Black Hole Entropy through Thin Matter Shells - Autenticação · Black Hole Entropy through Thin...

Black Hole Entropy

through Thin Matter Shells

Goncalo Martins Quinta

Dissertacao para obtencao do Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Professora Doutora Ana Maria Vergueiro Monteiro Cidade Mourao

Orientador: Professor Doutor Jose Pizarro de Sande e Lemos

Vogais: Professor Doutor Vilson Tonin Zanchin

Professor Doutor Vıtor Manuel dos Santos Cardoso

Doutor Antonino Flachi

Setembro 2013

Acknowledgments

I would like to start by thanking my supervisor Jose Sande Lemos from whom I have learned a lot,

not only about physics but about the life of a physicist as well. Thanks to him I have had opportunities

I didn’t think a master’s student could get. It has been a fun and rewarding experience working with

him.

I would also like to thank my family for their constant support, for always encouraging me to do what

I like to do regardless of anything else and for teaching me from an early time that true freedom only

comes with hard work.

I’m grateful as well for all my friends with whom I have had so many fun moments together. Their

ability to keep a cheerful mood even under working pressure is truly inspiring and their natural tendency

to fuel my sometimes exotic conversations about physics is really stimulating.

Finally, I save for last my special thanks to Sofia, for always knowing how to keep me happy and sane

in the real world of low energies.

i

Abstract

In this thesis we study the thermodynamics of thin matter shells in the context of general relativity,

as well as the implications of such study for the entropy of black holes.

We follow the same work methodology in a variety of cases, which starts by using Israel’s formalism

to obtain the necessary pressure and energy of the thin shell in order for it to be static. Those quantities

are then used to find the entropy of the shell which naturally possesses degrees of freedom that can be

parametrized by phenomenological functions which contain free parameters. These last are then subject

to an analysis which determines the interval of values for which the thermodynamic stability of the system

is guaranteed. This process is repeated for 4 different situations: a shell in 2+1 dimensions; a shell in 2+1

dimensions with a non-null cosmological constant; a shell in d dimensions; and an electrically charged

shell in 4 dimensions.

Lastly, we use the results obtained for the shells in the limit where they are taken to their gravitational

radius, where it is shown that they can behave like black holes. In particular, the case of a charged shell

in the extremal limit suggests a solution for the ongoing debate concerning the value of the entropy of

an extremal black hole.

Keywords

Thin shell, thermodynamics, black hole entropy, extremal black hole entropy

iii

Resumo

Nesta tese estuda-se a termodinamica de camadas finas de materia no contexto de relatividade geral,

assim como as implicacoes que esse estudo tras para a entropia de buracos negros.

E seguida a mesma metodologia de trabalho numa variedade de casos, que comeca com a utilizacao do

formalismo de Israel para obter a pressao e energia necessarias para manter uma camada fina de materia

no regime estatico. Essas quantidades sao entao usadas para se obter a entropia da camada, onde surgem

naturalmente graus de liberdade que podem ser parametrizados por funcoes fenomenologicas contendo

parametros livres. Estes ultimos sao analisados de forma a averiguar os intervalos de valores para os

quais a estabilidade termodinamica do sistema e garantida. Este processo e repetido para 4 situacoes

diferentes: uma camada em 2+1 dimensoes; uma camada em 2+1 dimensoes com constante cosmologica

nao nula; uma camada em d dimensoes; e uma camada electricamente carregada em 4 dimensoes.

Por ultimo, utilizam-se os resultados obtidos para as camadas no limite em que estas sao levadas ate ao

seu raio gravitacional, onde se mostra que estas se podem comportar como buracos negros. Em particular,

o caso de uma camada carregada no limite extremal sugere uma solucao para o debate existente sobre o

valor da entropia de um buraco negro extremal.

Palavras Chave

Camada fina, termodinamica, entropia de buracos negros, entropia de buracos negros extremos

v

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thin matter shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Black holes and black hole entropy 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Black hole entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Several methods to calculate the black hole entropy . . . . . . . . . . . . . . . . . . . . . . 6

3 Entropy of a thin shell in a (2+1)-dimensional spacetime 9

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 The thin shell spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Thermodynamics, entropy equation for the shell, and stability . . . . . . . . . . . . . . . . 13

3.4 Two specific equations of state for the thin shell matter: Entropy and stability . . . . . . 14

3.4.1 The simplest equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.2 A more contrived equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Entropy of a BTZ black hole through thin matter shells 17

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17



4.4 A specific equation of state for the thin shell matter: Entropy and stability . . . . . . . . 20

4.5 Thin shell entropy in the BTZ black hole limit . . . . . . . . . . . . . . . . . . . . . . . . 21

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Entropy of a d-dimensional black hole through thin matter shells 25

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25




5.5 Thin shell entropy in the limit of a black hole . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Entropy of a non-extremal charged black hole through thin matter shells 33

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33




6.5 Thin shell entropy in the limit of a black hole . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Entropy of an extremal black hole through thin matter shells: a solution to the debate 43

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.2 Entropy of an extremal black hole through a non-extremal shell . . . . . . . . . . . . . . . 43

7.3 Entropy of an extremal black hole through an extremal shell . . . . . . . . . . . . . . . . . 44

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8 Conclusions 47

Appendix A Thin Shell Formalism 49

A.1 Introductory definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.2 First junction condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.3 Second junction condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Appendix B Equations of thermodynamic stability for an electrically charged system 55

Appendix C Newton’s gravitational constant in d dimensions 59

Bibliography 61

viii

Preface

This work was supported by Fundacao para a Ciencia e Tecnologia (FCT), through projects

PTDC/FIS/098962/2008 and PEst-OE/FIS/UI0099/2011. The research included in this thesis has been

carried out at Centro Multidisciplinar de Astrofısica (CENTRA) in the Physics Department of Instituto

Superior Tecnico.

Chapters 3 to 5 were done in collaboration with Professor Jose Sande Lemos. Chapters 6 and 7 were

done in collaboration with Professor Jose Sande Lemos and Professor Oleg Zaslavskii.

A list of the works published and to be published soon in major international journals included in

this thesis are listed below.

- Jose P. S. Lemos, Goncalo M. Quinta, Thermodynamics, entropy and stability of thin shells in 2+1

flat spacetimes, Physical Review D 88, 067501 (2013), (Chapter 3) [1].

- Jose P. S. Lemos, Goncalo M. Quinta, Entropy of thin shells in 2+1 dimensions and the BTZ black

hole limit, in preparation for submission (Chapter 4).

- Jose P. S. Lemos, Goncalo M. Quinta, Entropy of thin shells in d dimensions and the Schwarzschild

black hole limit, in preparation for submission (Chapter 5).

- Jose P. S. Lemos, Goncalo M. Quinta, Oleg B. Zaslavskii, Entropy of charged thin shells and the

non-extremal black hole limit, in preparation for submission (Chapter 6).

- Jose P. S. Lemos, Goncalo M. Quinta, Oleg B. Zaslavskii, Entropy of the extremal black hole through

thin matter shells, in preparation for submission (Chapter 7).

Goncalo Quinta also presented the work Extremal versus non-extremal black hole entropy: a thermody-

namic approach related to Chapters 6 and 7 , in the 20th International Conference on General Relativity

and Gravitation (GR20), in July 11, 2013, in the Parallel Session - D4: Quantum fields in curved space-

time, semiclassical gravity, quantum gravity phenomenology, and analog models, chaired by C. Fewster

and S. Liberati, see for the abstract http://gr20-amaldi10.edu.pl/index.php?id=49&abstrakt=90

and for the slides http://gr20-amaldi10.edu.pl/index.php?id=18. The abstract is in the Book of

Abstracts:

- J. P. S. Lemos, Goncalo M. Quinta, Oleg B. Zaslavskii, Extremal versus non-extremal black hole

entropy: a thermodynamic approach, in GR20/Amaldi10 Abstract Book, eds. J. Lewandowski and Local

Organizing Committee (Warsaw, 2013), p. 246 (http://gr20-amaldi10.edu.pl/index.php?id=50) [2].

ix

http://gr20-amaldi10.edu.pl/index.php?id=49&abstrakt=90

http://gr20-amaldi10.edu.pl/index.php?id=18


In addition, the following work in portuguese dealing with aspects of chapters 6 and 7 is about to

be published in Tecnica (the journal of the Associacao dos Estudantes do Instituto Superior Tecnico

(AEIST)):

- Goncalo M. Quinta, Uma abordagem termodinamica a entropia de buracos negros, Tecnica, in press,

(2013) [3].

x

1Introduction

1.1 Motivation

The theory of thermodynamics is largely successful in describing systems which have well defined

macroscopic variables such as temperature, internal energy and pressure. Indeed, all the engineering

problems concerning heat and its relation to work and energy can be solved by making use of thermody-

namics alone. However, on a physical perspective, the knowledge of the macroscopic physics of a system

does not suffice. To fully understand a system means to be able to perceive the microscopic dynamics

underlying it and not just the behaviour of a very large number of its microscopic constituents. The

theory of statistical mechanics is an example of this, where the laws of thermodynamics are derived from

first principles using only the microscopic physical laws of the components of the system. The study of

statistical mechanics also emphasizes the fundamental aspect of the notion of degree of freedom, which is

an independent physical parameter appearing in the formal description of the state of a physical system.

