^r-iMf^r- ic/85/100

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

STATIONARY AXIALLY SYMMETRIC EXTERIOR SOLUTIONS

IN THE FIVE DIMENSIONAL REPRESENTATION

OF THE BRANS-DICKE-JORDAN THEORY OF GRAVITATION

William Bruckman

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1985 MIRAMARE-TRIESTE

IC/85/1OO

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CEJJTRE FOR THEORETICAL PHYSICS

STATIONARY AXIALLY SYMMETRIC EXTERIOR SOLUTIONS

IN THE FIVE DIMENSIONAL REPRESEIWATION

OF THE BRASS-DICKE-JORDAN THEORY OF GRAVITATION •

William Bruckman **

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

July

* To be submitted, for pub] i cation.

** Permanent address: Depart amenta de Fisica, Colegio Universitario deHumacao, P.O. Box 10136, Huinacao, Puerto Rico 00661,

ABSTRACT I , INTRODUCTION

The Inverse Scattering Method of Belinsky and Zakharov i s used to

investigate axially symmetric stationary vacuum soliton solutions in the

five-dimensional representation of the Brans-Dicke-Jordan theory of

gravitation, where the scalar field of the theory is an element of a five-

dimensional metric. The resulting equations for the space-time metric are

similar to those of solitons in general re la t iv i ty , while the scalar field

generated is the product of a simple function of the co-ordinates ana an

already known scalar field solution. Kerr-like solutions are considered

that reduce, in the absence of rotation, to the five-dimensional form of

the well-known Weyl-Levi Civita axially symmetric s tat ic vacuum solution.

Hence these metrics can he interpreted as the exterior gravitational

fields of rotating non-spherical configurations in the Brans-Dicke-Jordan

theory. With a proper choice of the parameters in the solutions, the

deviation from spherical symmetry will he due only to the angular momentum

of the system, as i t is with the Kerr metric in general re la t iv i ty . An

explicit exact /solution i s given.

The Brans-Dicke-Jordan (BDJ) scalar tensor theory was f i rs t in-

vestigated in connection with Dirao's large number hypothesis, by Jordan ,and, in connection with Maoh's principle, by Brans and Dicke 2),3) The

theory was originally expressed in a representation in vhich the local

measurable value of the gravitational "constant", G, is a function of a1) 2)

scalar field * . It can also be put in a form which is Einstein's

genera], relativity with the scalar field of the theory acting as an additional

external non-gravitational field . Another way of representing the BDJ

theory is with a five-dimensional field equation ) > where the metric

is independent of the fifth co-ordinate, and the elements g , , where

V = (0,1,2,3} is a space-time index, vanish. In vacuum, this is just a

special case of the Klein-Jordan-Thiry theory, where the g , components

are identified with an "electromagnetic" vector potential A . The Klein-

Jordan-Thiry theory is in turn a generalisation of the original Kaluza-Klein

unified theory of gravity ana eleetromagnetism, In which g, , = constant.

In order to study the physical implications of a theory we must find

solutions of its field equations. Fortunately, it has been possible to find

exact solutions of relativistic gravitational theories, like general

relativity and BDJ, when physically reasonable symmetries have been assumed.

For instance, many astrophysical systems of interest are approximately

stationary and axially symmetric, and can be very well described with metrics7)

having these symmetries. The Kerr solution ' is perhaps the moat important

known eact solution to the vacuum Einstein field equations. It represents

the exterior gravitational field of stationary axially symmetric rotating

systems, which In the absence of rotation reduces to the Schwarzschild

spherically symmetric solution. That is, the Kerr solution does not possess

deformations from spherical symmetry, other than those caused by the angular

velocity.

There are also axially symmetric stationary solutions in the BDJp V

theory representing the exterior metric of rotating configurations , butthey can only become spherically symmetric If the scalar field 1B a constant.On the other hand, there i s a simple extension of the Cosgrove-Tomimatsu-Sato

o\solutions of general relat ivi ty yl to the BDJ theory that allows the con-struction of solutions in which the system i s spherically symmetric In theabsence of rotation. However, the Cosgrove-Tomimatsu-Sato solutions involve,in general, transcendental func^ionc.

-1- -2-

In this paper the inverse scattering meth of Belinsky and Zakharov

' is used to construct axially symmetric stationary vacuum

solutions to the BDJ field equations. In particular, I shall consider

solutions that reduce to the well-known spherically symmetric static EDJ

vacuum metric in the absence of rotation, and therefore can be thought of as

generalizations of the Kerr solution in the presence of a scalar field. This

solution could be a starting point for the study of external gravitational'

fields of rotating stars in the BDJ theory. The analysis is carried out in

a five-dimensional represenLation of the BDJ theory, where the application of

the BZ technique is generalized in a straightforward way. Furthermore, this

representation provides a basis for future possible applications of the BS

formalism to the problem of finding classical solutions of, Kaluza-Klein type,

field theories of more than four dimensions. We point out, however, that the

BZ formalism could be applied directly to a particular four-dimensional

representation of the BDJ theory (the Einstein-scalar theory), since for an

axially symmetric stationary metric in vacuum or with an electromagnetic field

source, the scalar field decouples from the second-order field equations for

the space-time metric, and therefore these equations are identical to those

of general relativity 12'<5J_

In fact, Belinsky studied exact solutions of the Einstein-scalar

theory that describe the evolution of gravitational soliton waves against the

background of Friedmann cosmologieal models. A very interesting feature of

this work is that the energy-momentum tensor of the scalar field is now

representing the matter field of a perfect fluid with an equation of state

pressure = energy density.

