complete classifica-

tion of the stable configurations represents a non trivial task; it will be object of a future

work. Hereafter, we limit ourselves in presenting the discussion of a list of qualitatively

different situations.

A. Equal signed charges: e1 > 0 , e2 > 0 .

This is the only case in which we found also unstable equilibria.

A.1 m1 < m2 : b.h. smaller than n.s..

If m1 < m2 then, necessarily from(7.21), e1 < e2 : consequently X > 0 is always

satisfied and the equilibrium is always stable.

A.2 m1 > m2 : b.h. larger than n.s..

111

A.2.1 m1 > m2 and e1 < e2 .

Numerically we found only stable equilibrium (when it exist).

A.2.2 m1 > m2 and e1 > e2 .

In this case X is always negative and thus the equilibrium unstable.

A particular situation of this sub-case is the small-particle limit (that is

m2 = α e2 , e2 → 0 , with 0 < α < 1 constant); it agrees with the in-

stability found by Bonnor [21] .

B. Opposite signed charges: e1 < 0, e2 > 0 .

In this case we suspect that the equilibrium is always stable (however there is one subcase

in which we was not able to demonstrate it analytically).

Since X now is

X = (m2 −m1)(e2 − |e1|) + (e2 + |e1|) l0 , (7.30)

then we can consider the two different sub-cases: m1 < m2 and m1 > m2 .

B.1 m1 < m2 : b.h. smaller than n.s.

If m1 < m2 , then from condition (7.21) we have necessarily e1 < e2 , which implies

that X is always positive.

B.2 m1 > m2 : b.h. larger than n.s.

In this case we need to consider the two different configurations: |e1| ≷ e2 .

B.2.1 m1 > m2 , |e1| > e2 .

If |e1| > e2 , then one can see at first sight from (7.119) that X is always

positive. Anyway, we was not able in finding any numerical configurations

satisfying the separability condition (7.22).

B.2.2 m1 > m2 , |e1| < e2 .

If |e1| < e2 we are not able to demonstrate that X is always positive, but

we can say that at least for enough large values of e2 this is true, because

lime2→∞

l0 ' m1

|e1| e2 , (7.31)

and thus X → m1

|e1| e22 → +∞ .

C. One charge only e1 = 0

This case is always stable. Indeed, considering e1 = 0 , the stability condition (7.28)

112

becomes:

l0 + m2 > m1 . (7.32)

Then, considering the separability condition, which is now:

l0 > σ1 = m1 , (7.33)

it is immediate to see that (7.125) is always true.

7.3 Final remarks

The above analysis, despite it does not give a complete classification of all possible

equilibrium configurations, yields the following results.

1. Most of the cases are stable.

2. The only unstable case we found is that describing two sources with equal signed

charges and with the parameters of mass and charge of the black hole being both

greater of the corresponding ones of the naked singularity. This result agrees with

Bonnor limit.

3. The one-charge case is always stable.

4. In our previous chapter, we drawn some plots of the electric force lines. It results

that all that cases (taking the same numerical values of the parameters) are all

stable, except the case plotted in figures (6.1) and (6.2). They corresponds to the

case pointed out above at item number two.

5. This criterion of stability, although very peculiar, can be considered at least a

necessary condition for the stability in general sense. Anyway, it becomes also a

sufficient criterion in the limit in which one source is much smaller than the other.

Conclusions and prospects

In the first part of this thesis, in the framework of the solitonic Alekseev technique,

generating exact solutions of the electrovacuum symmetry reduced Einstein-Maxwell

equations, we have presented a procedure to construct approximate terms of such kind

of exact solutions. These terms are relative to the expansions respect to a control pa-

rameter interpretable as the Newtonian gravitational constant or, equivalently, as the

amplitude of the electromagnetic fields. Therefore this perturbative tool gives a way to

compute approximate solutions, belonging to such class of symmetries, both for weakly

coupled gravitational and electromagnetic fields and for weak electromagnetic fields over

curved gravitational backgrounds.

We have tested this procedure computing these approximate terms for the well known

one-soliton stationary exact solution representing a Kerr-Newman black hole. We have

hence deduced the lowest order terms relative to the fields generated by two Kerr-

Newman sources placed at fixed distance on the axis of symmetry. The achieved results

are the weak fields generated by two spinning charged sources in Minskowsky space-time

and the reparameterization between the mathematical parameters and the physical one.

This first deduction represents the first step of a work, still in progress, consisting in the

derivations of the next order terms. Its hoped task is to find a way to find a repara-

meterization for such weak fields solutions, the consequent equilibrium conditions and

thus, some sufficient conditions about the physical configurations to obtain regular static

fields; that is, fields generated by sources in equilibrium without any conical singularity

presents on the axis. Investigations of the solitonic technique, from the prospective of

this approximate approach, could also give some further indications about some peculiar

aspects of its. For example, the problems of the analytical continuations of the poles,

the relation between the Alekseev technique and the Belinski-Zakharov one.

The second part of the thesis has concerned the analysis of some features of the

Alekseev-Belinski solution describing the electrovacuum fields generated by two Riessner-

Nordstrom like sources in static equilibrium. In particular we have presented the be-

haviors of the electric field force lines relative to qualitatively different configurations,

following a classification with respect to the signs and the magnitude of the physical

113

114

parameters of mass and charge of the two sources. We have singled out those cases

which can not exist, like a small charge black hole near a naked equally signed charged

singularity or a small charge naked source near a opposite signed charged black hole.

Besides, we have obtained, and hence confirmed exactly, some results found in the past

by Bonnor and Hanni-Ruffini by means of test particle approximation approaches.

We have then performed an analysis of the stability of these configurations respect small

displacements from the equilibrium distance between the two sources. We have showed

that most of the configurations discussed are stable and singled out cases which are al-

ways unstable. However, due to the complexity of the stability conditions, which consist

in a system of irrational inequalities, the analysis performed is not complete. Namely,

we have found cases for which, until now, we have not been able in finding the discrim-

inating conditions between stable and unstable configurations. A complete analysis of

stable (in this sense) configurations could be an interesting task for a future research.

Again, with respect to this stability analysis, we have found and confirmed previous

results found by Bonnor by means of the test particle approximation.

It is a well known thing that the the Reissner-Nordstrom solution posses a slight physical

significance because, from an astrophysical point of view, celestial bodies are practically

uncharged. Besides, it is rather senseless to use this solution as a framework to find

model for elementary particles. We know that even for an electron, the particle with the

largest charge per unit mass, the gravitational corrections are significant at scales much

smaller respect those at which quantum field theories dominate. However, this does not

exclude the possibility to look for some results, at least at a semiclassical level of approx-

imation; a certain number of works dealing with such kind of investigations has been

produced1. The interest for investigations on the Reissner-Nordstrom solution is given

principally by the fact that it is a simple example of a solution of the Einstein-Maxwell

equations. Besides, as a remarkable fact, it has given new significant insights for the

development of the black holes theory. We believe that, in the context of this theory, the

new solution of Alekseev-Belinski could yield further new interesting results. We recall

that the analysis relevant to this solution that we have presented is restricted to fields

generated by separate sources. At this regard, an interesting topic for future research

could be, for example, the investigation on the possibility to have non-separate sources

configurations; at present, it is still an open question.

1See, for example, the bibliography cited in the introduction of [13]. The reader can find it belowas the first work added in the ”Additional contributions” section.

Additional contributions

Hereafter two works produced in collaboration with V.A. Belinski and M. Pizzi during

the last two years are added in their preprint-version.

In the first one [12, 13], we demonstrate an alternative (with respect to the ones ex-

isting in literature) derivation of exact solution of the Einstein-Maxwell equations for

the motion of a charged spherical membrane with tangential tension. We stress that the

physically acceptable range of parameters for which the static and stable state of the

membrane producing the Reissner-Nordstrom repulsive gravity effect exists. The con-

crete realization of such state for the Nambu-Goto membrane is described. The point is

that membrane is able to cut out the central naked singularity region and at the same

time to join in appropriate way the Reissner-Nordstrom repulsive region.

We have obtained a model of an everywhere-regular material source exhibiting a repulsive

gravitational force in the vicinity of its surface: this construction gives a more sensible

physical status to the the naked singularity part of the Reissner-Nordstrom solution.

In the second one [75], we describe the equation of motion of two charged spherical

shells with tangential pressure in the field of a central Reissner-Nordstrom source. We

solve the problem of determining the motion of the two shells after the intersection by

solving the related Einstein-Maxwell equations and by requiring a physical continuity

condition on the shells velocities.

We consider also four applications: post-Newtonian and ultra-relativistic approxima-

tions, a test-shell case, and the ejection mechanism of one shell.

This work is a direct generalization of Barkov-Belinski-Bisnovati-Kogan [7] paper.

115

I. Charged membrane as a sourcefor repulsive gravity

Introduction

One of the interesting effects of relativistic gravity which has no analogue in the New-

tonian theory is the presence of gravitational repulsive forces. The classical example is

the Reissner-Nordstrom (RN) field in the region close enough to the central singularity.

Indeed, in the RN metric

−ds2 = −f c2dt2 + f−1 dr2 + r2(dθ2 + sin2 θdφ2) (7.34)

where

f = 1− 2kM

c2r+

kQ2

c4r2, (7.35)

the radial motion of a test neutral particle follows the equation:

d2r

ds2= −1

2

df

dr=

k

c4r2

(Q2

r−Mc2

)(7.36)

from where one can see the appearance of repulsive force in the region of small r. In this

zone the gradient of the gravitational potential f(r) is negative and the gravitational

force in Eq.(7.36) is directed toward the outside of the central source.

For the RN naked singularity case (Q2 > kM2), in which we are interested in the

present paper, the potential f(r) is everywhere positive and has a minimum at the point

r = Q2/Mc2. Therefore at this point a neutral particle can stay at rest in the state of

stable equilibrium (the detailed study can be found in [27, 21]).