The set of all degrees of freedom of a system is contained in the so called phase space, and the under-

standing of the laws which govern the phase space leads to the full comprehension of how the microscopic

behaviour of the system averages out to the measurable thermodynamic quantities.

On a macroscopic scale, the microscopic degrees of freedom are encoded in the entropy, which quanti-

fies the number of a specific ways in which a system can be arranged. Therefore, if a microscopic theory

for a system if not available, one may use the entropy to try and find some clues. A special example

of such system is a black hole, whose thermodynamic properties alone were a particular challenge to

find. Bekenstein [4–6] found that the entropy should be proportional to the area of the black hole up

to a proportionality constant, which Hawking [7] fixed when he discovered that black holes radiated like

black bodies at a certain temperature, called Hawking temperature. These discoveries led to the definite

1

entropy of a black hole, also called the Bekenstein-Hawking entropy. This is an example in which there

is no known microscopic theory describing the system, since a complete theory of quantum gravity is not

available yet, but there is already a known expression for the entropy. The next step would be to use the

entropy to find clues about the underlying theory of quantum gravity but the problem is more complex

than it seems. In the case of statistical mechanics, the degrees of freedom are well know and localized, i.e.,

they are the velocities and positions of the constituent particles and the question is a matter of finding

the dynamics that involve them, whereas for a black hole that is not so straightforward.

When a black hole is formed by collapsing matter inside its gravitational radius, the information

about the microscopic configurations of the matter, i.e. the degrees of freedom, is lost inside the event

horizon, something which is also known as the ”no-hair theorem” [8, 9]. In other words, it is unknown

what becomes of the matter and the degrees of freedom associated with it once the black hole is created.

The entropy of a black hole does quantify the information of the black hole but it does not specify where

and what is the nature of the degrees of freedom. A variety of attempts have been made to find an answer

to this problem [10–14], although there is no agreement as to what degrees of freedom contribute to the

black hole entropy.

Another unanswered question relating black holes and entropy is what entropy should an extremal

black hole have. On one side we have the Bekenstein-Hawking entropy which should be valid for any

black hole, but on the other side the extremal black hole has zero temperature so it should have zero

entropy. This debate, along with the question of where are the degrees of freedom located, is the main

motivation for this work.

1.2 Thin matter shells

Despite the fact that a quantum theory of gravity should not create any distinction between grav-

itational and material degrees of freedom, it is still a subject of study at the phenomenological level

nonetheless. The reason for this is that its investigation may clarify some properties of the thermody-

namics of the gravitational field [15, 16], which in turn could shed some light on the features of a definite

unified treatment of quantum interactions. We are thus interested in a system which contains both grav-

itational and material degrees of freedom but which does not introduce too many complexities due to

the matter constitution. A particularly simple system which satisfies these requirements is a spherically

symmetric self-gravitating thin matter shell at a finite temperature.

A thin shell is an infinitesimally thin surface which partitions spacetime into an interior region and

an exterior region. Since it corresponds to a singularity in the metric of the spacetime, the thin shell

must satisfy some conditions in order for the entire spacetime to be a valid solution of the Einstein

equations. Such conditions are called junctions conditions, and relate the stress-energy tensor of the

shell to the extrinsic curvature of the spacetime through Israel’s massive thin shell formalism [17–22] (see

Appendix A). Thus, the material degrees of freedom of the shell are related to the gravitational degrees

of freedom through the gravitational field equations, and so the thermodynamics of the shell is deeply

connected to the structure of spacetime. Indeed, Davies, Ford and Page [23] and Hiscock [24], have

shown the usefulness of studying thin shells in (3+1)-dimensional general relativistic spacetimes from a

2

thermodynamic viewpoint.

Another reason which motivates the use of thin shells is the fact that they can be taken to their grav-

itational radius, i.e., the black hole limit. One can, for example, calculate the entropy of a shell for given

spacetimes and see what value it assumes in the black hole limit. Thus, the black hole thermodynamic

properties can be studied by a much more direct computation than the usual black hole mechanics if

thin shell are used, an idea which was developed by Brown and York [25] and Martinez [26] and which

is going to be used throughout this work. A similar approach used for the study of black holes through

quasi-black holes has also been proposed by Lemos and Zaslavskii [27–29].

1.3 Thesis outline

In this thesis we always use the same line of work concerning the computations involved with thin

shells. The strategy is the following. We directly integrate the first law of thermodynamics to obtain

the entropy of the shell, which requires a priori its pressure and total energy, which in turn are fixed by

junction conditions. In other words, the self-gravitating character of the system itself imposes physical

conditions on the matter of the shell, fixing completely its pressure and partially its temperature, as

we shall see. The entropy of the shell is then obtained up to a function which depends at most on the

gravitational radius r+ of the shell (or the rest mass M of the shell, in the case of 2+1 dimensions without

cosmological constant), which is essentially the inverse of its temperature. To advance further, one needs

to specify the form the undetermined function, which is physically equivalent to specifying the matter

fields forming the shell. In the spirit of the usual procedures in thermodynamics, we will adopt the most

simple phenomenological choice for the thermal equation of state, which will naturally come with free

parameters encoding the details of the matter fields. With an explicit expression for the entropy in our

hands, we end with a stability analysis of the parameters which will determine the conditions that the

free coefficients must satisfy in order for the shell to be thermodynamically stable, according to the formal

structure of thermodynamics developed by Callen [30].

As to the organization of the thesis, it is as follows. Chapter 2 begins with a summary of the

properties of black holes in general, in particular their entropy, which we will reproduce using a thin shell

approach. In Chapter 3 is computed the entropy of a general thin shell in 2+1 dimensions, followed by a

thermodynamic stability analysis of the free parameters. Chapters 4 to 6 follow the same line of work as

in Chapter 3 but with different spacetime choices: 2+1 dimensional spacetime with non-null cosmological

constant in Chapter 4; d-dimensional Schwarzschild spacetime in Chapter 5; and a Reissner-Nordstrom

spacetime in Chapter 6. Additionally, Chapters 4 to 6 analyze as well the limit of a black hole in the

results obtained, where the entropy of the corresponding black holes in each spacetime is recovered.

Chapter 7 is dedicated to the case of the extremal limit of a charged shell, in which a solution to the

debate of the entropy of a an extremal black hole is obtained. Chapter 8 sums up the conclusions of the

work done.

Throughout this work, we will use units of c = 1 and kB = 1, where c is the speed of light and kB is

Boltzmann’s constant, while keeping explicit Newton’s gravitational constant G and Planck’s constant ~.

3

2Black holes and black hole entropy

2.1 Introduction

One of the most simple and profound consequences of Einstein’s theory of general relativity is the

existence of black holes. On their final state, they can be characterized by their mass, angular momentum

and charge, something which is also known as the ”no-hair theorem” [8, 9]. A large amount of work and

progress in this branch of physics has been going on in the last fifty years, and tools developed for their

understanding have improved greatly [31].

The importance of black holes has been increasingly recognized, playing important roles in cosmology,

astrophysics and fundamental physics. When taken together with quantum field theory, new effects arise,

namely the emission of Hawking radiation and the existence of an associated black hole entropy.

In this chapter we intend to summarize the main properties of black holes and the results we wish to

recover using a thin shell approach.

2.2 Black hole entropy

The fact that black holes must have an associated entropy is clear from the second law of thermo-

dynamics: if a black hole had no entropy, then by throwing an object inside a black hole the entropy of

the system would decrease and so the second law of thermodynamics would be violated. Furthermore,

the study of black hole mechanics showed that there were three laws any black hole satisfied which were

surprisingly similar to the three laws of thermodynamics [32]. First of all, the surface gravity κ of a black

5

hole with mass m and electric charge Q, given by

κ =r+ − r−

2r2+

, (2.1)

is constant throughout the event horizon, analogously to the constancy of the temperature throughout a

body at thermodynamic equilibrium. Secondly, the energy conservation of a black hole is stated as

dm =κ

8πGdA+ ΦBHdQ (2.2)

similarly to the statement of conservation of energy in thermodynamics

dE = TdS + ΦdQ . (2.3)

Finally, the area of a black hole never decreases (∆A ≥ 0) much like the entropy in thermodynamics which

never decreases as well (∆S ≥ 0). By noting the similarity between the laws of black hole mechanics and

thermodynamics, in particular Eq. (2.3) and Eq. (2.2), one might suggest that the entropy of a black hole

should be proportional to it’s area, i.e.

dSBH =κ

8πGTBHdABH. (2.4)

Indeed, Bekenstein proposed in 1973 that it should be the case, leaving the constant of proportionality -

essentially the temperature of the black hole - undetermined [4].