In the next section, we will describe the Belinsky and Zakharov method

in i t s essentials, vithout restricting the number of dimensions to four. In

Sec.I l l , the BDJ theory in i t s five-dimensional representation (BDJ-5) Is

discussed in detail . Next, in Se~.IV we apply the BZ technique to the BDJ-5

theory, while in Sec.V the diagonal form of the solutions Is i-onnidered. ri'!;e

BZ technique relies on the knowledge of a seed solution, g , from which a

more general one, g is generated, hence in Sec.VI we •.-• • •§:M.ir.-reu conditions

for the BZ solution, g . , to become equal to g . Sees.VII &nd VIII deal

with specific applications of the formulas developed in previous sections

to the two cases where the background, g , represents a flat space-time, and

a Weyl-Levi Civita static metric. The solution .-unstruoted with the Weyl-

Levi Civita metric may be thought of as a particular generalization of the

Kerr solution to the BDJ theory, since i t is possible to obtain the limit

where, in the absence of rotation, g , approaches the static spherically

symmetric solution of the theory. The paper ends with an explicit exact

solution of this type .-3-

II. THE INVERSE SCATTERING METHOD OF BELINSKY AND ZAKHAROV

This method allows the generation of large classes of new solutions

from old, when the metric tensor depends only on two variables. Belinsky

and Zakharov * developed and employed the method to obtain exterior

solutions in general relativity. In particular, they have obtained the

Kerr solution starting from a flat space-time metric background. The

technique can also be used in relativistic theories of more than fourI1*)

dimensions. This was done by Belinsky and Ruffini , in the framework

of the five-dimensional Jordan-Thiry-Kaluza-Klein theory, to generate

stationary axially symmetric solutions starting from a constant metric.

In what follows we will briefly describe the Belinsky and Zakharov

technique. Their notation will be followed in its essentials. For

the sake of generality, we will not, yet, restrict the number of dimensions

considered, since the generalization only introduces trivial modifications

to the BZ formulas. Hence, let us consider a m+2 dimensional metric that

depends only on two co-ordinates p and Z. The line element can be written

in the Lewis form: ''

dS2 - R

f(p, 2) [dp2 + dZ2] + g (p, Z) dXa dXb;ab

i, b = 1 - m . (2.1)

Furthermore, the source-free Einstein equations in m+2 dimensions

admit the following co-ordinate condition-

(2.2)

det g = det g « -p' ,ab

(2.3)

With the metric (2.1) and the condition (2.3) the field equation (2.2) takes

the form

CP8»

. z]. P

0,

-h-

(In f) = -1/p + l/4p Trace [u - V ] , (2.5)

(In f ) , = l /2p Trace UV, (2.6)

where ve used the conventional notation ( )3X

Eq. (2.M is the compatibility condition for the following system of

linear equations:

D.Oi = — - (2.7)

V- I-'(2.8)

where \ is a complex variable. Moreover, when X= O, Eqs.(2.7) and (2.8)

are just

P*, (2.9)

*PU,

(2.10)

vhich imply that

X - 0) - g (2.11)

The solitonic solution i(i is given as a function of a particular

solution ty , corresponding to a given background metric g , in the following

way:

where

(2.12)

(2.13)

- 5 -

.> -. Jc (2.H)

C. : arbitrary constants.

i(il (k) is the inverse of i|i_ evaluated at

- \ ~ Z ± (2.15)

W : arbitrary constants and T are the elements of the inverse of the

following matrix:

(2.16)

The metric g is then given hy

0) - g -(2.17)

It also follows, from Eqs.(2.5) and (2.6) that

f - cn

Pn n

nnk«l

r , (2.13)

where f is the solution corresponding to g , and C is a constant.

Even though g satisfies Eq.(2.i*), it is not a solution of the field

equations (£.2), since now det g is not equal to det g = -p , but instead

equal to

det g - (-l)n n (P/uk)2 dec g <

(2.19)

where n is taken to be an even number in order to preserve the signature

of g. However, the new metric

• *(2.20)

-6-

is still a solution of Eq.. (2.1+) and, furthermore, satisfies the condition

det gph - V . ( 2 > 2 1 )

The funct ion f i s a l so modified by the t ransformat ion ( 2 . 2 0 ) .

The new func t ion , f , i s

phf ,

where

Const. Pn1 (l

k=!

n - l

\ >

n

k=i

2 n

nk>

(2.22)

(2.23)

The power 1/m is the only explicit reference to the dimensionality

of the geometry. The Belinsky and Zalcharov formulas of Eef.ll are recovered

simply by putting m = 2.

III. THE feDJ THEORY IN FIVE DIMEHSIOHS

The EDJ field equations are conventionally given in the form 2)

WH »

SITE,, ,Qt

L_1 ™ ™tn8ir tf

(3.1)

( 3 .2 )

where g is the metric for the four-dimensional space-time, and u> isuv

the coupling constant for the scalar field + . The comas and the semi-

colon denote par t ia l and covariant derivatives, respectively. The energy-

momentum tensor, T , includes a l l non-gravitational fields and matter,

and sat isf ies the conservation law

TWV = 0 . (3.3)

- 7 -

Units are chosen such that the gravitational constant and the speed of light

are equal to one.

The theory has also "been expressed in the more general form :

G rr—y v ij)

(2u+3-3cQ |Y T

2a $2

- 2 - 8TT a _• C2oo+3)

(3.M

(3.5)

where the new metric g and the scalar field

g by the conformal transformation:

are related to if and

a = constant, (3.6)

8uv (3.7)

The Einstein tensor 3 and the covariant derivatives are built with the

metric g , and the new energy momentum tensor T is given in terms of

T by(IV

T (3.8)

The above transformation, Eqs.(3.6) and (3.7), have been interpreted

by Dicke as a space-time dependent change in the units of measurement.