It is an interesting and nontrivial fact that the same sort of stationary equilibrium state

due to the repulsive gravity exists also as an exact asymptotically flat two-body solution

of the Einstein Maxwell equations which describes a Schwarzschild black hole situated at

rest in the field of a RN naked singularity without any strut or string between these two

objects [4, 6]. However, solutions of this kind have the feature that the object creating

the repelling region has naked singularity and this last property has no clear physical

interpretation. Consequently the pertinent question is whether the repelling phenomenon

116

117

around a charged source arises only due to the presence of the naked singularity or it

can be also a feature of physically reasonable structure of the space-time and matter.

By other words the question is whether or not it is possible to construct a regular material

source which can block the central singularity and join the external repulsive region in

a proper way. Then we are interested to construct a body with the following properties:

1. inside the body there are no singularities;

2. outside the body there is the RN field (7.34)-(7.35), corresponding to the case

Q2 > kM2;

3. the radius of the body is less than Q2/Mc2, so between the surface of the body

and the sphere r = Q2/Mc2 arises the repulsive region;

4. such stationary state of the body is stable with respect to collapse or expansion.

In this paper we propose a new model for such body in the form of spherically symmetric

thin membrane with positive tension. We assert that there exists a physically acceptable

range of parameters for which all the above four conditions (1)-(4) can be satisfied. We

illustrate this conclusion by the especially transparent case of a Nambu-Goto membrane

with equation of state ε = τ .

Then the existence of everywhere-regular material sources possessing RN “antigravity”

properties in the vicinity of their surfaces attribute to this phenomenon and to the RN

naked singularity solution more sensible physical status.

It is necessary to mention that at least two exact solutions of Einstein-Maxwell equations

representing a compact continuous spherically symmetric distribution of charged matter

under the tension producing the gravitationally repulsive forces inside the matter as

well as in some region outside of it already exist in the literature. These are solutions

constructed in [26] and [85]. A more detailed study of these two results can be found

in [42]. An interesting possibility to have a gravitationally repulsive core of electrically

neutral but viscous matter has been communicated in [77]. It is worth to remark that

the first (to our knowledge) mentioning of the gravitational repulsive force due to the

presence of electric field was made already in 1937 in [48] in connection to the nonlinear

model of electrodynamics of Born-Infield type. One of the first paper where a repulsive

phenomenon in the framework of the conventional Einstein-Maxwell theory has been

mentioned is [32]. The general investigation of the different aspects of this phenomenon

apart from the already mentioned references can be found also in the more detailed works

[51, 52, 38, 78]. Some part of these papers is dedicated to a possibility of construction a

classical model for electron. This is doubtful enterprise, however, because the intrinsic

structure of electron is a matter out of classical physics. Nonetheless the mathematical

118

results obtained are useful and can be applied to the physically sensible situations, e.g.

for construction the models of macroscopical objects.

Equation of motion of a membrane with empty space

inside

The equation of motion for the most general case of a thin charged spherically symmetric

fluid shell with tangential pressure moving in the RN field have been derived 38 years

ago by J.E. Chase [24]. The corresponding dynamics for a charged elastic membrane

with tension follows from his equation simply by the change of the sign of the pressure.

We derived, however, the membrane’s dynamics again using a different approach.

Chase used the geometrical method which have been applied to the description of sin-

gular surfaces in relativistic gravity in [53] and have been elaborated in [32, 56] for some

special cases of charged shells. An essential development of the Israel approach in appli-

cation to the cosmological domain walls can be found in the series of works of V.Berezin,

V.Kuzmin and I. Tkachev, see [16] and references therein. Our treatment follows the

method more habitual for physicists which have been used in [7], where the motion of

a neutral fluid shell in a Schwarzschild field was derived by the direct integration of the

Einstein equations with appropriate δ-shaped source. Now we generalized this approach

for the charged membrane and charged central source.

Of course, the membrane’s equation of motion that we obtained coincides with that of

Chase. Nonetheless the different approach to the same problem often has a methodolog-

ical value and gives new details. We hope that our case makes no exception, then for

an interested reader we put the main steps of our derivation in Appendix A (where we

considered a general case with central source).

In this section we study only the particular solution in which there is no central body,

that is inside the membrane we have flat space-time.

Although the basic formulas of this section follow from the Appendix A under restriction

Min = Qin = 0 the exposition we give here is more or less self-consistent. Only the defi-

nitions of 4-dimensional membrane’s energy density and tension need some clarification

which can be found in Appendix A.

For the thin spherically symmetric membrane with empty space inside and with radius

which depends on time the metrics inside, outside and on membrane are:

− (ds2)in = −Γ2(t)c2dt2 + dr2 + r2(dθ2 + sin2 θdφ2) (7.37)

− (ds2)out = −f(r)c2dt2 + f−1(r)dr2 + r2(dθ2 + sin2 θdφ2) (7.38)

− (ds2)on = −c2dη2 + r20(η)(dθ2 + sin2 θdφ2) (7.39)

119

In the interval (7.113) η is the proper time of the membrane. The factor Γ2 in (7.110) is

necessary to ensure the continuity of the global time coordinate t through the membrane.

The metric coefficient f(r) in the region outside the membrane is given by Eq.(7.35).

Matching conditions for the intervals (7.110)-(7.113) through the membrane’s surface

are:

[(ds2)in]r=r0(η) = [(ds2)out]r=r0(η) = (ds2)on (7.40)

If the equation of motion of the membrane r = r0(η) is known, then from these conditions

the connection t(η) between global and proper times and factor Γ(t) follow easily:

Γ(t) =f(r0)

√1 + c−2(r0,η)2

√f(r0) + c−2(r0,η)2

(7.41)

d t

dη=

√f(r0) + c−2(r0,η)2

f(r0)(7.42)

The differential equation for the function r0(η) follows from Einstein-Maxwell equations

with energy-momentum tensor and charge current concentrated on the surface of the

membrane. It is:

Mc2 = µ(r0)c2

√1 +

(d r0

c dη

)2

+Q2

2r0

− kµ2(r0)

2r0

(7.43)

Here µ(r0) > 0 is the effective rest mass of the membrane in the radially comoving

frame. This quantity includes the membrane’s rest mass as well as all kinds of interaction

mass-energies between membrane’s constituents, that is those intrinsic energies which are

responsible for the tension. The constants Q and M are the total charge of the membrane

and total relativistic mass of the system. These are the same constants which appeared

earlier in Eq.(7.35). The membrane’s energy density ε and tension τ are (see Appendix

A for a further clarification):

ε = ε0(r0)δ[r − r0(η)] τ = τ0(r0)δ[r − r0(η)] (7.44)

where

ε0 =µ(r0)c

2

8πr20

[1√

1 + c−2(r0,η)2+

f(r0)√f(r0) + c−2(r0,η)2

](7.45)

τ0(r0) =dµ(r0)

dr0

r0ε0(r0)

2µ(r0)(7.46)

The electromagnetic potentials have the form Ar = Aθ = Aφ = 0, At = At(t, r) and for

the electric field strength ∂At/∂r the solution is

∂At

∂r=

Qr2 for r > r0(η)

0 for r < r0(η)

(7.47)

120

The formulas (7.110)-(7.126) give the complete solution of the problem for the case of

empty space inside the membrane.

Finally we would like to stress the following important point. As follows from discussion

in Appendix A, the signs of the square roots√

1 + c−2(r0,η)2 and√

f(r0) + c−2(r0,η)2

coincide with the signs of the time component u0 of the 4-velocity of the membrane

evaluated from inside and outside of the membrane respectively. The component u0 is

a continuous quantity by definition and can not change the sign when passing through

the membrane’s surface. Besides, for macroscopical objects we are interested in in this

paper u0 should be positive. Consequently the both aforementioned square roots should

be positive. From another side it is easy to show that equation (7.119) can be written

also in the following equivalent form

Mc2 = µc2

√f(r0) +

(d r0

c dη

)2

+Q2

2r0

+kµ2

2r0

(7.48)

Then from this expression and from (7.119) follows that both square roots will be positive

if and only if

Mc2 − Q2

2r0

− kµ2

2r0

> 0 (7.49)

This is unavoidable constraint which must be adopted as additional condition for any

physically realizable solution of the equation of motion (7.119) in classical macroscopical

realm.

Nambu-Goto membrane with “antigravity” effect

To proceed further we must specify the function µ(r0), which is equivalent to specifying

an equation of state, as can be seen from (7.125).

Let us analyze the membrane with equation of state ε = τ . This model can be inter-

preted as “bare” Nambu-Goto charged membrane [55, 49], or as Zeldovich-Kobzarev-

Okun charged domain wall [90]. It follows from (7.125) that for such type of membrane

we have:

µ = γr20 (7.50)

where γ is an arbitrary constant. In this and next section we consider only the case of

positive constants γ and M :

γ > 0 , M > 0 . (7.51)

121

The sign of Q is of no matter since the charge appear everywhere in square. Now we

write the equation of motion (7.119) in the following form:

4

(d r0

c dη

)2

−(

kγr0

c2+

2M

γr20

− Q2

c2γr30

)2

= −4 . (7.52)

Formally this can be considered as the equation of motion of a non-relativistic particle

with the “mass” equal to 8 moving in the potential U(r0),

U(r0) = −(

kγr0

c2+

2M

γr20

− Q2

c2γr30

)2

(7.53)

and under that condition that particle is forced to live on the “total energy” level equal

to minus four.

For the existence of the stable stationary state we are interested in, the following condi-

tions should hold:

1. The gravitational field in the exterior region should correspond to the super-

extreme RN metric:

Q2 > kM2. (7.54)

2. The potential U(r0) should have a local minimum at some value r0 = Rmin. The

form (7.53) of U(r0) permit this if and only if

kγ2Q6 < (Mc2)4. (7.55)

Under this restriction the potential U(r0) has three extrema, two maxima at points

r0 = R(1)max and r0 = R

(2)max and a minimum which is located between them: R

(1)max <

Rmin < R(2)max. We show the shape of the potential U(r0) for this case in Fig.1.