In 1975, Hawking fixed the constant of proportionality by finding the explicit form of the black hole

temperature [7]

TBH =~κ2π. (2.5)

This expression was the result of the application of general relativity together with quantum field theory.

By inserting (2.5) in (2.4) and integrating, the entropy of a black hole immediately comes out as

SBH =1

4

ABH

l2p(2.6)

where lp ≡√G~ is the Planck length. This is the result which we will replicate using a thin shell

approach. It is important to note that this result is valid for any dimension, except that the definition of

the Planck length will vary according to the dimension involved.

On a final remark, if the black hole has angular velocity Ω and angular momentum J , the formulas

for the first law of thermodynamics and for the entropy are in the same spirit [33].

2.3 Several methods to calculate the black hole entropy

There are number of distinct ways to compute the black hole entropy. Bekenstein [4, 5] used a

thermodynamic approach together with information theory to obtain an entropy for the black hole,

which was correct up to a proportionality constant. Hawking [7] first obtained the exact formula for

the entropy by using the formalism of second quantization in quantum field theory in curved spacetime.

Later, he also obtained the same entropy but from a path integral approach of quantum field theory in

curved spacetime [34]. York [25] also derived the black hole entropy, from the perspective of a grand

6

canonical ensemble, where the quantities of interested are specified at a finite boundary. Finally, there

is an approach to compute the black hole entropy through quasi-black holes, proposed by Lemos and

Zaslavskii [27–29]. A review on the subject can be found in [35], for example.

The Bekenstein-Hawking entropy also led to the holographic principle of ’t Hooft [36], which states

that the degrees of freedom of a region of space circumscribed within an area A are in the area itself. For

a more detailed exposition of the holographic principle see for instance [37–40]

7

3Entropy of a thin shell in a

(2+1)-dimensional spacetime

3.1 Introduction

The relevance in the study of three-dimensional gravity started with the work of Deser, Jackiw, and ’t

Hooft [41], where it was shown that, though the corresponding vacuum solution is trivial since it consists

of Minkowski spacetime, a point particle distorts it into a conical space with the particle being located at

the vertex of the cone, and moreover, moving point particles display nontrivial dynamics. The next most

simple object, beyond a point a particle in 2+1 dimensions, is a thin shell, i.e, a ring dividing two vacua

regions, the interior and exterior to the ring itself. In this chapter we study the spacetime generated

by such a ring as well a the thermodynamic properties of such gravitational systems using the junction

conditions formalism.

By making use of the appropriate junction conditions for general relativity [17], one can determine the

pressure and rest mass of the shell in order for it to be static with interior and exterior spacetimes both

flat. Using then the formalism developed by Martinez [26], with the thermodynamic theory as presented

in [30], one can find generic expressions for the shell’s entropy, which upon some minimal assumptions

about the structure of the matter fields making the system, i.e., an ansatz for the shell temperature in

terms of the gravitational quantities that characterize the system, yields a definite expression for the

entropy of the shell and permits a stability analysis.

9

3.2 The thin shell spacetime

In 2+1 dimensions Einstein’s equation takes the form

Gab = 8πG3Tab , (3.1)

where Gab is the (2+1)-dimensional Einstein tensor, Tab is stress-energy tensor, and G3 is the gravitational

constant in 2+1 dimensions. G3 has units of inverse mass.

To find the solution for a thin shell in a (2+1)-dimensional spacetime, we follow the formalism devel-

oped in [17] (see Appendix A) and start by considering a one dimensional timelike hypersurface Σ that

partitions spacetime into two spherically symmetric regions: an inner region V − and an outer region V+.

Inside the hypersurface we use flat-polar coordinates xα− = (t, r, θ) for a flat metric with line element

ds2 = −dt2 + dr2 + r2dθ2 , r < R , (3.2)

where r = R is the radius of the thin-shell hypersurface.

On the outside of the shell, we will consider the spacetime to be again flat but the presence of matter

justifies the use of conical-polar coordinates xα+ = (t, r, θ), allowing the line element to be written as

ds2 = −β2dt2 + dr2 + r2α2d θ2, for r > R, and some constant α. It is preferable to make the change

α r → r, and without loss of generality one can put β = α, so that the metric takes the form

ds2 = −α2dt2 +dr2

α2+ r2d θ2 , r > R . (3.3)

The metric (3.3) has a conical singularity at r = 0 if the thin-shell hypersurface has a radius R(τ)→ 0,

i.e., it turns into a point particle.

The parametric equations of the thin shell hypersurface Σ are described by r = R(τ) and t = T (τ),

where τ is the proper time on the thin shell hypersurface. Choosing the coordinates ya = (τ, θ), we have

an induced metric hab given by

ds2Σ = −dτ2 +R2(τ)dθ2 (3.4)

which is simply the metric of a 1-sphere.

We are now ready to calculate all the necessary geometric quantities. Starting with the tangent

vectors, it is immediately seen that they will be the same viewed from both sides of the shell, according

to (A.2), since the metrics in both regions use the same coordinate system. Calculating explicitly, we get

eα±τ = eατ =∂xα±

∂τ= (T , R, 0) (3.5)

eα±θ = eαθ =∂xα±

∂θ= (0, 0, 1) (3.6)

where the dot denotes differentiation with respect to τ . Using the tangent vectors and the components

of the metrics inside and outside the shell, we are able to calculate the components of the induced metric

through the use of (A.8). However, instead of calculating them separately for each part of spacetime, it

is easier to note that (3.2) and (3.3) are of the form

ds2± = −F±dt2 +

dr2

F±+ r2dθ2 (3.7)

10

where F± are different constants, defined by

F+ = α2 (3.8)

F− = 1. (3.9)

This makes it clear that the calculations involved in the computation of the induced metric components

need only to be done once, since a simple substitution of (3.8) or (3.9) will automatically return the

induced metric in the corresponding side of the shell.

Therefore, using the metric components of (3.7), we arrive at the following non-null components of

the induced metrics

h±ττ = −F±T 2 +R2

F±(3.10)

h±θθ = r2 (3.11)

From the first junction condition (A.18), we know that h+ab = h−ab. On the other hand, the components

of the induced metric can also be read from (3.4), which are

hττ = −1 (3.12)

hθθ = R2 (3.13)

and since h+ab and h−ab are to be evaluated at the shell, they must also be equal to the hab directly read

from (3.4). Thus, we are led to the relations

−F±T 2 +R2

F±= −1 (3.14)

r2 = R2 (3.15)

The last equation doesn’t add anything new since it was already known that r = R(τ) but the first

equation can be manipulated to obtain

F±T =

√R2 + F± ≡ β±(R, R) (3.16)

where the new variable β will simplify the notation considerably. The components of the induced metric

with upper indexes must also be calculated since they will also be needed in calculations to follow. To

do so, first recall that

habhbc = δac = hac (3.17)

and because hab = 0 if a 6= b, we have

hτbhbτ = hττhττ = 1 (3.18)

which will also be true for the variable θ. This implies that hii = h−1ii where i = τ, θ, and so we have

hττ± = hττ =1

−F±T 2 + R2

F±

= −1 (3.19)

hθθ± = hθθ =1

r2=

1

R2(3.20)

11

We now turn to the calculation of the components of the normal vector which begin by noting that

it must be perpendicular to the 4-velocity

uα± ≡∂xα±∂τ

= eα±τ (3.21)

in each side of the shell. In other words, we require that uα+n+α = uα−n

−α = 0, which is satisfied by

n±α = (−R, T , 0), (3.22)

for example. With Eq. (3.22) and Eq. (3.15), it is possible to work out the non-null components of the

extrinsic curvature by using the definition (A.34), which are

K±ττ = − β±R

(3.23)

K±θθ = β±R (3.24)

By raising the first index through contraction with the induced metric with upper indexes hab, these

equations assume the particularly simple form

Kτ±τ =

β±

R(3.25)

Kθ±θ =

β±R. (3.26)

The components of the stress tensor can now be calculated, a task most simplified if we use one con-

travariant and one covariant index. Since we will only consider timelike shells (ε = 1), we can use the

results (3.25), (3.26) and (A.39) to obtain the non-null components

Sτ τ =β+ − β−8πG3R

(3.27)

Sθθ =β+ − β−8πG3R

. (3.28)