Note also that in vacuum, and if m = -3/2 the field equations (3.^) and

(3.5) are conformally invariant, a result pointed out by Anderson

In the limit a -+ 0 we obtain the field equations

G = 8it T + ( U 1 + , 2 ) f a ; 6; - % g *>'•% ] ,UV v-v ^ L V v y^ ;a J '

(3.9)

(3.10)

where now the bars mean that the tensors are built with the metric

(3.11)

This representation, Eqs.(3.9) and (3.10), is just Einstein's equations with

a scalar field, $, as external source; -the so-called Einstein-scalar theory.

"The case when

2(D+3 (3.12)

is of,particular interest. In this situation, Eqs.(3.M and (3.5) become

_ yvG = B I T -=^

UVg

J 3{2<u+3)

I t also follows from £qs.(3.13) and (3. lit) that

-8TT |1-

-V

V - K. • 3

Let us introduce the following five-dimensional metric:

g = * 2 ( x y ) ,

(3.13)

(.3.Ill)

(3.15)

(3.16)

Note that g is "static" with respect to the additional fifth dimension

X . It is straightfoward to show that

GUV = Guv - f t* • v " 8 v D *] • (3.17)

G* •= %Ry - (3.IS)

V = 5,y - 0 . (3.19)

Therefore Eqs.(3.13), (3.11') ani (3.15) can be put in the form

- 9 -

g - Sir (3.20)

"1] 4 (3.21)

Finally, defining

and

we obtain the concise form

- ^

(3.22)

0.23)

GAB ° TAB (3.21*)

The field equations (3.2U) are the five-dimensional representation of the

BDJ theory that is equivalent to Einstein's equations for a five-dimensional

metric which is "static" with repsect to the additional non-space-tine

dimension (BDJ-5 theory).

For a specific application of the five-dimensional BDJ theory in

vacuum, where the effect of a scalar field on the cosmological singularityh)

is studied, see Belinsky and Khalatnikov .

IV. APPLICATION OF THE BZ METHOD TO THE BDJ-5 THEORY

We can apply the BZ technique, described in Sec.II, to a five-

dimensional vacuum metric g , when thi3 is a function of only two

variables. Thus let us assume that the metric g , Eq.{3.l6) have the

form (2.1):

f[dp2 + dZz]+g dXadXb; a, b - 1, 2, 3, (U.I)

where we made the identification

-10-

i f a = 3

5 , - 0, if a 4 3) '

(1*.?)

Then i t f o l l o w s t h a t t h e m e t r i c (2 .SO) and ( ? . ? ? ) , w i t h m = 3 :

f p h

is a solution of the BDJ-5 theory in vacuum, 5 = 0, if the -background

metric, g , is also a solution, and, furthermore,

? - * M d g d a M c 8C J

t.5)

We now consider a sufficient condition for Eq,(lt.5) to be valid.

We see that if

M - 0, k > q; a 4 3,

(It.6)

(It.7)

then the matrix

takes the block form

vi o

Ct.8)

where

-11-

r n r l 2 . . . r,(It.9)

Hence

r r rV q j ' • ' qq

q+i " • •rq+m

r E (it.10}

(it. I D

vhich implies that

or, equivalently^

g - g33 a

0 ; a i i 3 . CW13)

Therefore, we will assume that M^ satisfies Eqs.(U.6) and (I*.7).

We now explore further consequences of this assumption. We knov that

*ocb

Then the conditions (I4.6) and Ct.7) can be implemented if we choose

- 0; C 4 3, (lt.15)

and

-12-

C- = 0; k < q,

<T = 0; k > q; a + 3, l % i | )

Note tha t the assumption in Eq.Ct.15) i s consistent with £qs.(2.7) and (2 .8 ) ,

i f the matrix g sa t i s f i e s Eq.Ct,13). Hence, we wi l l assume the va l id i ty

of Eqs.Cul5), U.l6) and C*.17).

The metric (U.3) takes the form

= 21 "z

0 0

With the metric in the form Ct.l8), the field equations (2.M tecome

(PS 1 l) + (fig,, ¥ "' ) z " 0.

[tatBp^.J.p^ = 0 . (It.20)

Eq.(lt.2O) is equivalent to Laplace's equation with cylindrical symmetry.

Eq.Ct.19) has the same form that the second-order equation for a metric

of the type Eq.(2.1) in general relativity. However, we see, from Eqs.(2.2l)

and (!*.l8) , that

(h.21)

while a oorres.ponding 2x2 metric in general relativity satisfies

det gg

ggr

(It. 22)

Therefore, a solution g in general relativity is not a solution in the

BDJ-5 theory, nevertheless, we can use g to Duild a solution g, in

the following way:

-13-

Cfc.23)

Then we see that indeed g, as given 'by the expression ft,23), is & solution

of Eq.(l4.19)» if g is also, since

<Pg 8'P >P ,z

B;1>

0 ,

where we used Eq.Ci.20). Furthermore, Eq.Ct.23) implies that

det g

(!t.25)

and therefore

det g p h = (g p h) 3 3 det g - - P% ^ - ^

as required by Eq.(2.3).

Summarizing, we have the following result . Given a solution, g ,

of the general relat ivi ty axially symmetric and stationary vacuum field

equations, we get a corresponding solution in the BDJ-5 theory

v»

where Hn(g )^T is a solution of Laplace's equation Ct.20). The equivalent

result in the four-dimensional representation has teen given 'by Sned ion

and Mclntosh . It is also possible to generalize this conclusion to

the case when electromagnetic field sources are present ' .

- l i t -

Let us return to Eq.Ct.3) and make use of the implications of

conditions (4.15)-(U.17). We obtain

C,Jl=l

Thus ve have

(4.28)

' a b = n ab

q v k » ..*.