The equation U(r0) = 0 has only one real root and this is also the first local

maximum R(1)max. The minimum and the second maximum are coming as two other

roots of the equation dUdr0

= 0.

The equation for Rmin is:

kγ2R4min − 4Mc2Rmin + 3Q2 = 0 . (7.56)

This fourth order equation has only two real solutions and Rmin is the smaller one.

3. For the stationary position of the membrane at the minimum of the potential we

must ensure the relation U(Rmin) = −4 which is:

kγ

c2Rmin +

2M

γR−2

min −Q2

c2γR−3

min = 2 (7.57)

(the minus two in the r.h.s. of (7.131) would be incompatible with Eq.(7.56) under

condition (7.127)).

122

Figure 7.1: The membrane’s motion can be described as the motion of a non-relativisticpoint particle in the potential U(r0).

4. To have repulsive region it is necessary for the membrane’s radius Rmin to be

less than the minimum of the gravitational potential f(r), that is less than the

quantity Q2/Mc2. In this case outside of the membrane surface in the region

Rmin < r < Q2/Mc2 we have the repulsive effect. Then we demand:

Rmin <Q2

Mc2. (7.58)

5. Also the additional constraint (7.49) should be satisfied. This means that for our

stationary solution we have to satisfy the inequality:

Mc2 − Q2

2Rmin

− kγ2

2R3

min > 0 . (7.59)

6. We have also another condition: that the electric field nearby the membrane

should be not too large, otherwise the stability of the model would be destroyed

123

by the strong macroscopical consequences of quantum effects, e.g. by the inten-

sive electron-positron pair creation. This condition (which was suggested by J.A.

Wheeler long time ago [8]) is:

Q

R2min

<< Ecr , Ecr =m2

ec3

ee~, (7.60)

where me and ee are the electron’s mass and charge). Ecr is the well known critical

electric field above which the intensive process of pair creation starts.

To satisfy these six conditions we have to find a physically acceptable domain in the

space of the four parameters M , Q, γ and Rmin. The point is that such domain indeed

exists and it is wide enough. If we introduce the dimensionless radius of the stationary

membrane x as

kγ

c2Rmin = x , (7.61)

then one can check directly that the first five of the above formulated conditions will be

satisfied under the following three constraints:

x < 1 (7.62)

M =c4

k2γ(3x2 − 2x3) (7.63)

Q2 =c8

k3γ2(4x3 − 3x4) (7.64)

The last two of these relations are just the equations (7.56) and (7.131) but written in

the form resolved with respect to M and Q2.

The formulas (7.138)-(7.141) shows that for the first five conditions it is convenient to

take x < 1 and γ as independent parameters, and then to calculate the mass and charge

necessary to obtain the model we need.

As for the last constraint (7.136) it gives some restriction also for parameter γ:

kγ2 <<x

4− 3xE2

cr . (7.65)

The energy density ε for the stationary state at r0 = Rmin, expressed in terms of para-

meters x and γ, is:

ε =γc2

8π(1 +

√x2 − 2x + 1)δ(r −Rmin). (7.66)

124

Appendix

For the spherically symmetric case the metric6 is:

−(ds)2in = g00c

2dt2 + g11dr2 + r2(dθ2 + sin2 θdφ2) (7.67)

where g00 and g11 depend only on t, r and the standard notation for the coordinates is:

(x0, x1, x2, x3) = (ct, r, θ, φ) . (7.68)

The Electromagnetic tensor Fik has the form:

Fik = Ak,i − Ai,k (7.69)

and the Einstein-Maxwell equations are:

Rki −

1

2Rδk

i =8πk

c4T k

i (7.70)

(F ik);k =4π

cρui (7.71)

The energy-momentum tensor for a spherical charged membrane with energy density ε

and tangential tension τ is:

T ki = ε uiu

k − (δ2i δ

k2 + δ3

i δk3)τ +

1

4π(FilF

kl − 1

4δki FlmF lm) (7.72)

and for the membrane’s 4-velocity ui we have:

u0 = u0(t, r), u1 = u1(t, r), u2 = u3 = 0 , uiui = −1 . (7.73)

The main step is to define the 4-invariant charge and energy densities ρ and ε. After

that, the tension τ follows automatically from the Einstein-Maxwell equations and from

the equation of state. To construct ρ and ε we apply the Landau-Lifschitz procedure

[57].

The charge dq in the 3-volume element dV =√

g11g22g33 dx1dx2dx3 is a 4-invariant quan-

tity by definition (although dV is not a 4-scalar). The three-dimensional charge density

ρ(3) can be introduced by the relation dq = ρ(3)dV . Consequently, for the spherically

symmetric membrane case it is:

ρ(3) =Qδ(r − r0)

4πr2√

g11

, (7.74)

6We use the notations in which the interval is written as −ds2 = gikdxidxk and metric signature is(−, +, +,+), i.e. the time-time component g00 is negative. The norm of a time-like vector is negative.The Roman indices take values 0, 1, 2, 3. The Newtonian constant is denoted by k. The simple partialderivatives we designated by a comma, while covariant derivatives by semicolon.

125

where Q is the electric charge of the membrane and r0 is the membrane’s radius. Indeed

it is easy to check that Q =∫

ρ(3)dV as it should be7.

Since ρ(3)dV is a 4-scalar the quantities ρ(3)dV dxi represent a 4-vector. With the use of

the previous formula we obtain:

cρ(3)dV dxi =cQδ(r − r0)

4πr2u0√−g00g11

ui√−g d4x , (7.75)

where g is the 4-metric’s determinant. The last formula shows that the factor in front

of ui√−gd4x is a 4-scalar. This scalar is nothing else but the 4-invariant charge density

ρ which appeared in the Maxwell equation (7.71):

ρ =cQδ[r − r0(t)]

4πr2u0√−g00g11

. (7.76)

For the electric current jk we have jk = ρuk.

The 4-scalar energy density ε which figure in the energy-momentum 4-tensor (7.72) can

be constructed exactly in the same way if we observe that the rest energy of the matter

in a 3-volume element dV (i.e. the sum of the all kinds of the internal energies of this

element in the reference system in which this element is at rest) is a 4-invariant quantity

by definition. Then we can introduce the 3-dimensional rest energy density (the direct

analogue of the previous charge density ρ(3)) which under integration over 3-volume gives

the total rest energy µc2 of the membrane. Then µc2 is the sum of the all kinds of internal

energies of the membrane in the radially comoving system in which membrane is at rest.

In this way we obtain:

ε =µc2δ[r − r0(t)]

4πr2u0√−g00g11

. (7.77)

Clearly the effective rest mass µ of the membrane in the presence of a tension depends

on the membrane radius r0(t).

In the case of spherical symmetry the electromagnetic potentials Ai can be taken in the

form:

A0 = A0(t, r), A1 = A2 = A3 = 0, (7.78)

which gives only one nonvanishing component for the electromagnetic tensor Fik, namely

F10 (and its antisymmetric partner F01):

F10 = A0,1 . (7.79)

Now, we enter with definitions (7.67)-(7.69) and (7.72)-(7.79) into the Einstein-Maxwell

equations (7.70)-(7.71) to calculate the solution. These calculations need special care

7The δ-function in curved metric (7.67) is defined by the usual relation∫

δ(r − r0)dr = 1. Suchδ-function has dimension cm−1.

126

since we are dealing with distributions in application to the non-linear theory. In general

this is not a trivial task (see e.g. [84, 39, 82]), however, for particular case of spherical

symmetry everything is tractable and can be done easily thanks to the specially simple

structure of the field equations. The resulting solution contains four arbitrary constants

of integration Min, Qin and Mout, Qout which have an obvious interpretation as mass and

charge of a central RN source and the total mass and charge of the whole system (the

central body together with the membrane) respectively. The membrane’s charge Q is

simply the difference of Qout and Qin:

Q = Qout −Qin . (7.80)

To represent the solution in compact form we use the proper time η of the membrane,

denoting the membrane’s equation of motion as r = r0(η), and introducing the following

notations:

φin(r) = 1− 2k Min

c2r+

kQ2in

c4r2

φout(r) = 1− 2k Mout

c2r+

kQ2out

c4r2

(7.81)

Sin(η) =√

φin(r0) + c−2(r0,η)2

Sout(η) =√

φout(r0) + c−2(r0,η)2

(7.82)

We consider the global time t in (7.67) as continuous quantity when passing through the

membrane. Then the intervals inside, outside and on the membrane are:

− (ds2)in = −Γ2(t)φin(r)c2dt2 +dr2

φin(r)+ r2(dθ2 + sin2 θdφ2) (7.83)

− (ds2)out = −φout(r)c2dt2 +

dr2

φout(r)+ r2(dθ2 + sin2 θdφ2) (7.84)

− (ds2)on = −c2dη2 + r20(η)(dθ2 + sin2 θdφ2) (7.85)

The matching conditions for these intervals through the membrane are:

[(ds2)in]r=r0(η) = [(ds2)out]r=r0(η) = (ds2)on (7.86)

Using the relations (7.86), the factor Γ(t) in (7.83) and the connection t(η) between

global and proper times can be expressed through the membrane’s radius r0(η):

dt

dη=

Sout

φout(r0), (7.87)

Γ(t) =φout(r0)Sin

φin(r0)Sout

. (7.88)

127

Namely the continuity conditions (7.86) and continuous character of the time variable t

are responsible for the appearance of the term Γ2(t) in g00 in Eq.(7.87). Since this term

depends only on time, it can be easily removed by passing to the internal time variable

tin by the transformation

Γdt = dtin , (7.89)

which can be found with the help of (7.87) after the function r0(η) became known. In

terms of the variables (tin, r) also the internal metric (7.83) takes the standard RN form.