Inserting the particular choice of metrics (3.8) and (3.9), we arrive at the components of the stress-energy

tensor

Sτ τ =1

8πG3

√α2 + R2 −

√1 + R2

R(3.29)

Sθθ =1

8πG3

(R√

α2 + R2− R√

1 + R2

). (3.30)

On the other hand, we will assume that the shell is made of a perfect fluid, so it must have a surface

stress-energy tensor of the form

Sab = (λ+ p)uaub + phab (3.31)

where λ is the superficial mass density of the shell, p is its superficial pressure and ua its velocity vector,

given explicitly by

ua ≡ ∂ya

∂τ= (1, 0) (3.32)

or in covariant components

ua = habub = (−1, 0). (3.33)

12

The restriction of a perfect fluid thus results in the non-null components of the stress-energy tensor

Sτ τ = −λ (3.34)

Sθθ = p. (3.35)

Thus, comparing these components with Eqs. (3.29)-(3.30), it follows that

−λ =1

8πG3

√α2 + R2 −

√1 + R2

R(3.36)

p =1

8πG3

(R√

α2 + R2− R√

1 + R2

)(3.37)

Taking the static limit R = R = 0 in the above equations and using the definition of the shell’s rest mass

M = 2πRλ , (3.38)

we have immediately

M =1− α4G3

(3.39)

p = 0. (3.40)

In order to have a properly defined shell radius one has to impose α > 0. Imposing positive mass it

follows 0 < α < 1, whereas negative masses appear in the range 1 < α < ∞. From Eq. (3.40) we see

as well that in order for the shell to be static in a (2+1)-dimensional spacetime, its linear pressure must

vanish.

3.3 Thermodynamics, entropy equation for the shell, and sta-bility

We make the assumption that the shell possesses an entropy S which is a function of its rest mass M

and radius R. In this case, it is more fruitful to work with the variables M and A, where A = 2πR is the

perimeter of the shell, so

S = S(M,A) . (3.41)

Using these variables and assuming further that the shell has everywhere a local temperature T , the first

law of thermodynamics can be written as

TdS = dM + pdA . (3.42)

By putting p = 0 as given in Eq. (3.40), the first law for the (2+1)-dimensional thin shell spacetime

simplifies to

TdS = dM . (3.43)

Equation (3.43) can be integrated provided that the integrability conditions are satisfied. In this case

there is only one condition. It states that T = T (M), i.e., the temperature T is a function of the mass

M alone. This simple dependence stems from the fact that p = 0. Then, the most general expression for

S is

S(M) =

∫β(M)dM + S0 (3.44)

13

where β ≡ 1/T is the local inverse temperature of the shell at the equilibrium position r = R, and S0

is an integration constant. Note that β(M) is an arbitrary function of the mass which can be specified

once the specific matter fields that constitute the shell are known.

It is then possible to study the the local intrinsic stability of the shell at the thermodynamic level,

which is guaranteed as long as the inequalities(∂2S

∂M2

)A

≤ 0 , (3.45)

(∂2S

∂A2

)M

≤ 0 , (3.46)

(∂2S

∂M2

)(∂2S

∂A2

)−(

∂2S

∂M∂A

)2

≥ 0 , (3.47)

are satisfied, according to [30] (see Appendix B).

3.4 Two specific equations of state for the thin shell matter:Entropy and stability

3.4.1 The simplest equation of state

The simplest form that can be considered for β(M) is a power-law,

β(M) = γ G(1+u)3 Mu (3.48)

where we are assuming M ≥ 0, γ and u are free parameters with γ > 0 to guarantee positive temper-

ature and the factor G3 must be present for dimensional reasons. As shall be shown in Chapter 4, in

3+1 dimensions the Planck’s length lp, with lp =√G4 ~ (G4 being the gravitational constant in four-

dimensional spacetime and ~ Planck’s constant), appears naturally in the temperature of a thin shell,

since for a given mass there is always an intrinsic length associated to it (the gravitational radius of the

system), and so to have the correct units for the entropy one must resort to lp. However, here in flat 2+1

dimensions there is no intrinsic spacetime radius, and so Planck’s length does not appear in this analysis

at all. This problem is thus purely classic and G3, with units of inverse mass, suffices to set the scale.

When β(M) has the form given in Eq. (3.48), one can substitute this in (3.44) to get

S(M) =γ

u+ 1(G3M)

(1+u)+ S0 , for u 6= −1 , (3.49)

and

S(M) = γ ln(G3M) + S0 , for u = −1 . (3.50)

Although the values of the parameters γ and u cannot be calculated without first specifying the nature

of the matter fields, it is possible to constrain them such that physical equilibrium states of the shell are

possible. Starting with S0, it is natural to assume that a zero mass shell should have zero entropy, i.e.,

S(M → 0) =∫β(M)dM + S0 → 0. It is seen directly from Eq. (3.48) that the entropy diverges when

M → 0 for u ≤ −1. Therefore the above normalization condition can only be satisfied for u > −1 and

S0 = 0.

14

In relation to stability, it is seen that conditions (3.46) and (3.47) are automatically satisfied for any

u. On the other hand, one can find that condition (3.45) can only be satisfied provided that u ≤ 0. Thus

we conclude that assuming a power law equation of the form (3.48), stability of the shell is possible for

any M ≥ 0 as long as the parameter values of u are restricted to

− 1 < u ≤ 0 . (3.51)

One can also consider negative values of M , i.e., α > 1. The relation (3.48) would be of the same

form with the proviso that one takes the absolute values of M . The same results would follow.

3.4.2 A more contrived equation of state

Another possibility for β(M) could be a quadratic function in M , of the form β(M) = δ G(1+a)3 (M +

CM2)a, where δ and a are some parameters with δ > 0 to guarantee positive temperature, and C is

some constant. The constant C, however, has a natural connection to Eq. (3.39). Indeed, defining

m ≡M + CM2 we can solve for M , obtaining the physical solution M = −1+√

1+4Cm2C . Comparing with

Eq. (3.39), we can make the association C = −2G3 and so α =√

1 + 4Cm =√

1− 8G3m, thus arriving

at the natural quadratic expression for β

β(M) = δ G(1+a)3 (M − 2G3M

2)a. (3.52)

Equation (3.52) is most easily integrated in the variable m. Changing from M to m in Eq. (3.44), defining

the parameter η = δ/α, and changing back again to M , we obtain the entropy

S(M) =η

a+ 1G

(1+a)3 (M − 2G3M

2)(1+a) + S0 , for a 6= −1 (3.53)

and

S(M) = η ln(G3(M − 2G3M

2))

+ S0 , for a = −1 . (3.54)

Again, although the values of the parameters η and a cannot be calculated without first specifying

the nature of the matter fields, it is possible to constrain them such that physical equilibrium states of

the shell are possible. Using the same reasoning as in the last section, we assume that it is natural to

consider that S(M → 0) → 0. Looking at (3.52), it is seen that the entropy diverges when M → 0 for

a > −1, so the above normalization condition can only be satisfied for a > −1 and S0 = 0.

From Eq. (3.52), the stability equations (3.46) and (3.47) are automatically satisfied since the entropy

does not depend on A. Also, as mentioned above, we must have α > 0 to have a physical acceptable

solution. Considering Eq. (3.45), it is possible to show that it implies the inequality

(2a+ 1)α2 − 1 ≤ 0 . (3.55)

We can study the two cases, M > 0 and M < 0.

For M > 0, α is in the range 0 < α ≤ 1. In this case Eq. (3.55) is automatically satisfied if the

exponent a obeys a ≤ 0. It is also satisfied for a > 0 but only if 0 < α ≤√

12a+1 . Equivalently, this

means that the rest mass M of the shell must be within the range

1−√

1/(2a+ 1)

4G3≤M <

1

4G3, for a > 0. (3.56)

15

For M < 0, we know that α > 1, and Eq. (3.55) requires additionally that α ≤√

12a+1 . In terms of

rest mass M , this represents the range

1−√

1/(2a+ 1)

4G3≤M < 0. (3.57)

However, since√

12a+1 ≥ 1 and 2a+ 1 > 0, we see that the analysis for M < 0 is only valid for parameter

values −1/2 < a < 0.

3.5 Conclusions

General relativity in a (2+1)-dimensional spacetime has no curvature in empty space but in matter

distributions curvature may still exist. Einstein’s equation thus still plays a role in determining the

required pressure and energy of a static thin shell (or ring in this (2+1)-dimensional setting). Indeed,

we have seen that in this situation the pressure must be zero and the rest mass of the shell must satisfy

Eq. (3.39).