I r1 ^ . a> w 3i(U.29)

k,£>q

(U.30)

*.31)

The component (g h> can be re-expressed in a simpler form. In order to dot h i s , we note that

(It. 32)

which implies that

3 * 3 3 - 3

Consequently, Eq.(4.3O) becomes

C.33)

where

- 1 5 -

( U . 3 5 )

On the other hand, we know the following general result for a non-singular

matrix H, „:

det [1 + Ny] - det N^ + (det N (k.36)

from which we obtain

"k j

c-Dn"q nn>q (It. 37)

Therefore, substituting Eq. (4.37) into Eq^lt.S1*), we get

k>q \ J 3 3 k>q [P j k>q U(U.38)

where n and q have been chosen as even numbers, in order to preserve

the signature of the metric. Thus, we see that the expression for the

scalar field (g u)-3-i involves in a very simple way the background S -it

the poles u. , and is independent of the matrix +^.

V. THE DIAGONAL FORM OF THE SOLUTION

In order to study the non-rotational limit of the solutions,

we will consider in this section the case where g = g = 0. We can

diagonalize the metric g using the same procedure that led to the

"block form (U.18). That is, let us assume that § and ty^ are diagonal

and furthermore,

0 ; i f k >

C = 0 ; i f k < s o r k > q .

(5-1)

(5.2)

-16-

vhere s < q, is a positive integer. Then it follows that

= ck i/r1 = ck ir1 •= o, if k > s,C DCl 1 O H

Mk = C k If "' - 0, if k < s, or k > q,

and therefore

(5.3)

(5.1*)

8 i l

= l U ^ t5.ll)

(5.12)

(5.5)and also, from Eq.(1*.38),

where

and

'kj.

0, if k or Ji > s

r i u ' i f k a t l d ^ -

'•22Jk£

0, if (k or I) < s , or, (k or i.) > q ]

F „ , i f s < (k and t ) < q IikS. ~ '

(5.6)

(5.T)

( 8 . ) , , - 1 I " 4

where the expressions at the extreme right in Eqs.(5.1l) and (5.12) follow

in the same way we obtained Eq.U.38), in Sec.IV, and we have chosen 's1 an

even number to preserve the signature of g b.

To get an explicit expression for the metric component f h ,

corresponding to the above diagonal metric, we will calculate the determinant

of r k r We find,

Hence, T takes the block form

r n o o

o o rwhere we put

(5-S)

(5.9)

det T = det rtl det V22 det r3J,

dec T]E = det

det T_ = det | —•q

nk>s

det

+ p2)

(5.13)

(

(5.15)

We find, from Eq.(lt.29), that g h simplifies todet r 3 3 = det | ^ - ; — j !1 (Kk) I33| det

k

-i • (5.16)

(5-10) Using the following resul t , shown in Appendix A

-17--18-

d e t

1 + V (5.17)

We can see t h a t , for m poles \i ,

Therefore

(5.18)

det r - n (Mk) | _k - i

s(s-l)

• «. ( V * +(5.19)

r « • i i CM*)2 '•n (p u

,£ > s k H(5.20)

(5.21)

Sutistitutior, of Eqs.(5.19)-{5.21) in Eq.(5-13), and th i s in Eq.(it.li), gives

fk" 1

P£

1+

3-1)

3 2 )IT

Bn

(uk-u

(P U1 k ^

+ P 2 ) ~n~7wf+p2) nkk>s

p(n-qXn-q-l) I (

(5.22)

-19-

VI. THE LIHTT Cj • g

I t i s important to studj under wlii1 ri roLonstanccs ttic solution

g reduces to the "background metric g . For example, i t i s useful

to invest igate the new solut ion, g , "near" the background when the

physical in terpreta t ion of g i s wel.l known.

We can easily show that the diagonal element,, Eqs.(U.38), (5.1l)

and (5-12), reduce to g if ei ther one of 1 >>e following two conditions

are sa t i s f i ed :

= g-B = n-q and ^ = u s + k = U ? s + k ; k < s = .

i i ) In each group of poles , l < k < s , s < k < q , q < k < n ,

the poles come in pairs u < U, t , such that

- z (6.D

We see that the condition (i) implies the following results:

(6.2)

n pn - 2

k>q k

•5 2•= I! £j-

k>s uk

s= n

k=l

o ^ l

k-l yk n(6.3)

Also from the assumption (ii) it follows that

k k'

and therefore, using Eq.(6,10,

(6.5)

n 4k=l \

' 4k>S Uk

" 4k>q Uk

(6.6)

Hence, in both cases (i) and (ii), g » | , since the coefficients of

eab l n Eqs.(lt.36), (5.11) and (5.12) are unity.

-20-

We also expect that, when 5 p hthen f

ph = V In

Appendix B, it is shown explicitly that, when g is diagonal, then for

each one of the conditions (i) and (ii), indeed we have f = fph 0

VII. SOLUTIONS WITH PSEUDO-EUCLIDEAN BACKGROUND

The simplest application of the Belinsky and Zakharov method to

the EDJ-5 theory is the generation of solutions using a flat space-time

metric with a constant scalar field as background:

' P2

0

0

f

0

-1

0

= 1.

0

0

1

(7.1)

(7.2)

where the constant scalar field, g , has been normalised to unity. A

particular solution iK, corresponding to the metric (7.1) is

-XV

0

0

0

- 1

0

0

0

1

XW = (7.3)

j

We can get new soliton solutions g , f . in the BDJ-5 theory in aph ph

straightforward way, using Eqs.(T.l)-(7.3) in Eqs.(lt.U), (14.29) and (It.38).

The problem of creating new soliton solutions in general relativity

from a flat space-time metric has been discussed in detail by Belinsky

and Zakharov ' . In particular, they considered the two soliton

case to build the Kerr metric. These solutions can readily be transformed

into BDJ-5 solutions using the prescription in Eq.Ct.27). Hence an

alternative way of generating solutions from flat space-time is to construct

them as in general relativity and then use Eq.C4.27), with the scalar field

33 by

k>q \= 1,

-21-

or any other solution of Laplace's equation (it. 20). For instance, the

Kerr solution can be transformed to get a rotating solution in the BDJ theory.