As it was already mentioned, the membrane’s effective rest mass µ which appeared in the

energy density (7.77) depends on the membrane radius. The concrete form of the function

µ(r0) is not known in advance and its specification is equivalent to the specification of the

equation of state. For an arbitrary µ(r0) the Einstein-Maxwell equations (7.70)-(7.71)

give the following equation of motion for the membrane:

Moutc2 −Minc

2 =1

2(Sin + Sout)µc2 +

QQin

r0

+Q2

2r0

, (7.90)

together with the condition that both square roots Sin and Sout defined by (7.82), should

have the same sign. The provenance of this condition is due to the fact that the signs of

Sin and Sout are nothing else but the signs of the time-component of u0 of the membrane’s

4-velocity when it is seen from the inside (r → r0 − 0) and outside (r → r0 + 0) of the

membrane surface respectively. In our approach (with continuous coordinates t, r) we

can consider the 4-velocity ui as a field continuous through the surface of the membrane.

We can define ui everywhere in space-time simply by smooth parallel transport from the

membrane’s surface, no matter that the membrane is concentrated only at the points

r = r0. This concentration is ensured not by ui but due to the δ-functions in the densities

ρ and ε. Since u0 can not change sign passing through the membrane, Sin and Sout should

have the same sign.

Of course, we need to know the fields u0 and u1 only on the membrane, and there they

are:

u0 = t,η , u1 = c−1r0,η . (7.91)

It is easy to check that the matching conditions (7.87) and (7.88) are nothing else but

the demand that the normalization constraint uiui = −1 should hold independently from

which side we approach the surface of the membrane.

It is worth to be remarked that the Einstein-Maxwell equations also demand for the

trajectory r0(η) the second order (in time) differential equation of motion. However,

this last one represents simply the result of the differentiation in time of the first order

128

equation (7.90). Then this second-order equation we can forget safely.

The resulting expressions for the energy density and tension are:

ε =µc2

8πr20

[φin(r0)

Sin

+φout(r0)

Sout

]δ[r − r0(η)] (7.92)

τ =r0

2µ

dµ

dr0

ε . (7.93)

The electric field F10 outside the membrane is:

F10 =Qout

r2, r > r0 . (7.94)

Inside the membrane we have:

F10 =Qin

r2

dtindt

, r < r0 , (7.95)

where the factor dtindt

depends only on time and can be calculated from the relations

(7.88) and (7.89). The origin of this factor is due to the fact that we use the time t as

continuous global time including the region inside the membrane. If we describe the in-

ternal metric in terms of internal time tin the field strength F10 would be simply Qin/r2.

The formulas (7.81)-(7.85), (7.87), (7.88) and (7.90)-(7.95) provide the complete solution

of the problem. It is worth explaining briefly the main steps of our integration procedure

that we applied to the Einstein-Maxwell equations.

As in any spherically symmetric problem it is convenient to use, instead of the full original

Einstein equations (7.70), only its (00), (1

1) and (10) components, and the hydrodynamical

equations T ki;k = 0. All the remaining components of equations (7.70) after that will be

satisfied identically either due to the Bianchi identities or due to the symmetry of the

problem. Then the solution for g11 together with the basic eq.(7.90) follows from (00) and

(10) components of Einstein equations (7.70), and after that the solution for g00 follows

from the difference of the (00) and (1

1) components of (7.70). The solution for the electric

field F10 is the result of the Maxwell equations (7.71). The hydrodynamical equations

T ki;k = 0 give only two relations. The first one simply express the tension τ in terms of

other quantities and this is the formula (7.93). The second one results in the already

mentioned second order differential equation for r0(η) which represents the differentia-

tion in time of the first order equation (7.90). Then this second order equation is of no

importance.

We remark also that the procedure described above need a caution because the sym-

bolic function are involved. Nevertheless everything going well under the following three

standard operation rules with such functions:

1. ddx

θ(x) = δ(x) ,

129

2. F (x)δ(x) = 12[F (−0) + F (+0)]δ(x) ,

3. ddx

θ2(x) = 2θ(x)δ(x) = δ(x) .

(To call the third rule as the standard one is a little exaggeration; however it works

well and final results indeed coincide with those obtained in literature by different ap-

proaches). Originally we obtained the solution in global form using the step function

θ(x) and only after that we represented the results separately in the regions r > r0 and

r < r0. However, since θ(x) is defined also at the point x = 0 [θ(0) = 1/2], we found by

the way the values for the metric and electric field also at the points of the membrane’s

surface. Such global form is:

1

g11

= 1− 2kMin

c2r− 2k(Mout −Min)

c2rθ[r − r0(η)] +

k

c4r2Qin + Qθ[r − r0(η)]2 (7.96)

1√−g00g11

=1

Γ+

(1− 1

Γ

)θ[r − r0(η)] (7.97)

F10 =

√−g00g11

r2Qin + Qθ[r − r0(η)] , (7.98)

to which should be added the equation (7.90). This equation arise as self-consistency

condition for the (00) and (1

0) components of Einstein equations, which can be verified by

the direct substitution into these components of the above global expressions together

with eqs. (7.91)-(7.93).

Finally it should be mentioned that the membrane’s equation of motion (7.90) can be

written in the following two equivalent forms:

µc2Sin = Moutc2 −Minc

2 − QinQ

r0

− Q2

2r0

+kµ2

2r0

(7.99)

µc2Sout = Moutc2 −Minc

2 − QinQ

r0

− Q2

2r0

− kµ2

2r0

. (7.100)

Each of these two equations is equivalent to (7.90) which can be checked easily by simple

algebraic manipulations. For practical calculations we can use only one of these equa-

tions, however, in addition it is necessary to ensure the same sign for both quantities

Sin and Sout. (For a membrane with empty space inside they both should be positive).

More convenient is relation (7.99) which we write as

Moutc2 =Minc

2 + µc2√

φin(r0) + c−2(r0,η)2

+QinQ

r0

+Q2

2r0

− kµ2

2r0

(7.101)

130

This is the equation obtained by Chase [24] with the aid of a different derivation proce-

dure which makes use of Gauss-Codazzi equations (see Israel [53]).

Eqn.(7.101) is interesting because in spite of the fact that µ depends on time (or on r0)

this equation looks like an usual integral of motion, that is as if µ was a constant. Rela-

tion (7.101) expresses the conservation of the total energy Moutc2 of the system which is

the sum of the five familiar constituents: 1) the rest energy of the central body, 2) the

kinetic energy of the membrane together with its gravitational potential energy in the

gravitational field of the central body, 3)the electric interaction energy between mem-

brane and central source, 4) the positive electric self-interaction energy of the membrane,

and 5) the negative gravitational self-interaction energy of the membrane.

Conclusions

1. We showed that exists a possibility to have a spherically charged membrane in sta-

ble stationary state producing RN repulsive gravitational force outside its surface and

having flat space inside. To construct such model one should take a pair of constants

0 < x < 1 and γ > 0 satisfying the inequality (7.142) and calculate from (7.137) and

(7.140)-(7.141) the membrane’s radius Rmin, total mass M and charge Q.

2. The equation of motion (7.119) can be used also for the description of the oscil-

lation of the membrane in the potential well ABC (see fig.1) above the equilibrium point

C. If we slightly increase the total membrane’s energy Mc2 then the potential U(r0)

around its minimum (i.e. the point C and its vicinity) will be shifted slightly down but

he level ”minus four” in Eq.(7.53) on which the system lives will remain at the same

position. Then the membrane will oscillate between the new shifted walls AC and CB.

3. It is easy to see that in the general dynamical state the membrane can live only

inside the potential well ABC. All regions outside ABC are forbidden. In the region to

the right from the point R(2)max and above the potential U(r0) any location of the mem-

brane is impossible due to the fact that inequality (7.49) is violated there.

This means that a membrane of considered type in principle can not have the radius (no

matter in which state) greater than R(2)max. In turn for R

(2)max it is easy to obtain from the

potential (20) the upper limit R(2)max < c2

kγ

(4k2γM

c4

)1/3

.

The same violation of the inequality (7.49) take place in the domain between R(1)max and

R(2)max and above the segment AB. The motion in the region to the left from the point

R(1)max and above the curve U(r0) is forbidden again due to the same violation of the con-

dition (7.49). This means that a membrane of considered type in principle can not have

131

the radius less than R(1)max. In particular there is no way for a membrane with positive

effective rest mass µ to collapse to the point r0 = 0 leaving outside the field correspond-

ing to the RN naked singularity solution. This conclusion is in agreement with the main

result of the paper [7].

4. Although we claimed that the stationary state of a membrane constructed is sta-

ble this stability should be understood in a very restrict sense, that is as stability in the

framework of the dynamics described by the equation (7.119). We do not know what

will happen to our membrane after the whole set of arbitrary perturbations will be given.

5. In general the arbitrary perturbations will change also the equation of state. We

investigated a membrane with equation of state ε = τ . However this case can be con-

sidered only as “bare” Nambu-Goto membrane, by other words as a toy model. In the

papers [55, 49, 76, 23, 87, 50] it was shown that arbitrary perturbations essentially renor-

malize the form of the equation of state of the strings and membranes. Moreover for

the membranes [49] (differently from the strings) the fixed points of the renormalization

group for the transverse and longitudinal perturbations does not coincide, which means

that for the general “wiggly” membrane there is no equation of state of the type ε = ε(τ)

at all.

6. We also would like to stress that for appearance of repulsive force the presence of

electric field is of no principal necessity. For example the repulsive gravitational forces

arise also in neutral viscous fluid [77] and in the course of interaction between electrically

neutral topological gravitational solitons [11].

7. From the conditions (21)-(26) also follows that in addition to the inequality (25)

the radius Rmin of the shell in the stable stationary state cannot be less than Q2

2Mc2. A

simple analysis shows that there is no way for Rmin to be arbitrarily small keeping some

finite non-zero value for M and Q.