Upon using the first law of thermodynamics we have found a specific differential equation for the

entropy of the ring which contained a degree of freedom encoded in the inverse temperature β. We have

chosen the two most simple ansatz for the inverse temperature, a power law on the shell’s rest mass

and a quadratic form of it, obtaining two distinct expressions for the ring’s entropy. This shell entropy

is purely classic, the only fundamental constant that enters into the problem is the (2+1)-dimensional

gravitational constant G3, which has unit of inverse mass. A thermodynamic stability analysis yielded

the range for the allowed parameters, revealing that the shell’s rest mass must be confined to a given

interval if the shell is to be stable.

We also note that these ring-shell spacetimes when extended into 3+1 dimensions, through the use of

a trivial coordinate z say, represent infinite cylinders. Thus this thermodynamic study also holds in 3+1

general relativity for those cylindrical thin shells.

16

4Entropy of a BTZ black hole through

thin matter shells

4.1 Introduction

The study of general relativity in 2+1 dimensions plays an important role on the analysis of systems

in curved spacetime. The decrease in dimensionality with respect to the usual 3+1 spacetime is relevant

for the simplification of calculations, leaving needless complications aside while keeping the essential

physical features. Black holes stand as specially important example of this, namely the BTZ black hole

[42] which was the first black hole solution discovered in a 3-dimensional spacetime. Such solution exists

only when a negative cosmological constant in present since, in the case where it is null, no curvature

exists in vacuum and therefore no gravitational attraction is present outside the matter distribution. The

discovery of this solution drew much attention throughout a variety of research fields [43–55], and is thus

considered to be an important example to be studied in this thesis as well.

Despite existing in 3 dimensions, BTZ black holes have an associated Hawking temperature as well as

a Hawking-Bekenstein entropy, just as in the case of 3+1 dimensions. In this chapter, we are interested

in a particular king of gravitating system, namely a thin shell with its surrounding spacetime, which can

be taken to the BTZ black hole limit. Since the shell is prone to a direct attack of its thermodynamic

properties, we can obtain the thermodynamic quantities of interest of a black hole through a method

entirely different from the usual black hole mechanics.

The calculations of the first three sections resemble those of Chapter 3. Indeed, the same methodology

is used, calculating the differential of the entropy of the shell by using the junctions conditions and first

law of thermodynamics and making a stability analysis of the parameters present in the ansatz for the

17

temperature of the shell. The fifth and last section takes the black hole limit of the results obtained,

where it is regained the Hawking-Bekenstein entropy of a BTZ black hole.


We consider a scenario similar to the one presented in Chapter 3 namely a ring situated in a hyper-

surface Σ which partitions a (2+1)-dimensional spacetime into two regions.

Inside the shell we use the coordinates xα− = (t, r, θ), where the metric has a non-null cosmological

constant Λ

ds2 = −(Λr2 + 1)dt2 +dr2

(Λr2 + 1)+ r2dθ2. (4.1)

On the exterior of the shell, the spacetime is described by the line element

ds2 = −(Λr2 − 8G3m)dt2 +dr2

(Λr2 − 8G3m)+ r2dθ2 (4.2)

written in the same coordinates as (4.1), where m the total energy of the shell. Since the coordinates

used in both sides of the shell are the same, so will the parametric equations that describe it, which will

be denoted again by r = R(τ) and t = T (τ).

For the coordinates on the hypersurface itself, we will choose once more ya = (τ, θ), which allows the

line element of the hypersurface to be written as

ds2Σ = −dτ2 +R2(τ)dθ2. (4.3)

Following the same calculations as in the previous chapter, we begin by computing the tangent vectors.

Since the coordinates used are the same in both cases, the resulting tangent vectors in this case are also

given by Eqs. (3.5)-(3.6).

Moving next to the components of the induced metric, one starts by noticing that the line elements

(4.1) and (4.2) are of the form

ds2± = −F±(r)dt2 +

dr2

F±(r)+ r2dθ2, (4.4)

similarly to Eq. (3.7), except that in this case F± are no longer constants but functions of the radial

coordinate, defined by

F+(r) = Λr2 − 8G3m (4.5)

F−(r) = Λr2 + 1. (4.6)

These definitions give cleaner and more general results. Calculating the components of the induced

metric, we find the same results as in Chapter 3

h±ττ = −F±(r)T 2 +R2

F±(r)(4.7)

h±θθ = r2. (4.8)

In fact, all the computations leading to Eqs. (3.27)-(3.28) in Chapter 3 are exactly the same as in

this case, so those results are also valid in this situation. Hence, the stress-energy tensor components are

18

obtained by inserting the definitions (4.5) and (4.6) on Eqs. (3.27)-(3.28), resulting in

Sτ τ =

√−8G3m+ ΛR2 + R2 −

√1 + ΛR2 + R2

8πG3R(4.9)

Sθθ =ΛR+ R

8πG3

(1√

−8G3m+ ΛR2 + R2− 1√

1 + ΛR2 + R2

). (4.10)

On the other side, assuming the shell is made of a perfect fluid, we also know that the stress-energy

tensor must have the form of Eq. (3.31) and so we have again that Sτ τ = −λ and Sθθ = p, since the

4-velocity vector ua is the same. Comparing the components of Sab obtained in both ways, we get

−λ =

√−8G3m+ ΛR2 + R2 −

√1 + ΛR2 + R2

8πG3R(4.11)

p =ΛR+ R

8πG3

(1√

−8G3m+ ΛR2 + R2− 1√

1 + ΛR2 + R2

). (4.12)

Finally, taking the limit of a static shell R = R = 0 and using the definition of the shell’s rest mass

(3.38), it follows that

M =

√1 + ΛR2 −

√−8G3m+ ΛR2

4G3(4.13)

p =ΛR

8πG3

(1√

−8G3m+ ΛR2− 1√

1 + ΛR2

). (4.14)

From Eq. (4.13) it is also possible to extract an explicit expression for m in terms of the rest mass of the

ring

m =−1− 16G3M

2 + 8G3M√

1 +R2Λ

8G3(4.15)

which will be useful in the sections to follow.


Proceeding now to the calculation of the entropy of the shell, we follow the same line of reasoning

as in Chapter 3, i.e. we express the entropy S as a funcion of the shell’s rest mass M and perimeter A,

while assuming a constant temperature T throughout the shell that validates the existence of the first

law of thermodynamics (3.42) of the system.

Inserting Eq. (4.14) and the differential of Eq. (4.13) in the first law, we obtain

dS = β(M,R)r+

√Λ

4G3Rkdr+. (4.16)

where r+ ≡√

8G3m/Λ is the gravitational radius of the ring, β ≡ 1/T is the inverse of the temperature

at the equilibrium position r = R and

k ≡√

1−r2+

R2(4.17)

is the redshift factor. Eq. (4.16) is integrable as long as the appropriate form for β is given. To find the

correct expression for the inverse of the temperature, it is necessary to note that the temperature plays

a role of an integration factor in the first law of thermodynamics, which implies that there will be an

19

integrability condition that must be specified in order to guarantee the existence of an expression for the

entropy, i.e. that the differential dS is exact. Such integrability condition states that(∂β

∂A

)M

=

(∂βp

∂M

)A

(4.18)

or, changing to the variables (r+, R), (∂β

∂R

)r+

=β

Rk2. (4.19)

It can be shown that the differential equation (4.19) has the analytic solution

β(r+, R) = Rk√

Λb(r+) (4.20)

where b(r+) is an arbitrary function of the gravitational radius. The presence of this function is due to

the underspecification of the matter fields which constitute the shell. Note that b(r+) has units of inverse

temperature and can be interpreted as the inverse of the temperature the shell would possess if located

at R =√

1/Λ + r2+, which can be seen from Eq. (4.20).

Inserting Eq. (4.20) in Eq. (4.16), one is led to the specific form for the differential of the entropy

dS = b(r+)r+Λ

4G3dr+ (4.21)

and integrating yields

S(M,R) =Λ

4G3

∫b(r+) r+ dr+ + S0 (4.22)

where S0 is an integration constant. Eq. (4.22) opens the possibility of studying the local intrinsic stability

of the shell at the thermodynamical level, which is guaranteed as long as the inequalities (3.45), (3.46)

and (3.47) introduced in Chapter 3 are satisfied.

4.4 A specific equation of state for the thin shell matter: En-tropy and stability

The most simple suggestion for b(r+) is a power-law equation of the form

b(r+) = α4G3

Λ

ra+

l(2+a)p

(4.23)

where α and a are free parameters and the factors 4G3/Λ and lp = G3~ must appear for dimensional

reasons, with lp being the Planck’s length in a 3-dimensional spacetime and ~ Planck’s constant. Inserting

Eq. (4.23) in (4.22), we get

S(M,R) =α

a+ 2

(r+

lp

)(a+2)

+ S0 . (4.24)

For a = −2 expression (4.24) is still valid by considering it as yielding a very low power in r+, i.e., it

gives a logarithmic function, S(M,R) = α ln(r+) + S0 for a = −2. Using these formulas for the entropy,

one is able to find the stability conditions imposed on the free parameters.