However, this solution does not become spherically symmetric in the absence

of rotation, if the scalar field is non-constant, as we shall see below.

The Brans-Dicke-Jordan spherically symmetric static vacuum solution,

t, tai

(Appendix C)

g , f , takes the following form in the five-dimensional representation

s p

-ai1-*-" [i-f *ij

where fi and v are constants related by

(7.5)

(7.6)

(7.7>

(7.8)

(7.9)

and 6(5 must be identified with the gravitational mass of the system

(Appendix C). The spherical co-ordinates r , 8 , are related to the

cylindrical co-ordinates p, z in the following way:

Z = (r - S)cos 9 .

We can re-express g in terms of p and z:

T- -, S-v

(7.11)

(7.12)

(7.13)

(7.Ht)

where the functions p and u are just the poles, with W = -S and +S,

-22-

u = -(6 + z) + + p2 - (r-2g) [l- cos e ] ,

V = (3 - z) - /(@-z)2 + p2 --(r-28) [l+ cos 9 ] ,

(7.15)

(7.16)

Conversely, if the scalar field (&^),-, is not a constant the solution (7.20)

does not represent a spherically symmetric configuration.

More generally, if half of the poles, w (k odd) are chosen equal2 - 2 -

to M (or -p /ii) and the other half equal to u (or -p /v) then the

diagonal solution, Eqs.(U.38), (5.U), (5.12), takes the form

On the other hand, according to Eq.(lt.27) the F, , BDJ-5 solution generatedpr:

from the Kerr solution background, g , j s given t>-

-It

(7.17)

(7.22)

(7.23)

ana since In the absence of rotation the Kerr solution 'becomes the Schwarzschild

solution

r2 sin 2 6 0

0 -fl- •?

then we have that, without rotation.

Ph(7.19)

To compare the metric (7.5)-(7.7) with Eqs.(T.19) we must identify (g )

with [1-2 p/r] . Thus we get

gph rJ

o (7-20)

which is Just like g but with 6 = 1 . Therefore the condition i2 + 3v2 = 1,

can only be satisfied if v = 0, or equivalently when

V " = constant. ( T_ 2 l )

which is similar to the functional form of g . Nevertheless, in order for

g to be equal to g we must identify

Ks — ) ,

6 - v - ± ( H - q + s) ,

2v - ± (-|n+ q ) ,

or, eq^uivalently,

5 ' ± (s - * q) ,

V - ± ftq-jn),2 2

but again, 6 + 3v" cannot be unity un.esr. v vani shes.

(7.25)

(7.26)

Thus it seems that to obtain the spherically symmetric solution as

a diagonal limit of a soliton metric we must start with a non-flat metric

g . In particular, if we put § = g , it ia possible to obtain the reduction

g _> g when either of the conditions (i) or (ii), discussed in Sec.VI,

are satisfied. Or, another possibility is to start with a static axially

symmetric but not necessarily spherical background solution, and, instead of

imposing the conditions (i) or (ii), require that the diagonal metric

generated is equal to g . The study of solutions of this type, where g

is given by the BDJ-5 generalization of the well-known Weyl - Levi Civita

solution of general relativity, is discussed in the next section.

-23-

fm

VIII. SOLUTIONS WITH WEfL-LEVI CIVITA BACKGROUND

A solution of the BDJ-5 field equations that describe the exterior

gravitational field of a static axially symmetric configuration is given by

(Appx .C )

(8.1)

(8.2)

(8.3)

where

[1 - -2a 3

[l- f |JSin

o 2 = <52 + 3v2 (8.5)

When V = 0, this solution is the well-known Weyl-Levi Civita s ta t ic metric.

We can verify that a particular solution of ECLS.(2.7) a nd (2.8)

when g = g is

= - Awl (-6-v

(8.6)

r ex -u) ( A - u) i 2 v

* . , » • L xw J 'The evaluation of <J* at A = u gives

U K

2W, M,k k

-S-\j

(8.7)

(8.8)

(8.9)

(8.10)

-23-

- U)(8.11)

Hote that

T4J0 CX> •> g ,L i m X -> 0

as required by Eq.(2.1l). An alternative form for the solution can te

obtained using the expression

(8.12)

XW = X2 + 2X z - pz = (X - u )<X - p0) , (8.13)

where the functions yQ and jj are poles with Wfc = 0:

" o =z 2 + P J (8.lit)

(8.15)

It is worth noticing that, as follows from Eq_s.(8.l)-(8,3)_,

Lim

pz 0 00 - 1 00 0 1 (8.16)

ana furthermore, using Eqs. (8.6)-(8.8) and (7.15), (7.1

(X)l im

-AW00

0-I

0

001

(8.17)

Thus, when £ = 0, the seed solutions g , ipQ are the same aa for the pseudo-

Euclidean case (Sec.VII).

Let us consider the diagonal form, Eqa.(5.1l)) (5.12) and (It.33).

We see that a Weyl-Levi Civita type solution can "be obtained when the poles

are chosen in the following way:

- 2 6 -

- — =- r [ l - cos 6] , k odd I

- 4r = - r [ l + cos 6] , k evenV >

(6.18)

since, in th is case,

(8.19)

(8.20)

(8.21)

where

• = a + f - » , (6.22)

(8.23)

In other words, if we s tar t with g(6,\i) as background, the new diagonalmetric will be g(J ,v) . Furthermore, if the parameters 6 and v are

2 "2 2 o - -chosen such that 5 = & +3v = 1 , we will obtain gU.u) = g and*M) -t .

spFor these types of solutions, the functions iji (k), Eqs.(8.9)-(8.1l)

take the form, for k odd

*C)^ = 2Br(l - cos 6)[2{1 - - -(1 + cos 9)}]"S 'V , ,„ , .mi r to. 24)

"5"V

• J I J - - c « i - ! (8.25)

(8.26)

and for k even,

-27-

M t SS * . , « • . ; ! '::

= 2Br(l + cos 6)[2{1 - | (1 - cos

- 7 (i - cos e))] S-v

e)} ] / v .