II. Intersection of self-gravitatingcharged shells in aReissner-Nordstrom field

Introduction

The mathematical model that we analyze in this paper describes the dynamic evolu-

tion of two spherical shells of charged matter which freely move outside the field of a

central Reissner-Nordstrom (RN) source. Microscopically these shells are assumed to

be composed by charged particles which move on elliptical orbits with a collective vari-

able radius. The angular motion, distributed uniformly and isotropically on the shell

surfaces, is mathematically described by a tangential-pressure term in the energy mo-

mentum tensor of the Einstein equations. The definition of the shell implies that all the

particles have the same following three ratios: energy/mass, angular momentum/mass,

and charge/mass. Indeed, since the equations of motion for any singled-out particle ”a”

are

dtads

=1

−mac2gtt(ra)(Ea + eaA0(ra)) (7.102)

(dra

ds

)2

=1

m2ac

4(Ea + eaA0(ra))

2

(1

−gtt(ra)grr(ra)

)−

(l2a

m2ac

2

1

r2+ 1

)1

grr(ra)(7.103)

(dθa

ds

)2

=l2a

m2ac

2

1

r4− k2

a

m2ac

2

1

r4 sin2 θa

(7.104)

dϕa

ds=

ka

mac

1

r2 sin2 θa

(7.105)

(gtt and grr are the components of a spherical symmetric metric and A0 is the electric

potential; ka and la are arbitrary constants), it is easy to see that the radial motion for

all particles is the same if

Ea

ma

= const,ea

ma

= const,|la|ma

= const, ∀a, (7.106)

132

133

where each const. does not depend on the index a. Therefore, if at the beginning the

particles are on the same radius ra = R0, then the shell will evolve “coherently”, i.e. all

particles will evolve with the same radius.

Now the problem we are interested in is to find the exchange of energy between the

two shells after the intersection. Indeed the motion of the shells before and after the

intersection can be easily deduced from the equation of motion for just one shell, which

equation has been found many years ago by Chase [24] with a geometrical method first

used by Israel [53].

What we achieve in the present paper is the determination of the constant parameters

after the intersection knowing just the parameters before the intersection. Actually the

unknown parameter is only one, m21, which is the Schwarzschild mass parameter mea-

sured by an observer between the shells after the intersection. This parameter is strictly

related to the energy transfer which takes place in the crossing, and it is found imposing

a proper continuity condition on the shells velocities.

In the model we assume that the emission of electromagnetic waves is negligible, and

that there are no other interactions between the two shells apart the gravitational and

electrostatic ones. In particular the shells, during the intersection, are assumed to be

”transparent” each other (i.e. no scattering processes).

The paper is divided as follows: in Sec.2 we preliminarily discuss the one-shell case;

in Sec.3, which is the central part of this article, we find the unknown parameter m21;

then, Secs.4-7 are devoted to some applications: post-Newtonian approximation, zero

effective masses case (i.e. ultra-relativistic case), test-shell case, and finally the ejection

mechanism.

In this paper we deal only with the mathematical aspects of the problem; some astro-

physical applications of charged shells in the field of a RN black hole have been considered

in [25].

A gravitating charged shell with tangential pressure

The motion of a thin charged dust-shell with a central RN singularity was firstly studied

by De La Cruz and Israel [32], while the case with tangential pressure was achieved by

Chase [24] in 1970. All these authors used the extrinsic curvature tensor and the Gauss-

Codazzi equations. However we followed a different way, indeed the same solution can be

found also by using δ and θ distributions and then by direct integration of the Einstein-

Maxwell equations (see [7] and the appendix in [13]). This method has the advantage

of a clearer physical interpretation, and it is also straightforward in the calculations;

134

however in the following we will give only the main passages.

Let there be a central body of mass min and charge ein and let a spherical massive

charged shell with charge e move outside this body. It is clear in advance that the field

internal to the shell will be RN, while externally we will have again a RN metric but

with different mass and charge parameters mout and eout = ein +e. Using the coordinates

x0 = ct and r, which are continuous when passing through the shell, we can write the

intervals inside, outside, and on the shell as

− (ds)2in = −eT (t)fin(r)c2dt2 + f−1

in (r)dr2 + r2dΩ2 (7.107)

− (ds)2out = −fout(r)c

2dt2 + f−1out(r)dr2 + r2dΩ2 (7.108)

− (ds)2on = −c2dτ 2 + r0(τ)2dΩ2 (7.109)

where we denoted

dΩ2 = dθ2 + sin2 θdφ2

and

fin = 1− 2Gmin

c2r+

Ge2in

c4r2, fout = 1− 2

Gmout

c2r+

G(ein + e)2

c4r2. (7.110)

In the interval (7.109), τ is the proper time of the shell. The “dilaton” factor eT (t) in

(7.107) is required to ensure the continuity of the time coordinate t through the shell. If

the equation of motion for the shell is

r = R0(t), (7.111)

then joining the angular part of the three intervals (7.107)-(7.109), one has

r0(τ) = R0[t(τ)], (7.112)

where the function t(τ) describes the relationship between the global time and the proper

time of the shell. Joining the radial-time parts of the intervals (7.107)-(7.108) on the

shell requires that the following relations hold:

fin(r0)

(d t

dτ

)2

eT (t) − f−1in (r0)

(d r0

cdτ

)2

= 1 , (7.113)

fout(r0)

(d t

dτ

)2

− f−1out(r0)

(d r0

cdτ

)2

= 1 . (7.114)

If the equation of motion for the shell, i.e. the function r0(τ), is known, then the function

t(τ) follows from (7.114) and consequently T (t) can be deduced by (7.113). Thus the

problem consist only in determining r0(τ), which can be done by direct integration of

the Einstein-Maxwell equations

Rki − 1

2Rgk

i = 8πGc4

T ki

(√−gF ik),k =

√−g 4πcρui

(7.115)

135

with the energy-momentum tensor given by:

T ki = ε uiu

k + (δ2i δ

k2 + δ3

i δk3)p + T (el) k

i (7.116)

T (el) ki =

1

4π(FilF

kl − 1

4δki FlmF lm) . (7.117)

Here on we employ the following notations:

−ds2 = gikdxidxk, gik has signature (−, +, +, +)

xk = (ct, r, θ, ϕ) i, j, k... = 0, 1, 2, 3

p ≡ p(R0) = pθ = pϕ =tangential pressure (pr = 0)

Fik = Ak,i − Ai,k

The above equations are to be solved for the metric

−ds2 = g00(t, r)c2dt2 + g11(t, r)dr2 + r2dΩ2, (7.118)

and for the potential

A0 = A0(t, r), A1 = A2 = A3 = 0. (7.119)

As follows from the Landau-Lifshitz approach [57] (see [7]) the energy distribution of the

shell is

ε =M(t)c2δ[r −R0(t)]

4πr2u0√−g00g11

, (7.120)

while its charge density is

ρ =c eδ[r −R0(t)]

4πr2u0√−g00g11

, (7.121)

where δ is the standard δ-function. In the absence of tangential pressure p, the quantity

M in Eqn.(7.120) would be a constant, but in presence of pressure, Mc2 includes the rest

energy along with the energy (in the radially comoving frame) of the tangential motions

of the particles that produce this pressure.

It can be checked that the Einstein part of (7.115) actually lead to the solution (7.107)-

(7.109) with, in addition, the “joint condition”

√fin(r0) +

(d r0

cdτ

)2

+

√fout(r0) +

(d r0

cdτ

)2

= 2(mout −min)

µ(τ)− e2 + 2eein

µ(τ)c2r0

, (7.122)

where we denoted

µ(τ) = M [t(τ)], (7.123)

136

while

mout −min = E/c2 (7.124)

is a constant which can be interpreted as the total amount of energy of the shell. Then,

from the Maxwell side of (7.115) the only non-vanishing component of the electric field

is

F01 = −√−g00g11

r2ein + eθ[r −R0(t)] (7.125)

(θ(x) is the standard step function). Finally, the equations T ki;k = 0 can be reduced to

the only one relation:

p = −dM

dt

c2δ[r −R0(t)]

8πru1√−g00g11

(7.126)

We will not treat here the steady case (i.e. r0 = const) which should be treated sepa-

rately; thus in the following we will assume always r0 6= const. .

The joint condition (7.122) can be written in several different forms: two of them, which

will be useful in the following, are

√fin(r0) +

(d r0

cdτ

)2

=(mout −min)

µ(τ)+

Gµ2(τ)− e2 − 2eein

2µ(τ)c2r0

(7.127)

and √fout(r0) +

(d r0

cdτ

)2

=(mout −min)

µ(τ)− Gµ2(τ) + e2 + 2eein

2µ(τ)c2r0

. (7.128)

As in [7], all the radicals encountered here are taken positive, since for astrophysical

considerations only these cases are meaningful. To proceed further, we must specify the

equation of state, i.e. the function µ(τ). Here we consider a particle-made shell, therefore

the sum of kinetic and rest energy of all the particles is

Mc2 =∑

a

(mac

2

√1 +

p2a

m2ac

2

), (7.129)

where pa is the tangential momentum of each particle (the electric interaction between

the particles is already taken into account by the self-energy term of, e.g., (7.127), thus

one has not to include it in M too). From the definition of the shell (see Introduction)

it follows:p2

a

m2a

=l2a

m2aR

20

=const

R20

, (7.130)

the square root in (7.129) does not depend on the index a; then defining

∑a

mac2 = mc2,

∑a

|la| = L,

137

formula (7.129) can be re-written (remembering definition (7.123) too) as

µ(τ) =

√m2 +

L2

c2r20(τ)

. (7.131)

Thus, now, one can determine the function r0(τ) from equation (7.122) (or from one of the

equivalent forms (7.127)-(7.128)) if the initial radius of the shell and the six arbitrary

constants min, mout, m, ein, e and L are specified. Accordingly with (7.120), (7.126),

(7.123) and (7.131), the equation of state that relates the shell energy density ε to the

tangential pressure p is

p =ε

2

L2

m2c2R20

(1 +

L2

m2c2R20

)−1

(7.132)

as in the uncharged case, i.e. the presence of the charges do not modify the relation

between energy density and pressure (indeed the presence of the charge is hidden in the

equation of motion). Note that when the shell expands to infinity (R0 →∞) the angular

momentum becomes irrelevant and the equation of state tends to the dust case p << ε.