Despite the fact that the parameters α and a do not have specific values as long as the nature of the

matter fields is not specified, it is possible to constrain them nonetheless, such that physical equilibrium

states of the shell are possible. We start by noting that a zero mass (equivalantely r+ = 0) shell should

20

naturally have zero entropy, i.e., S(M → 0) → 0. Since Eq. (4.23) implies that the entropy diverges for

a ≤ −2, we can only have the above normalization if a > −2 and S0 = 0.

As for the stability equations, it is possible to show that Eq. (3.45) implies the inequality

m ≥ a

a+ 1

R2Λ

8G3(4.25)

which, together with the condition 8G3m ≤ R2Λ (from the square root in Eq. (4.13)), sets up the

restricted values for ma

a+ 1

R2Λ

8G3≤ m ≤ R2Λ

8G3. (4.26)

for a ≥ 0. If −1 ≤ a < 0, the lower limit will assume negative values, but the inequality is satisfied

nonetheless. For a < −1, the left half of the inequality will exceed the right half and thus a < −1 is

excluded. Turning now to Eq. (3.46), it leads to the relation

8G3m+ aR2Λ(1 +R2Λ) ≤ aR2Λ√

1 +R2Λ√−8G3m+R2Λ (4.27)

which, when used in conjuntion with Eq. (4.25), leads to

a(a+ 1)Λ2R4 + (a+ 1)(2a+ 3)R2Λ + (a+ 2)2 ≤ 0. (4.28)

The above relation can only be verified for a certain set of values for R, namely

0 ≤ R ≤

((2a+ 3) +

√(5a+ 9)/(a+ 1)

2(−a)Λ

)1/2

(4.29)

and the parameter a must also be restricted to −1 < a < 0. This automatically establishes a limit on m

as well through Eq. (4.26). Finally, regarding Eq. (3.47), one obtains the inequality

(a+ 1) 8G3m(1 +R2Λ)√

1 +R2Λ + (−(a+ 1) 8G3m+ aR2Λ)√−8G3m+R2Λ ≤ 0. (4.30)

Imposing at the same time the condition Eq. (4.26) in Eq. (4.30), we are left with

aR2Λ(1 +R2Λ)3/2 + (a− 1)R3Λ

√Λ

a+ 1≤ 0 (4.31)

which is satisfied in the parameter region of interest −1 < a < 0 and (4.29).

4.5 Thin shell entropy in the BTZ black hole limit

Given the differential of the entropy (4.21), it is interesting to ask oneself if in the limit R → r+ the

shell will possess an entropy equal to the Hawking entropy of a BTZ black hole

SBTZ =π

2

r+

lp. (4.32)

In order to answer such question, one begins by noting that it is expected that quantum fields are

inevitably present and that the back-reaction will diverge unless one chooses the matter to be at the

Hawking temperature [42]

TH =~

2πr+Λ (4.33)

21

which will fix the function b as

b(r+) =2π

~1

r+Λ. (4.34)

This is equivalent to choosing a = −1 and α = π/2 in the ansatz (4.23). Inserting Eq. (4.34) in Eq. (4.21)

and integrating, the entropy of the shell gives

S(M) =π

2

r+

lp(4.35)

which is exactly the same as the entropy of a BTZ black hole.

From the stability conditions one sees that the case where a shell is taken to the limit of a BTZ

black hole is thermodynamic unstable since Eq. (4.28) is not satisfied despite the fact that Eq. (4.26) and

Eq. (4.30) are. This is expected in other physical grounds: the shell emulates the behavior of a black

hole and the latter radiates with a spectrum of a black body, increasing its temperature as the mass

decreases. Therefore, the shell is also expected to radiate, thus rendering it impossible any chance of

thermodynamic stability since the temperature varies with time.

Finally, it is also interesting to note the case where we take the shell to it’s gravitational radius but

leave the value of a undetermined. In that situation, the shell may be thermodynamically stable if the

parameter satisfies all the stability conditions and if the total energy m of the shell is within the range

0 ≤ m ≤(2a+ 3) +

√(5a+ 9)/(a+ 1)

16(−a)G3(4.36)

as can be seen from Eq. (4.29) after substitution of the radius R = r+.

4.6 Conclusions

In this chapter we have considered the metric for a BTZ black hole outside the shell and a vacuum

with cosmological constant inside it. The thin shell formalism allowed us to compute the necessary mass

and pressure of the shell required to achieve a static equilibrium, which in turn gave us the differential

of the entropy the shell upon substitution in the first law of thermodynamics. The expected degree of

freedom, due to the arbitrariness of the matter fields from which the shell is made of, was encoded in the

temperature b(r+). To parametrize said temperature, we used a power law of the Hawking temperature

of a BTZ black hole, resulting in the entropy (4.24).

The computation of the specific form of the entropy led to an analysis of the parameter regions for

which the ring is thermodynamic stable. We have seen that the radius must be within the interval (4.29)

and the parameter a must assume values between −1 < a < 0, from which we see immediately that the

BTZ black hole limit is automatically excluded as a thermodynamic stable system since it corresponds

to the case a = −1.

The presence of an interval for the parameter a has a natural physical explanation as well. When a

system has a given temperature, it radiates energy at a given rate thus decreasing its total energy m. At

the same time, since the gravitational radius r+ is proportional to√m, the inverse of the temperature

would decrease or increase, depending on the sign of a. However, in the case where a > 0, the temperature

would increase and the shell would forever be releasing energy to the point where it would be radiated

22

away and no thermodynamic equilibrium could be achieved. This shows that the parameter a must be

negative if there’s to be any hope of achieving thermodynamic equilibrium. On the other hand, the

temperature cannot keep decreasing for all times since we want it to have a constant value and not one

changing with time. This is where the pressure of the shell comes into play. When m decreases, so does

the shell’s mass M , while the pressure p gets greater, as can be seen from Eq. (4.13) and Eq. (4.14). In the

first law of thermodynamics (3.42), the term dM would then decrease, contributing to a decrease in the

temperature T , while the term pdA contributes to an increase. Therefore, while the energy radiated away

lowers the temperature, the pressure p increases it, thus making it possible to achieve a state where rates

of change in the temperature cancel each other, so that it remains constant. Finally, the parameter a must

have a minimum negative value, since the pressure cannot be guaranteed to compensate an arbitrarily

high change in the temperature. Hence we reach the conclusion that a must assume negative values

between d < a < 0, where d is some negative value, which we determined to be −1.

Finally, we found that if one used the Hawking temperature for the shell it would give an entropy

equal to the Bekenstein-Hawking entropy of a black hole, a result which is by no means trivial since there

is no reason a priori for a particular distribution of matter to have the same entropy as a black hole if it

has a constant Hawking temperature.

23

5Entropy of a d-dimensional blackhole through thin matter shells

5.1 Introduction

The idea that spacetime may be composed of more than 4 dimensions has been central to many

theories which attempt to unify fundamental forces. Examples of this are string theory [56, 57], which

proposes a quantum theory of gravity by assuming that particles are strings at the Planck length; the

Kaluza-Klein theory [58] which tried to unify the gravitational and electromagnetic fields by introducing

a fifth spatial dimension; or the Arkani-Dimopoulos-Dvali model, which eliminates the hierarchy problem

by postulating the existence of large extra dimensions [59–62], amongst many others. It is compelling,

therefore, to study a thin shell in a Schwarzschild spacetime with any number of dimensions d, which is

what is going to be worked in this chapter.

The procedure will be the same as in the last two chapters: through the junction conditions we

will retrieve the pressure and total energy of the shell, which will used together with the first law of

thermodynamics to derive an expression for the entropy. Afterwards, the degree of freedom associated

with the temperature of the shell will be parametrized by the most simple phenomenological function

possible, in order to make a thermodynamic stability analysis of the coefficients related to the microscopic

composition of the shell. Finally, the shell will be taken to its gravitational radius in order to recover the

Bekenstein-Hawking entropy of the Schwarzschild black hole in a d dimensional spacetime.