(8.27)

(8.28)

(8.29)

Thus we can construct generalizations of the Weyl-Levi Civita metric by-

substituting the above matrix tyQ and the metric {8.1)-(8.U) in Eqs.(U.£9),

(1*.38) and (k,k). The case ti = q = s = 2, i s given explicitly below

& ~ * sia , v s v - f ,

(gp h)1 2 - ^ - [A(a - b cos 9 ) - a sin2 e ( a2 - b1 - MR)] ,

fP h - ^ ^ - , 6 E « . i ,

where

_ B -2

Mo (B +

2Mo)

1

1 -

r

r

4- =y Sin! 9

M = i o B )126

(8.

( 8 .

(B.

( 8 .

(8.

30)

31)

32)

33)

34)

(8.35)

C8.36)

3.37)

-28-

- 4rI"26 (bo+ a.)

8) + - — —

R = (r - &) + M,

E R2 + (h - a cos 6)2

-261

(8.. 39)

(8.40}

A E 1- [r(r _ 20)] = (R - H)' - y' .fl2

(8.U1)

The constants aQ, b Q, H , and 6 have been chosen such that M? - 8 = a -b 2,

in order to obtain the Kerr-IJut solution when fi = v = 0, as we shall see

below.

The study of the physical interpretation of the above solution is

under way, but we can already state the following results:

0, then Y » S , M * MQ, a = aQ, ~o = hQt and thei} If S«

solution becomes equivalent to the Kerr-Nut metric, since one can. verify that

the time co-ordinate transformation

T - t + 2aQ^

leads to the Boyer-Lindquist Kerr-Nut line element

cos 6 - a sin2 e[MR + bz])dT

S Jn2

If b = 0, we get the Kerr solution with mass M and geometric angular

momentum Ma .

ii) If a » b = 0, we have (g 1 1 ) - , 2 = 0, and therefore xero angular

momentum. The metric (8.3O)-(8.3't) becomes a Weyl-Levi Civita solution

2S-.1-S-V! (8.U)

(8.1*4)

-29-

. [! . 1 | ] 2V i (3.1*5)

{8.1*6)

where 5 : J U , depending on the choice MQ = +_ B. If, furthermore,

o = S + 3v = 1, we have spherical symmetry.

ACKNOWLEDGMENTS

The author would like to thank Professor Emil Kazes from the

Pennsylvania State University for his continuous encouragement and advice

in this work. Thanks are also due to Professor Belinsky for useful

discussions. He is grateful to Professor Atdus Salam, the International

Atomic Energy Agency and UNESCO for hospitality at the International

Centre for Theoretical Physics, Trieste. This work was supported in part

by a grant from the Resource Centre for Science and Engineering of the

University of Puerto Pico, and by the Humacao University College.

-30-

SUMMARY OF THE APPENDICES OF THIS WORK APPENDIX A

In Appendices A-F we show the following:

1Appendix A: Determinant

1 + L *t ( w <vv] [^.T1

In this appendix it is shovn that

det

(A.I)

Appendix B: If g is diagonal and any one of the conditions (i) or (ii),

defined in Sec.VI, are satisfied then f = f .ph 0

Appendix C: The tfeyl Levl Civita metric in the BDJ-5 theory is derived.

Appendix D: Determinant iJi(A) = \0{v), where G(w) i s a function only of

W = [X2 - p * ] A " 1 .

Appendix E:

Appendix F: If g and i|i are diagonal, then Ji{)i) g 1 IJJ(—p2/X) = F(Vf),

where F is an arbitrary function of W.

First note that

det H?i ^ m

{1 + det

where we used the fact that det

equivalent to

J =[H bj det

det R {1 + A ^ ) - K <Ak - AjXBt - Bk)

(A.2)

. Hence Eq_. (A.l) is

U.3)

The proof of Eq.(A.3) will be by induction. Hence let us assume that Eq.(A,3)

is valid for a. (n-l) x (n-l) matrix:

n-l - v-(A.5)

Then we have

n

k>£(Ak" V ( V

n-l n-l

"n-l

5

where we used Eq_. (A. 5). We know that

n_!C (A.6)

-31--32-

det C = f cn-l 1... cn

n-1

SI

(A,7)

Also, it is easy to show that

" " V - Vn-1

it (A.8)

for any particular choice of n-1, i . Consequently, using Eqs.(A.T) and

(A.8),

" n-I

n (AJ^-A^(BJJ-BJ^) = E. , n c

" (l+A.B ](1+U. )ij £ n * f

- K,t •

n-1

n

(A.9)

The terms

E (I+AnBn)(HAs,Bi^c"^ = (HAnBn) H (1 + A ^ J , (A..10)

are independent of 1 . Hence any term on the right-hand side of Eq.(A.9)

that contains a factor like K.K ...K is nymmet. '• i i a i under pernrutationp

of i., i ,.,.,i . Therefore these term;; do siot .•o::tribut!> t.n the sum inx. m s

Eq.(A.9). Thus, the only surviving terms are either independent of K

or linear in K . Consequently, Eq..(A,9) becomes

-33-

n-1 n-1

(A.U)

Using Eq.(A.10), we obtain after some rearrangement

n - 1 ''mn^in,n J I D c " - - — • -irm n i m j , H j , criCL SLi£

(A.12)

•where we use the definitions

C- 5n-1

(A.13)

ni

n-1

n-1

(A.15)