The shells intersection

Let us now consider the case of two shells which move in the field of a central charged

mass. The generalization from the previous (single-shell) case is straightforward if the

shells do not intersect: indeed the outer shell do not affect the motion of the inner one,

while the inner one appears from outside just as a RN metric. Therefore the principal

aim of this section is to consider the intersection eventuality and to predict the motion

of the two shells after the crossing, having specified the initial conditions before the

crossing. After the intersection one has a new unknown constant that has to be found

by imposing opportune joining conditions as now we are going to explain (the analysis

follows step by step the [7]’s one). Let us previously analyze the space-time in the (t, r)

coordinates (which are continuous through the shells). We define the point O ≡ (t∗, r∗)

as the intersection point; then the space-time is divided in four regions (see Fig.1):

COB (r > R1, r > R2),

COA (R1 < r < R2),

AOD (r < R1, r < R2),

BOD (R2 < r < R1).

(7.133)

138

Figure 7.2: The four region in which it is divided the spacetime; the two lines representthe trajectories of shell-1 and shell-2.

Correspondingly to these regions we have the metric in form (7.114) but with different

coefficients g00 and g11:

g(COB)00 = −fout(r) , g

(COB)11 = f−1

out(r) (7.134)

g(COA)00 = −eT1(t)f12(r) , g

(COA)11 = f−1

12 (r) (7.135)

g(AOD)00 = −eT0(t)fin(r) , g

(AOD)11 = f−1

in (r) (7.136)

g(BOD)00 = −eT2(t)f21(r) , g

(BOD)11 = f−1

21 (r) (7.137)

The dilaton factor T (t) allows to cover all the space-time with only one t-coordinate;

here, fin and fout are the same as those in (7.110) while f12 and f21 are given by similar

expressions:

f12 = 1− 2Gm12

c2r+

G(ein + e1)2

c4r2(7.138)

f21 = 1− 2Gm21

c2r+

G(ein + e2)2

c4r2(7.139)

As we said, the parameters min, m12, mout, ein, e1, e2 are assumed to be specified at the

beginning, while m21 is the actual unknown constant which has yet to be determined

from the joining conditions on (t∗, r∗).

139

Before the intersection

Let us write the equation of motion for the two shells before the intersection (shell-1

inner and shell-2 outer). This can be made easily adapting the (7.128) and (7.127) to

the present case:

√f12(r1) +

(d r1

cdτ1

)2

=(m12 −min)

M1

− GM21 + e2

1 + 2eine1

2M1c2r1

(7.140)

for shell 1, while for shell 2

√f12(r2) +

(d r2

cdτ2

)2

=(m12 −min)

M2

+GM2

2 − e22 − 2(ein + e1)e2

2M2c2r2

(7.141)

with

M1 =

√m2

1 +L2

1

c2r21

, M2 =

√m2

2 +L2

2

c2r22

. (7.142)

Here, τ1 and τ2 are the proper times of the first and second shells respectively, while

r1(τ1) = R1[t(τ1)] and r2(τ2) = R2[t(τ2)]. Now we have to impose the joining conditions

for the intervals on both the shells. For the first shell (on curve AO) one has:

eT1(t)f12(r1)

(d t

dτ1

)2

− f−112 (r1)

(d r1

cdτ1

)2

= 1 (7.143)

eT0(t)fin(r1)

(d t

dτ1

)2

− f−1in (r1)

(d r1

cdτ1

)2

= 1; (7.144)

while for the second shell:

fout(r2)

(d t

dτ2

)2

− f−1out(r2)

(d r2

cdτ2

)2

= 1 (7.145)

eT1(t)f12(r2)

(d t

dτ2

)2

− f−112 (r2)

(d r2

cdτ2

)2

= 1. (7.146)

If all free parameters and initial data to Eqs.(7.140)-(7.142) were specified and if the

functions r1(τ1) and r2(τ2) were derived, then their substitution in (7.143)-(7.146) gives

the functions τ1(t), τ2(t) and T1(t), T0(t), which is enough for determining the motion of

the shells before the intersection. Therefore the intersection point (t∗, r∗) can be found

by solving the system r∗ = r1(τ1(t∗))

r∗ = r2(τ2(t∗)) ,(7.147)

which we assume that has a solution.

140

After the intersection

The equation of motion for the shells after the intersection time t∗ can be constructed

in the same way again by turning to Eqns.(7.127) and (7.128), and introducing the new

parameter m21 which characterize the “Schwarschild mass” seen by an observer in the

region BOD. We use Eq.(7.127) for (now outer) shell 1 and Eq.(7.128) for (now inner)

shell 2:√

f21(r1) +

(d r1

cdτ1

)2

=(mout −m21)

M1

+GM2

1 − e21 − 2e1(ein + e2)

2M1c2r1

, (7.148)

√f21(r2) +

(d r2

cdτ2

)2

=(m21 −min)

M2

− GM22 + e2

2 + 2e2ein

2M2c2r2

. (7.149)

Naturally, M1(r1) and M2(r2) are given by the same expression of (7.142) but now they

have to be calculated on r1(τ1) and r2(τ2) after the intersection.

Joining the intervals on the first shell (on curve OB) yields

fout(r1)

(d t

dτ1

)2

− f−1out(r1)

(d r1

cdτ1

)2

= 1 (7.150)

eT2(t)f21(r1)

(d t

dτ1

)2

− f−121 (r1)

(d r1

cdτ1

)2

= 1. (7.151)

Then, joining the second shell (on curve OB) we obtain:

eT2(t)f21(r2)

(d t

dτ2

)2

− f−121 (r2)

(d r2

cdτ2

)2

= 1 (7.152)

eT0(t)fin(r2)

(d t

dτ2

)2

− f−1in (r2)

(d r2

cdτ2

)2

= 1. (7.153)

Since the initial data to Eqs.(7.148) and (7.149) have already been specified (from the

previous evolution), then the evolution of the shells after the intersection would be de-

termined from Eqs.(7.148)-(7.153) if parameter m21 were known. Thus we need an

additional physical condition from which we could determine m21.

This condition follows from the fact that the Christoffel symbols (i.e. the accelerations)

of the shells have only finite discontinuities (finite jumps), therefore the relative velocity

of the shells must remain continuous through the crossing point.

In the presence of two shells, we can construct one more invariant than in the single

shell case (where only uiui = −1 was possible): the scalar product between the two

4-velocities of the shells. We can also avoid to apply the parallel transport if we evaluate

141

the 4-velocities on the intersection point (t∗, r∗). The continuity condition can be found

imposing that the scalar product has to have the same value when evaluated in both the

two limits t → t−∗ and t → t+∗ .

Determination of Q

Let us start determining the quantity

Q ≡ g(COA)00 u0

AOu0CO + g

(COA)11 u1

AOu1COt=t∗,r=r1=r2=r∗ , (7.154)

which is the scalar product of the two 4-velocities evaluated in the intersection point from

the region AOC (along the curves AO and CO). Written explicitly, the unit tangent

vector to trajectory AO is

uiAO = (u0

AO, u1AO, u2

AO, u3AO)

=

(d t

dτ1

,d r1

cdτ1

, 0, 0

)

t≤t∗

, (7.155)

while for the trajectory CO we have

uiCO = (u0

CO, u1CO, u2

CO, u3CO)

=

(d t

dτ2

,d r2

cdτ2

, 0, 0

)

t≤t∗

. (7.156)

The fact that these are actually unit vectors follows from the joining equations (7.143)

and (7.146).

The components of the vector (7.155) can be easily expressed from Eqs.(7.140) and

(7.143) as

(d t

dτ1

)

t≤t∗

=e−T1(t)/2

M1(r1)f12(r1)

(m12 −min − GM2

1 (r1) + e21 + 2e1ein

2c2r1

)(7.157)

(d r1

cdτ1

)

t≤t∗

=

=δ1

M1(r1)f12(r1)

√(m12 −min − GM2

1 (r1) + e21 + 2e1ein

2c2r1

)2

−M21 (r1)f12(r1)

(7.158)

where

δ1 = sgn

(d r1

cdτ1

)

t≤t∗

. (7.159)

142

Analogously, for the components of vector (7.156), we obtain the following expressions

from Eqs.(7.141) and (7.146):

(d t

dτ2

)

t≤t∗

=e−T1(t)/2

M2(r2)f12(r2)

(mout −m12 +

GM22 (r2)− e2

2 − 2e2(ein + e1)

2c2r2

)(7.160)

(d r2

cdτ2

)

t≤t∗

=δ2

M2(r2)f12(r2)·

·√(

mout −m12 +GM2

2 (r2)− e22 − 2e2(ein + e1)

2c2r2

)2

−M22 (r2)f12(r2) (7.161)

δ2 = sgn

(d r2

cdτ2

)

t≤t∗

. (7.162)

Thus, from the preceding results, we obtain:

Q = −1M1M2f12

·

·(

m12 −min − GM21 +e2

1+2e1ein

2c2r∗

)(mout −m12 +

GM22−e2

2−2e2(ein+e1)

2c2r∗

)+

−δ1δ2

√(m12 −min − GM2

1 +e21+2e1ein

2c2r∗

)2

−M21 f12

√(mout −m12 +

GM22−e2

2−2e2(ein+e1)

2c2r∗

)2

−M22 f12

;

(7.163)

here and in the following we omit the coordinate dependence of fa, Ma etc., implicitly

assuming that they have to be evaluated on (t∗, r∗) where not differently indicated.

Determination of Q′

It is possible to apply the same procedure to the region BOD (i.e. after the intersection

time), finding the quantity

Q′ ≡ g(BOD)00 u0

OBu0OD + g

(BOD)11 u1

OBu1ODt=t∗,r=r1=r2=r∗ . (7.164)

Now the unit tangent vectors to trajectories OB and OD are8:

uiOB = (u0

OB, u1OB, u2

OB, u3OB)

=

(d t

dτ1

,d r1

cdτ1

, 0, 0

)

t≥t∗

, (7.165)

8Obviously, when we say t ≥ t∗, we tacitly assume before a (possible) second intersection.