25


Consider a spherically symmetric timelike (d− 1) hypersurface Σ in a d dimensional spacetime. The

interior of the shell is empty, with a flat metric

ds2− = −dt2 + dr2 + r2dΩ2

d−2 (5.1)

where

dΩ2d−2 = dθ2

1 + sin2 θ1dθ22 + sin2 θ1 sin2 θ2dθ3 + . . . = dθ2

1 +

d−3∏i=1

sin2 θidθ2d−2 (5.2)

is the differential of the solid angle in d dimensions and the Schwarzschild coordinates xα− = (t, r, θ1, . . . , θd−2)

were used. Outside the shell, we use again the coordinates xα+ = (t, r, θ1, . . . , θd−2) and assume a

Schwarzschild metric

ds2+ = −

(1− 8πGd

(d− 2)Ωd−2

2m

rd−3

)dt2 +

(1− 8πGd

(d− 2)Ωd−2

2m

rd−3

)−1

dr2 + r2dΩ2d−2 (5.3)

where m is the total energy of the shell and Gd is Newton’s constant in d dimensions, defined so that

Einstein’s equation can be written as (see Appendix C)

Gab = 8πGdTab. (5.4)

As before, the evolution of the shell is parametrized by the equations r = R(τ) and t = T (τ), which

leads to the (d− 1)-dimensional induced metric on the shell

ds2Σ = −dτ2 +R2dΩ2

d−2 (5.5)

where the coordinates ya = (τ, θ1, . . . , θd−2) have been used.

Now that we have all the necessary input information, we begin as usual by calculating the tangent

vectors to Σ, which we know from the start to be the same in each side of the shell since the coordinates

used are the same as well. The tangent d component vectors are explicitly given by

eα±τ = eατ =∂xα±

∂τ= (T , R, 0, . . . , 0) (5.6)

eα±θ1 = eαθ1 =∂xα±

∂θ1= (0, 0, 1, . . . , 0) (5.7)

...

eα±θd−2= eαθd−2

=∂xα±

∂θd−2= (0, 0, 0, . . . , 1) (5.8)

Using the tangent vectors and the components of the metrics (5.1) and (5.3), we can compute the non-null

induced metric components. However, following the example of the previous chapters, we note that the

metrics in each side of the shell can be written in the form

ds2± = −F±(r)dt2 +

dr2

F±(r)+ r2dΩ2

d−2 (5.9)

where we define

F+(r) = 1− 8πGd(d− 2)Ωd−2

2m

rd−3(5.10)

F−(r) = 1. (5.11)

26

Computing the induced metric components through Eq. (A.8), we obtain

h±ττ = −F±(r)T 2 +R2

F±(r)(5.12)

h±θ1θ1 = r2 (5.13)

h±θ2θ2 = r2 sin2 θ1 (5.14)

...

h±θd−2θd−2=

d−3∏i=1

r2 sin2 θi. (5.15)

From the first junction condition (A.18) and by using the induced metric components (5.5) on the shell

itself, we are able to derive the relations

−F±(r)T 2 +R2

F±(r)= −1 (5.16)

r2 = R2 (5.17)

r2 sin2 θ1 = R2 sin2 θ1 (5.18)

... (5.19)

r2 sin2 θd−2 = R2 sin2 θd−2. (5.20)

Only the first equation is of use, namely in defining the quantity (3.16) which will simplify again the

notation of the results obtained. It is also convenient to calculate the components of the induced metric

with upper indexes. Recalling (3.17) and (3.18), we arrive at

hττ± = hττ =1

−F±T 2 + R2

F±

= −1 (5.21)

hθ1θ1± = hθ1θ1 =1

r2(5.22)

...

hθd−2θd−2

± = hθd−2θd−2 =1

r2 sin2 θd−2

. (5.23)

Turning to the calculation of the normal vector, we first need the 4-velocity vector

uα± ≡∂xα±∂τ

= eα±τ (5.24)

to which the normal vector is perpendicular. Thus, the normal vector can be

n±α = (−R, T , 0, . . . , 0) (5.25)

for example. The non-null components of the extrinsic curvature are then found by using the definition

(A.34). In particular, raising the first index, we obtain

Kτ±τ =

β±

R(5.26)

Kθk±θk =

β±R

(5.27)

27

where k = 1, . . . , d − 2. From this point, we can calculate the non-null components of the stress-energy

tensor through Eq. (A.39), which turn out to be

Sτ τ =(d− 2)

8πGd

β+ − β−R

(5.28)

Sθkθk =1

8πGd

(β+ − β−

R+ (d− 3)

β+ − β−R

). (5.29)

In the case d = 3 we recover Eqs. (3.27)-(3.28). Assuming now that the shell is made of a perfect fluid,

it must have a surface stress-energy tensor of the form

Sab = (σ + p)uaub + phab (5.30)

where σ is the superficial mass density of the shell, p is its superficial pressure and ua its (d−1) component

velocity vector, given explicitly by

ua ≡ ∂ya

∂τ= (1, 0, . . . , 0) (5.31)

or in covariant components

ua = habub = (−1, 0, . . . , 0). (5.32)

Combining (5.30) with (5.28) and (5.29), we arrive at the relations

−σ =(d− 2)

8πGd

β+ − β−R

(5.33)

p =1

8πGd

(β+ − β−

R+ (d− 3)

β+ − β−R

). (5.34)

The specific form of the shell’s total energy m can be obtained by defining the shell’s rest mass

M = Ωd−2Rd−2σ, (5.35)

and inserting it in Eq. (5.33). Using the definition of β± and solving for m, we get

m = M√

1 + R2 − µM2

2Rd−3(5.36)

where the quantity

µ =8πGd

(d− 2)Ωd−2(5.37)

was defined for the sake of notation simplicity. Substituting Eq. (5.36) in Eq. (5.34) and taking the static

limit R = R = 0 in both equations, we arrive at

m = M − µM2

2Rd−3(5.38)

p =(d− 3)µM2

2(d− 2)Ωd−2Rd−2(Rd−3 − µM). (5.39)

Eq. (5.38) is the Arnowitt-Deser-Misner energy, being the sum of the proper energy M and the gravita-

tional biding energy related to the process of building the shell [23, 63, 64]. Using Eq. (5.38), we are able

to solve for the rest mass of the shell, obtaining

M =Rd−3

µ(1− k) (5.40)

28

where

k ≡(

1−(r+

R

)(d−3))1/2

(5.41)

is the redshift factor and r+ ≡ (2µm)1/(d−3) is the Schwarzschild radius of the shell. Using Eq. (5.40)

and the definition of the k, we can express the pressure in the simpler form

p =(d− 3)(1− k)2

2(d− 2)Ωd−2Rkµ(5.42)

which in the case d = 4 gives the same result as in [26]. The results in d dimensions are also in agreement

with [65].


Assuming that the shell has a well defined temperature T along the entire shell, it will obey the first

law of thermodynamics

TdS = dM + pdA. (5.43)

Inserting in this expression the differential of the area of the shell A = Ωd−2Rd−2, the differential of

(5.40) and equation (5.42) for the superficial pressure, it is readily shown that

TdS =(d− 3)

2µkrd−4+ dr+. (5.44)

Defining β ≡ 1/T , one sees that the temperature assumes the role of an integration factor which can

make dS an exact differential for the entropy as long as the integrability condition(∂β

∂A

)M

=

(∂βp

∂M

)A

(5.45)

is satisfied. Changing variables from (M,R) to (r+, R), this equation becomes(∂β

∂R

)r+

=(d− 3)(1− k2)

2k2Rβ (5.46)

which has the analytic solution

β(r+, R) = b(r+)k (5.47)

where b(r+) is an arbitrary function of r+ alone, which reflects the arbitrariness of the matter fields which

constitute the shell. It is also interesting to note that Eq. (5.47) is Tolman’s formula for the temperature

in curved spacetime [66]. Since k → 1 as R → ∞, this function is the inverse of the temperature if the

shell was located at infinity. Eq. (5.47) can now be substituted in (5.44) to give the differential of the

entropy

dS =(d− 3)

2µb(r+)rd−4

+ dr+, (5.48)

which can be integrated to obtain the total entropy

S(M,R) =(d− 3)

2µ

∫b(r+)rd−4

+ dr+ + S0 (5.49)

with S0 being a constant of integration. The results (5.47) and (5.48) in the case d = 4 are in agreement

with [26]. Once an expression for b(r+) is provided, a thermodynamic stability analysis becomes possible

by use of Eqs. (3.45)-(3.47).

29


To proceed to a more detailed study of the thermodynamic properties of the shell, assume that b(r+)

is given by a power law function of the form

b(r+) ≡ η0

~ra(d−2)+1+

la(d−2)p

(5.50)

where a and η0 are free parameters and lp = (Gd~)1/(d−2) is the Planck length in a d-dimensional

spacetime. Defining η = (d− 3)η0Gd/(2µ), the choice (5.50) leads to

S(M,R) =η

(a+ 1)(d− 2)

(r+

lp

)(a+1)(d−2)

+ S0 , for a 6= −1 (5.51)

and

S(M,R) = η ln(r+) + S0 , for a = −1. (5.52)

This last equation is automatically ruled out, as well (5.51) for parameter values a ≤ −1, since the

entropy diverges for r+ → 0 or equivalently M → 0 where it is intuitive that we should have S → 0 as

M → 0. Therefore, Eq. (5.51) is physically acceptable for a > −1 as long as S0 = 0. The final expression

for the entropy with the choice (5.50) is therefore

S(M,R) =η

(a+ 1)(d− 2)

(r+

lp

)(a+1)(d−2)

(5.53)

for parameter values a > −1.