Substituting Eq.U.12) and using again the definitions (A.13)-(A.15) in

Eq.(A.ll) we get

n-1 C ^ (? n-1 (A.16)

On the other hand, we know that

(A.IT)

Eq.(A,17) can te re-expressed in the following form:

n-1det H C

+ £. ki k snn

•'• n , i 2 , . . . i o k kifc i

?n- l

We see that

c". cn » eCkiv

CniCsn

uSi«,

Therefore, applying Eq.(A,19) to Eq. (A.l8),ve obtain

det < ^ - e. . V <£ cn - e

ni

(A.19)

<u

V TT

{A.20)

- 3 5 -

Comparing Eq.(A.2O) with Eq,(A,l6) we see that

d e t C° = det n (1 +k > J l

- Aj)(B r B ) ,

(A.21)

or, equivalently, from (A.2)

det(A.22)

Since A.22 is valid for a 2 * 2 matrix, then we have shown that is

valid for all n >_ 2. Q.E.D. In particular, if A = B = y , we have

det1 +

(A.23)

-36-

APPENDIX B

It is shown below that when g is diagonal, then for each one ofph

the conditions (i) or (ii), in Sec,VI, we have

f . " const. foph (B.I)

First consider the case (i). In this situation Eq.(5,22) becomes

fph3 Q ' fo P

k-1

!! k k k 2 •> •> •

n ( M IM 2M 3 ) g l l g 2 2g 3 3

k-1

P(P-l)

k=l

We know that

(uv+p2) nk J i

(B.2)

11 It 33det g - - Q2 , (B.3)

and

k Mk Mk ck ck ck - (k ck ck(B.U)

In Appendix D it is shown that

det i> (X) - XG(w(B.5)

where G(W) is only a function of

(B.6)

Since, using E<i.(2.15).

U2. + 2ukz - p2

(B.T)

we have

det <KUk) - Const. (B.8)

-31-

Furthermore, we will also use the following relation, which is shown in

Appendix E:

(M - p )k p2) j k (B.9)

Subs t i tu t ing Eqs . (B .3 ) , (B.I*), (B.8) and (B.9) in Eq.(B.2) we obtain

Const.

11

k.Jt(B.10)

On the other hand, from Eq.(2.23), we have

Q » Const. p

k=l j k.A Va (B.ll)

where we used Eq.(B.9). The last factor of Eq.(B.ll) can be re-expressed

in the following form:

(B.12)

where we use the condition (i). Thus we have, putting n = 3s.

Q'1 - Const, p(3s+l)2 f s

II Uk

k=ln (u u +p2)

(B.13)

Substitution of Eq..(B.13) into Eq.(B.lO) gives the desired result:

f - Const,ph (B.lU)

-38-

(6.5)

He now consider the condition (ii). Let us recall Eqs.C6.l4) and

Some consequences of Eqs.C6.lt) and (6.5) will be derived below. First

consider

it

(B.15)

but

" z> =

where we used Eqs.(6.1t), (6.5) and (B.7). Consequently,

(B.16)

)K -Const. II —i— lip, .

CB.17)

Also it can easily be shown that

II(u5 + p2) - H(y2 + p2)(u= + p2) = n p2(y5 - y.,)2

(B.18)

T h e r e f o r e v e h a v e , f o r m p o l e s u , o r e q u i v a l e n t l y ~ p a i r s u D » U a i

ji (u«2+ P 2 )Const. s

m

Const. Const, p

(E.19)

where we used Eq,(6.!*)

-39-

We can now simplify the last three factors of Eq.(5.£2) using

Eq.CB.19), since we have, for m poles y ,

pjnCm-1) TJ (yk-yj,) Const, p

2 (m-1)

k k

m(m-1)-3m

Const. P ,

CB.20)

where «e used Eqs.(B.9), (B.19) and Eq.(6.1t). On the other hand, with

-1EIJS.(B.19) and C6.it), the expression for Q~ also simplies to

Q T - Const, p ' .

Using the results (B.20) and CB.21) in Eq.(5.22), we get

CB.21)

f . =ph

Const. f o n(M ) .gq

n (k>s

k 2 •IT (Mj) g

k>q CB.22)

Finally, it is shown in Appendix F that, if g and ip are diagonal,

then the matrix

(B.23)

is only a function of W(JO, and therefore

= Const. ,CB.2U)

or, equivalently, for a = 1,2,3

C' * l ia °aa a a "i

(No s u m o v e r a r ) .

'aa «.' °aaConst.

(B.25)

-1*0-

This implies that APPEBDTX C

k>s k>qConst. (B.26)

A solution of the Einstein scalar field enunLionr. (3.9) and (3.1Q)

that represents the exterior gravitational field of a stat ic axially

symmetry configuration is

Hence again

gK.S l! ?l -o

sen^O ,(C.I)

f = Const, fph (C.2)

Q.E.D.

1 -

; G2 ? S2 + 3\J=, (C3)

(C.I+)

where B and 6 are constants, and the co-ordinate? r, S are related to

P and .z ty Eq.s.(7.1O) and (7.11). If '" = 0, the above equations transform

into the well-known Weyl-Levi Civita meLri'r- * of general re la t ivi ty .