143

and

uiOD = (u0

OD, u1OD, u2

OD, u3OD)

=

(d t

dτ2

,d r2

cdτ2

, 0, 0

)

t≥t∗

; (7.166)

from the joining conditions (7.151) and (7.152) it is possible to see that these are actually

unit vectors. The components of uiOB can be deduced from Eqs.(7.148) and (7.151),

while the components of uiOD from Eqs.(7.149) and (7.152). Then, using the metric in

the region BOD, it is possible to calculate the scalar product

Q′ = −1M1M2f21

·

·(

mout −m21 +GM2

1−e21−2e1(ein+e2)

2c2r∗

)(m21 −min − GM2

2 +e22+2e2ein

2c2r∗

)+

−δ′1δ′2

√(mout −m21 +

GM21−e2

1−2e1(ein+e2)

2c2r∗

)2

−M21 f21

√(m21 −min − GM2

2 +e22+2e2ein

2c2r∗

)2

−M22 f21

,

(7.167)

where δ′1 and δ′2 have been defined as in (7.159) and (7.162), but for t ≥ t∗. We introduced

these symbols only for generality, but actually we are interested only in the case with9

δ′1 = δ1, δ′2 = δ2 . (7.168)

The necessary continuity requirement is thus

Q = Q′, (7.169)

then, since r∗ is assumed to be known, this equation allows to find m21.

Physical meaning of Q and Q′

Using standard definition for the shell velocities before the intersection one has

(v1

c

)2

=g

(COA)11 (r1)

−g(COA)00 (r1)

(d r1

cdt

)2

(7.170)

(v2

c

)2

=g

(COA)11 (r2)

−g(COA)00 (r2)

(d r2

cdt

)2

, (7.171)

9This is the only possible case if one excludes v1(t∗) = v2(t∗) = 0, because there are non disconti-nuities in the velocities.

144

and similarly for the velocities after the intersection,

(v′1c

)2

=g

(BOD)11 (r1)

−g(BOD)00 (r1)

(d r1

cdt

)2

(7.172)

(v′2c

)2

=g

(BOD)11 (r2)

−g(BOD)00 (r2)

(d r2

cdt

)2

. (7.173)

Then it is easy to obtain from the definitions (7.154) and (7.164), that10

Q =

v1v2/c

2 − 1√1− v2

1/c2√

1− v22/c

2

t=t∗,r1=r2=r∗

(7.174)

and

Q′ =

v′1v

′2/c

2 − 1√1− (v′1)2/c2

√1− (v′2)2/c2

t=t∗,r1=r2=r∗

. (7.175)

Determination of P and P′

First of all it is convenient to introduce new symbols to simplify the expressions of Q

and Q’. With

q1 ≡ −GM21 + e2

1 + 2e1ein

2c2r∗

q2 ≡ GM22 − e2

2 − 2e2(ein + e1)

2c2r∗,

and

q′1 ≡GM2

1 − e21 − 2e1(ein + e2)

2c2r∗

q′2 ≡ −GM22 + e2

2 + 2e2ein

2c2r∗,

then Q and Q′ can be re-written as

Q = −1M1M2f12

·

·

(m12 −min + q1) (mout −m12 + q2) +

−δ1δ2

√(m12 −min + q1)

2 −M21 f12

√(mout −m12 + q2)

2 −M22 f12

(7.176)

10It is also worth noting that√

Q2 − 1/Q = −|v1/c−v2/c|/(1−v1v2/c2), which is the relative velocitydefinition of two “particles” in relativistic mechanics.

145

andQ′ = −1

M1M2f21·

·

(mout −m21 + q′1) (m21 −min + q′2) +

−δ′1δ′2

√(mout −m21 + q′1)

2 −M21 f12

√(m21 −min + q′2)

2 −M22 f12

,

(7.177)

Now, in principle is possible to find m21 by squaring and solving Q = Q′ (which is a

quartic equation). However the procedure is cumbersome and moreover it is not possible

with Eq.(7.169) alone to determine the sign of the roots. Fortunately, as in the non-

charged case, it is possible to follow another easier way. Indeed, it is possible to introduce

two other invariants, say P and P ′, similar to Q and Q′, which are constructed using the

scalar products of the 4-velocities of the two shell, but now taking the limit to (t∗, r∗)

from the AOD and COB regions respectively. More explicitly, we define

P ≡ g(AOD)00 u0

AOu0OD + g

(AOD)11 u1

AOu1ODt=t∗,r=r1=r2=r∗ , (7.178)

and

P ′ ≡ g(COB)00 u0

COu0OB + g

(COB)11 u1

COu1OBt=t∗,r=r1=r2=r∗ . (7.179)

Then, the same continuity requirement of Eq.(7.169) implies that it must hold also that

Q = P , P = P ′ . (7.180)

Following the same method used to find Q and Q′, after some calculations, one arrives

toP = −1

M1M2fin·

·

(m12 −min + p1) (m21 −min + p2) +

−δ1δ′2

√(m12 −min + p1)

2 −M21 fin

√(m21 −min + p2)

2 −M22 fin

(7.181)

146

andP ′ = −1

M1M2fin·

·

(mout −m21 + p′1) (mout −m12 + p′2) +

−δ′1δ2

√(mout −m21 + p′1)

2 −M21 fout

√(mout −m12 + p′2)

2 −M22 fout

,

(7.182)

where we have denoted

p1 ≡ GM21 − e2

1 − 2e1ein

2c2r∗

p2 ≡ GM22 − e2

2 − 2e2ein

2c2r∗,

and

p′1 ≡ −GM21 + e2

1 + 2e1(ein + e2)

2c2r∗

p′2 ≡ −GM22 + e2

2 + 2e2(ein + e1)

2c2r∗.

Determination of m21; the energy transfer

Thus the complete set of continuity conditions at the point of intersection can be written

as

Q = Q′, Q = P, Q = P ′. (7.183)

It turns out that this three quartic equations for the unknown parameter m21 have only

one common root. It is possible to find the solution using hyperbolic functions. The

final result is remarkably simple:

m21 = min + mout −m12 − e1e2

c2r∗− GM1M2

c2r∗Q , (7.184)

or equivalently, in terms of f21:

f21 = fin + fout − f12 + 2G2M1M2

c4r2∗Q . (7.185)

It can be easily seen from Eqn.(7.184) that the charge ein of the central singularity does

not affect the result (but it affects the equation of the motion of the shells and thus

Q). Formula (7.184) solves the problem of determining the mass parameter m21 from

the quantities specified at the evolutionary stage before intersection. It is then possible

147

to determine the energy transfer between the shells. Indeed the energy of shell 1 and 2

before the intersection are, respectively

E1 = (m12 −min)c2 , E2 = (mout −m12)c2 , (7.186)

while, after the intersection

E ′1 = (mout −m21)c

2 , E ′2 = (m21 −min)c2 . (7.187)

The conservation of total energy is automatically ensured by the above formulas, indeed

E1 + E2 = E ′1 + E ′

2 . (7.188)

Then it is natural to define the exchange energy as

∆E = E ′2 − E2 = −(E ′

1 − E1) . (7.189)

Then, from Eqn.(7.184) and the above definitions, it follows that

∆E = −e1e2

r∗− GM1M2

r∗Q . (7.190)

It is also useful (especially for the Newtonian approximation) to use Eqn.(7.174) and

re-express ∆E as:

∆E = −e1e2

r∗− GM1M2

r∗

v1v2/c

2 − 1√1− v2

1/c2√

1− v22/c

2

r=r∗

. (7.191)

Post-Newtonian approximation

For slow velocities of the shells it is interesting to consider the Post-Newtonian limit of

Eqn.(7.191):

∆E =Gm1m2 − e1e2

r∗+

+1

2c2

Gm1m2

r∗[v1(r∗)− v2(r∗)]2 +

Gm2L21

m1r3∗+

Gm1L22

m2r3∗

+ o

(1

c4

). (7.192)

It is worth noting that only the zeroth order in 1/c2 changes with respect to the un-

charged case (because of the Coulomb term −e1e2/r∗), while all the other orders remain

unchanged, being of kinetic origin; m1 and m2 are the rest masses of the shells, indeed

we have used for the masses M1 and M2 the definitions (7.142).

148

It can be also useful to re-express all the quantities in a Newtonian language and consider

only the zeroth order in 1/c2, e.g. we can expand the energy as

E = mc2 + E + o

(1

c2

), (7.193)

where m and E do not depend on c. Therefore, similarly, we can define at the first order

in 1/c2

m12 −min = m1 +E1

c2, mout −m12 = m2 +

E2

c2, (7.194)

mout −m21 = m1 +E ′1c2

, m21 −min = m1 +E ′2c2

. (7.195)

Then it follows also that the energy conservation law takes the form

E1 + E2 = E ′1 + E ′2 , (7.196)

and Eqn.(7.189) becomes

E ′1 = E1 −∆E , E ′2 = E2 + ∆E , (7.197)

where ∆E = (∆E)c→∞. Thus from the above formulas and definitions it is clear that

∆E =Gm1m2 − e1e2

r∗. (7.198)

Pressureless shells with zero effective masses (L1 =

L2 = 0 and M1 = M2 = 0)

It is interesting also to consider the case in which the motion of the particles of the

shells is only radial (i.e. L1 = L2 = 0) and the rest masses are negligible with respect to

the kinetic energies and to the charges —indeed this is the case for two shells composed

by (ultra)rela-tivistic electrons and positrons. In this case the effective masses can be

replaced by

M1 = M2 = λ , (7.199)

where λ is a parameter arbitrary small.

From Eqn.(7.190), with Q expressed by formula (7.163), it is easy to find that the energy

transfer in this case is

∆E = −e1e2

r∗+

c4r∗2Gf12

(fin − f12)(f12 − fout) + o(λ2) , (7.200)

149

having assumed that the shells have opposite-directed velocities, i.e.