The total gravitational mass of the syatem in (SB , whicti follows from the

requirement that the equations of no1 innsi become Mewt.nn Lan in the limit

The case a ' = i i« of pat IJi.ul ai interest , since we get a

spherically symnintrir1 gravitationa] field. The line element takes the form

d s " dr' + r ' + sin 0

(C5)

1 - 6(C.6)

A s o l u t i o n of t.hfi BDJ-S f i e l d e q u a t i o r j , g , i s " b u i l t , from t h e m e t r i c

(C.I. ' - ( C . 3 ) and t h e sca.la:r f i e l d [C,h] u s i n g t h e t r a n s f o r r a a t i o t i s ( 3 . 1 6 ) ,

( 3 . 6 ) , ( 3 . 7 ) , t o g e t h e r wi th E q . ( 3 . 1 1 ) :

- 4 1 -

* aV,

*>•>• • - I1 - T J

or, in the canonical form {k,1 },

E.S

(C.7)

(C.8)

(C.9)

( CIO)

(C.ll)

APPENDIX D

2Let us show that if det. g = -P , then

det *(*) - AG(H),(D.I)

vhere G(W) is a function only of W, subjected to the following condition:

(D.2)

We know, from Eq.(2.8) that

X2 + p2

Consider the trace of the above equation

(D.3)

Trace_! pTrace U+ X Trace V P2 T r a c e 8 g' 1 + PX Trace g > z g'

X2 + p2 X2 + p2

(D.I*)

Using the Hell-known resul t , valid for any non-singular matrix H,

Trace

we obtain, from Eq.(

D,(det ['

det N(D.5)

g) >z] (det g)"2 p

det

where we used

X2 +

(0.6)

-1*3-

det g = - p .

Similarly, from Eq.(2.T) it follows that

det V

On the other hand

D2(ln X) = \4-

+ p4) det g

-2\

, , -2XIn X =

2 2

(D.7)

(D.8)

(D.9)

(D.1Q)

particular solution is

G(W) = W,

det l)J( X ) = XW.

(D.16)

(D.IT)

Therefore

D In <i^Ji) - 0, (D.ll)

The solution to Eqa.(D.ll) and (D.12) Is

Furthermore, since ip(X » 0) = g, we must have

(D.12)

(D.13)

det Y - det g - - Plim X-fO

and consequently

XG(W)--p2

lim(D.15)

-1*6-

APPENDIX E

We w i l l see lielow tha t

2(Wk - (E. I )

where n is the number of poles, u^. Thus consider two poles u , \i .

They satisfy Eq.(£.15), which in turn implies

U2 - 2u.(W0 - z) - p2 = 0

(E.2)

(E.3)

The difference of these two equations gives, after some re-arrangement

Substitution of Eq.(E.lt) in Eq.(E.2) yields

CE.lt)

(E.5)

or, equiveilently,

(E.6)

Thus we have

(E.7J

-1*7-

APPENDIX F

We will prove that if lfi is diagonal then

* (X)g l U -y- ) - F{W),(F.I)

where FCW) is an arbitrary function of W. First i t i s shown that D^ and

D are invariant under the following transformation:

P - P

(F.2)

(P.3)

Hence ve have

_3_ _ _3_ + ZX _3_3p " 3p p j X "

(F.5)

3 _ 3 (F.T)

Then it follows that

3z ' 3X " Yl~ I2 + p2 (F.8)

: ) 3X 3 ^* 2 1T 1 J si

(x2+u2)D2(X) . (F.9)

The field equations(2.7) and (2.8) imply that

-U8-

T % I / T V I - i T , pv -= r ^

- [Di(X)i|)(X)]g , z g

(F.10)

and similarly,

[p U

\ z + p 2 'P

Us ing E t i s . ( F . a ) and ( 7 . 9 ) , i n E q s . ( F . l O ) and ( F . l l ) , we ofctain

l (X) - [ D j g j g " 1 - 0 , (F.1E)

)]i(j-I(X) - [D2g] g"1 = 0. CF.13)

If g and iji are diagonal, then Eqs.(F.12) and (F.13) simplify to

g"1) - 0,Dj(X)

D2(X) ln(((i(X)i|)(X) g"!) - 0 .

The solution to Eqs.tF.lM and (F.15) is

ili(A)i|J(X) g"' =' ( —^— ) ^(X) g"1 " F(W) ,X

where F{w) is an arbitrary function of

X2 + 2Xz - p2

(F.llt)

(F.16)

-1*9-

REFEBEMCES

1) P. Jordan, Z. Physik 151, 11? (1959).

2) G. Brans and R.H. Dicke, Phys. Rev. 2}i_, So.3, 925 (l96l).

3) R.H. Dicke, The Theoretical Significance of Experimental Relativity

(Gordan and Breach ig6U).

It) V.A.Belinsky. and I. Khalatnikov, Sov. Phys.-JETP 36, 591 (1973).

5) U, Eruckman, Thesis Diseertation, Dept. of Astronomy, The Pennsylvania

State University (1979).

6) A. Liehnnerowicz, Theories Relativlstes de la Gravitation et de

1'electromagnetistae (Mason Paris, 1955).

7) E. Kerr, Phys. Rev. Lett. 11_, 237 (1963).

8) C. Mclntosh, Commun. Math. Phys. 3J_, 335 (19T1*).

9) C M . Cosgrove, J. Phys. A 10, lWl (19T7).

10) V.A. Belinsky and V.E. Zakharov, Sov. Phys.-JETP U5_, 985 (1973).

11) V.A. Belinsky and V.E. Zakharov, Sov. Phys.-JETP ^0, 1 (1979).

12) A. Eris and M. Gurses, J. Hath. Phys.l8_ , 1303 (1977).

13) V.A. Belinsky, Sov. Phys.-JETP 5£, 623 (1979).

14) V.A. Belinsky and R. Ruffini, Phys. Letts. B 89, No.2, 195 (1980).

15) T. Lewis, Proc. Roy. Soc. (London) A136, 176 (1932).

16) J.L. Anderson, Phys. Rev. D3, 1689 (1971).

17) G. Sneddon and C. Mclntosh, Int. J. Theor. Phys. £J_, 1*11 (1971*).

IS) H. Weyl, Ann, Phys. Leipzig, 5}*., 117 (1917).

19) T. Levi Givita, Atti Acad. Haz. Lincei, %, 26 (1917).

-50-