δ1δ2 = −1 . (7.201)

Otherwise, if the shells goes in the same direction, i.e.

δ1δ2 = 1 , (7.202)

then Eqn.(7.190) becomes simply

∆E = −e1e2

r∗+ o(λ2) ; (7.203)

obviously the previous formulas make sense only if r∗ exists. We want to underline the

presence of the term o(λ2), because, strictly speaking, a charge cannot have zero rest

mass, therefore we are in the case of just small effective masses. As expected, in the

case of vanishing charges (e1 = e2 = 0), Eqn.(7.203) gives zero at λ = 0 because this is

the case of two photon-shells which go in the same direction and therefore cannot never

intersect.

The intersection of a test shell with a gravitating one

One-shell case

Let us consider firstly the case of a test shell on the RN field. This limit has the only

aim to show that the shell’s equation of motion (7.127) actually reduce to the simple

test-particle case; the limit can be obtained by putting

m → λm , e → λm , L → λL , (mout −min)c2 → λE (7.204)

with λ → 0. Then, considering also (7.131), we find that Eqn.(7.127) becomes

E = µc2

√fin(r0) +

(d r0

cdτ

)2

+eein

r0

− λGµ2 − e2

2r0

, (7.205)

now, putting λ = 0 the self-energy term is killed; then re-writing Eqn.(7.205) using the

more familiar Schwarzschild time t and Eqn.(7.112),

E = c2

√m2 +

L2

c2R20(t)

√f 3

in(R0)

f 2in(R0)−

(dR0

cdt

)2 +eein

R0

+ o(λ) , (7.206)

it is easy to recognize that Eqn.(7.206) coincides with the first integral of motion of a

test-charge particle on the Reinssner-Nordstrom background, where E is the conserved

energy of the particle, m the rest mass, e the charge and L the angular momentum.

150

Two-shell case, with one test-shell

Now we can deal with the more interesting two-shell case, in which shell-2 is considered

”test”. To gain this limit we have to put

m2 → λm2 , e2 → λm2 , L2 → λL2 , (7.207)

(mout −m12)c2 → λE2 , (m21 −min)c2 → λE ′

2 .

Then, using Eqn.(7.184) with Q given by formula (7.163), one obtains

∆E = − e1e2

r∗+ 1

r∗f12·

·(

E1 − GM21 +e2

1+2e1ein

2c2r∗

)(E2 − e2(ein+e1)

c2r∗+ λ

GM22−e2

2

2c2r∗

)+

−δ1δ2

√(E1 − GM2

1 +e21+2e1ein

2c2r∗

)2

−M21 f12

√(E2 − e2(ein+e1)

c2r∗+ λ

GM22−e2

2

2c2r∗

)2

−M22 f12

.

(7.208)

Thus, only the self-energy terms of shell-2 are killed by λ = 0.

Now, it is worth noting the following fact: shell-1 does not have any discontinuity when

it intersect the shell-2 (this is natural because shell-2 is “test” and does not affect the

metric), on the other hand shell-2 undergoes a discontinuity in the metric when it cross

shell-1 and consequently it has an actual discontinuity in the velocity. It is easy to

calculate this gap; indeed using the definition (7.171) of velocity v2 [with the time d tdτ2

given by the joint condition (7.146)], with metric coefficient (7.135), and with the help

the first integral of motion (7.141), one finds

v22(r2) = 1− fout(r2)

(E2

M2(r2)− e2(e1 + ein)2

M2(r2)r2

)−2

+ o(λ) , t ≤ t∗ , (7.209)

where we have used f12 = fout + o(λ); in the same way, using (7.173), (7.152), (7.137),

and (7.149), the velocity v′2 (after the intersection) is

[v′2(r2)]2 = 1− fin(r2)

(E ′

2

M2(r2)− e2(e1 + ein)2

M2(r2)r2

)−2

+ o(λ) , t ≥ t∗ , (7.210)

where E ′2 can be expressed in function of E2 with the help of (7.208). From the previous

formulas it is clear that in general

v′2(r∗)− v2(r∗) 6= 0 . (7.211)

151

Shell ejection

The exchange in energy of the shells during the intersection makes possible that one

initially bounded shell can acquire enough energy to escape to infinity.

The shell ejection mechanism can take place also in the Newtonian regime. In this case,

from Eqs.(7.197)-(7.198) it results that

E ′1 = E1 − Gm1m2 − e1e2

r′∗, E ′2 = E2 +

Gm1m2 − e1e2

r′∗, (7.212)

and then, after the first intersection

E ′′1 = E ′1 + Gm1m2−e1e2

r′′∗= E1 − (Gm1m2 − e1e2)

(1r′∗− 1

r′′∗

)

E ′′2 = E ′2 − Gm1m2−e1e2

r′′∗= E2 + (Gm1m2 − e1e2)

(1r′∗− 1

r′′∗

),

(7.213)

where we have denoted the radius of the first and second intersection with r′∗ and r′′∗respectively. In the following we will consider only the case

Gm1m2 − e1e2 > 0 , (7.214)

this is e.g. the case in which the two shells have opposite charges. Thus, also in the case

E1, E2 < 0, if

r′′∗ > r′∗ , (7.215)

and if the initial condition were in such a way that r′∗ is enough small and r′′∗ not too

much close to r′∗, then it is possible to have E ′′2 > 0, i.e. the ejection of the second shell.

Let us now assume that r′′∗ > r′∗, and consider a “semi-relativistic” case in which at the

first intersection we use the full relativistic formulas11,

E ′1 = E1 − M1(r′∗)M2(r′∗)

r′∗(−Q) + e1e2

r′∗

E ′2 = E2 + M1(r′∗)M2(r′∗)

r′∗(−Q)− e1e2

r′∗,

(7.216)

while at the second intersection we use the Newtonian approximation,

E ′′1 = E ′

1 + Gm1m2−e1e2

r′′∗

= E1 −[

M1(r′∗)M2(r′∗)(−Q)−e1e2

r′∗− Gm1m2−e1e2

r′′∗

]

E ′′2 = E ′

2 − Gm1m2−e1e2

r′′∗

= E2 +[

M1(r′∗)M2(r′∗)(−Q)−e1e2

r′∗− Gm1m2−e1e2

r′′∗

].

(7.217)

11Remember that −Q = 1 + o(1/c2).

152

This approximation is always justified if the radius of the second intersection r′′∗ is enough

large. Now, it is remarkable that whatever the value of r′∗ is, the first term in the square

brackets in Eqn.(7.217) satisfies the inequality

M1(r′∗)M2(r

′∗)(−Q)− e1e2

r′∗>

Gm1m2 − e1e2

r′∗. (7.218)

Comparing the expressions (7.217), (7.218) and (7.213) it is possible to see that in the

relativistic regime the shell ejection possibility is even greater than in the Newtonian

case. Furthermore, it is worth noting that the presence of the charge do not change

qualitatively the pure gravitational analysis, but just magnifies the ejection effect.

Gm1m2 − e1e2 < 0 case

Let us consider also briefly the case in which the shells are equal-signed charged and

the repulsion overcome the gravity attraction, i.e. Gm1m2 − e1e2 < 0. In this case the

ejection can happen only after an odd number of intersections.

E.g. after three intersections, from the previous formulas we have, in the Newtonian

approximation:

E ′′′1 = E1 − (Gm1m2 − e1e2)

(1

r′∗− 1

r′′∗+

1

r′′′∗

). (7.219)

Obviously this formula has a meaning only if

1

r′∗<

1

r′∗− 1

r′′∗+

1

r′′′∗, (7.220)

otherwise the ejection happens at the first intersection (and then there would not be

other crossings, and no r′′∗ , r′′′∗ ), or never more; if Eqn.(7.220) is true, then it means that

the barycenter of the two shells is falling into the center singularity.

Conclusions

We have found the energy exchange between two charged crossing shells (formula

(7.191)). Then we have studied special cases of physical interest in which the formu-

las simplify: the non relativistic case, the massless shells, the test shell, and finally the

ejection mechanism in a semi-Newtonian regime: we found that the ejection mechanism

is more efficient in the charged case than in the neutral one if the charges have opposite

sign (because the energy transfer is larger due to the Coulomb interaction).

Personal works

Publications

1. M. Pizzi and A. Paolino, Equilibrium configurations in the double Reissner-Nordstromexact solution, International Journal of Modern Physics A (IJMPA) 23-8 (2008), 1222.

2. A. Paolino and M. Pizzi, Electric force lines of the double Reissner-Nordstrom exactsolution, International Journal of Modern Physics D (IJMPD) 17-8 (2008), 1159.

3. V.A. Belinski, M. Pizzi and A. Paolino, A membrane model of the Reissner-Nordstromsingularity with repulsive gravity, accepted for pub. by International Journal of ModernPhysics (IJMPD).

4. V.A. Belinski, M. Pizzi and A. Paolino, Charged membrane as a source for repulsivegravity, proceeding of III Stuckelberg Workshop (2008).

5. M. Pizzi and A. Paolino, Intersections of self-gravitating charged shells in a Reissner-Nordstrom field, submitted to International Journal of Modern Physics D (IJMPD).

Talks and posters

1. A. Paolino and M. Pizzi, Electric force lines of the double Reissner-Nordstrom solution,at the II Stueckelberg Workshop, (2007); delivered by the coauthor M. Pizzi.

2. M. Pizzi and A. Paolino, Electric force lines in the Alekseev-Belinski solution, Poster-Section at the APS april meeting, St. Louis, Missouri (USA), (2008); presented by thecoauthor M. Pizzi.

3. V.A. Belinski, M. Pizzi and A. Paolino, Charged membrane and repulsive gravity, at theIII Stuckelberg Workshop (2008); delivered by the coauthor V.A. Belinski.

4. M. Pizzi and A. Paolino, Elettric Force Lines and Stability in the Alekseev-Belinski so-lution, at the III Stuckelberg Workshop (2008); delivered by the coauthor M. Pizzi.

153

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