With an explicit expression for the entropy, it is now possible to make a thermodynamic analysis of the

parameter regions which guarantee the thermodynamic stability of the shell by applying the inequalities

(3.45)-(3.47). Starting with Eq. (3.45), it is satisfied for any value of a, as long as k is within the range

k ≤

√d− 3

(2a+ 1)d− (4a+ 1). (5.54)

From condition (3.46) it is possible to obtain the condition

k ≥ a− 2α(d)

a+ 2. (5.55)

where

α(d) ≡ d− 3

d− 2(5.56)

is always smaller than 1. For parameter values a ≤ 2α(d), any 0 ≤ k ≤ 1 is physically acceptable. On

the other hand, if a > 2α(d) then k is only allowed to be in the range

a− 2α(d)

a+ 2≤ k ≤ 1. (5.57)

Finally, regarding equation (3.47), it is possible to show that it implies the condition

(−4 + 2a+ 2d− 3ad+ ad2)k2 + (−6 + 12a+ 2d− 10ad+ 2ad2)k + 6a− 5ad+ ad2 ≥ 0. (5.58)

30

Focusing our attention for d > 3, the above inequality is satisfied for values of k in the region k ∈

[0, k−[∪]k+, 1], where

k± =3− a(d− 3)(d− 2)− d±

√(3 + 2a(1 + a(d− 2))(d− 2)− d)(3− d)

(2 + a(d− 1))(d− 2)(5.59)

Since the quantity inside the square root must be greater than zero, the parameter a must satisfy the

condition−1−

√2d− 5

2(d− 2)≤ a ≤ −1 +

√2d− 5

2(d− 2), (5.60)

which also satisfies the requirement a > −1.

5.5 Thin shell entropy in the limit of a black hole

We are now interested in taking the shell to the black hole limit to see what value the entropy takes.

As was the case in Chapter 4, the shell must possess a temperature equal to the Hawking temperature

TH due to the presence of quantum fields in the vicinity of the shell, namely [67]

TH =~

4π

(d− 3)

r+(5.61)

which means that the function b must be

b(r+) =4π

~r+

(d− 3)(5.62)

which is the same as selecting the values a = 0 and η = (d − 2)Ωd−2/4 in the ansatz (5.50). Inserting

this in Eq. (5.48) and integrating, it is shown that the entropy for the shell is

S(M) =Ωd−2 r

d−2+

4Gd~=Ashell

4ld−2p

(5.63)

where Ashell is the area of the shell. Since the area of the shell in the limit R → r+ is equal to the

area of the black hole, we see that the shell has the same entropy as a the Schwarzschild black hole in a

d-dimensional spacetime.

5.6 Conclusions

This chapter was dedicated to the study of a thin shell in d-dimensional Schwarzschild spacetime,

which generalized the study performed in [26]. The rest mass and pressure required for the shell to be

static where found by using the junction conditions, which in turn made it possible to find an expression

for the entropy of the shell by direct integration of the first law of thermodynamics, where an arbitrary

function of the gravitational radius naturally appeared. Following the example of the previous chapters,

we parametrized the undetermined function, which corresponded to the thermal equation of state of the

shell, by means of a power law, resulting in an explicit expression for the entropy.

A study of the intrinsic stability of the shell revealed that the parameter a must be constrained to

the interval (5.60), although there is no unique allowed range for k since the intersections of the intervals

obtained for it depend on the values of a and d.

We also observed the same behaviour of the shell when substituting the temperature with the expres-

sion for the Hawking temperature, giving the Bekenstein-Hawking entropy of a Schwarzschild black hole

in d dimensions.

31

6Entropy of a non-extremal charged

black hole through thin matter shells

6.1 Introduction

All the chapters worked out so far involved at most two state variables in the entropy equation of the

shells, namely their rest mass and radius. In this chapter, we will introduce another state variable in the

thermodynamics of the thin shell system, by investigating electrically charged shells. As a consequence,

this will considerably complicate the thermodynamic analysis of the shell, in particular the computation

of the stability regions for the parameters contained in the thermal equation of state, which is shown to

be naturally undetermined as in the previous chapters.

Apart from the increased computational complexity, charged of shells are particularly important since

they allow us to investigate the entropy of extremal black holes in two distinct ways. However, we will

restrict ourselves to the non-extremal case in this chapter, leaving the pursue for the extremal limit to

Chapter 7.


Consider the thin shell formalism in the case of a 4-dimensional electrically charged spherically sym-

metric timelike thin shell. The metric that correctly describes the exterior spacetime to the shell is given

by the Reissner-Nordstrom line element

ds2+ = −

(1− 2Gm

r+GQ2

r2

)dt2 +

(1− 2Gm

r+GQ2

r2

)−1

dr2 + r2(dθ2 + sin2 θdφ2

)(6.1)

33

where m is the total energy of the shell, Q its electric charge and the coordinates xα+ = (t, r, θ, φ) have

been used. This line element has the two natural horizons

r± = mG±√m2G2 −GQ2 (6.2)

where r+ is an event horizon and r− is a Cauchy horizon. As for the interior of the shell, since there is

no energetic content inside it, the metric will be flat, i.e.

ds2− = −dt2 + dr2 + r2

(dθ2 + sin2 θdφ2

)(6.3)

where the coordinates xα− = (t, r, θ, φ) have again been used. Due to the spherical symmetry of the shell,

it is convenient to use the spherical coordinates ya = (τ, θ, φ) on it, where τ is the proper time of an

observer situated at the shell. If t ≡ T (τ) and r ≡ R(τ) are the parametric equations of the shell, then

the induced metric will be that of a 2-sphere

ds2Σ = −dτ2 +R2(τ)

(dθ2 + sin2 θdφ2

). (6.4)

We can now calculate all the geometric quantities of interest. Before proceeding, however, it is fruitful

to note that not only the metrics (6.1) and (6.3) are of the form (5.9) in the particular case d = 4, but

also the coordinates used in each side of the shell are the same. Hence, all the quantities that need to be

calculated are obtained by fixing d = 4 in the results of Chapter 5, except that now the functions F±(r)

are defined as

F+(r) = 1− 2Gm

r+GQ2

r2(6.5)

F−(r) = 1. (6.6)

Using Eqs. (5.28)-(5.29) with d = 4, the non-null components of the stress-energy tensor are shown

to be

Sτ τ =β+ − β−

4πR(6.7)

Sθθ = Sφφ =β+ − β−

8πR+β+ − β−

8πR. (6.8)

On the other hand, the assumption of a perfect fluid leads to Sτ τ = −σ and Sθθ = p, where as before σ

is the mass density of the shell and p its pressure. Comparing both ways of computing the components

of Sab, one is led to

−σ =β+ − β−

4πR(6.9)

p =β+ − β−

8πR+β+ − β−

8πR. (6.10)

We can further develop these equations by using the definition of the shell’s rest mass M , which is

M = 4πRσ. (6.11)

Inserting this result in (6.9) along with the definition of β±, we can solve for m

m = M√

1 + R2 − GM2

2R+Q2

2R(6.12)

34

which is a result with a clear physical interpretation. Since m is the total energy of the shell, the first

term is the total kinetic energy and the second and third terms are the energy required to construct the

shell subject to its own gravity attraction and electric repulsion, respectively.

Inserting (6.12) in (6.10) and taking the limit of a static shell R = R = 0, we obtain the equations of

state for the pressure and total energy

m = M − GM2

2R+Q2

2R(6.13)

p =GM2 −Q2

16πR2(R−GM). (6.14)

Equation (6.13) can be solved with respect to the rest mass M , returning

M =R

G(1− k) (6.15)

where the redshift factor k was introduced, defined by

k ≡√

1− 2Gm

R+GQ2

R2=

√(1− r+

R

)(1− r−

R

). (6.16)

Note that when expressing k in the variables (R, r+, r−) instead of (R,M,Q), a certain symmetry nat-

urally appears. Indeed, some calculations will become more straightforward and give rise to symmetric

results if we make that change of variables. Thus, it is important to express the pressure as a function of

r+ and r− as well, giving

p =R2(1− k)2 − r+r−

16πGR3k. (6.17)

The results for the pressure and total energy are in agreement with [65, 68]


We now turn to the calculation of the entropy of the shell. To determine it, one needs the first law of

thermodynamics for a charged 2-dimensional system

TdS = dM + pdA− ΦdQ (6.18)

where dA is the differential of the area of the shell A = 4πR2, dM is the differential of the rest mass,

dQ is the differential of the charge GQ2 = r+r− and Φ is the electric potential of the shell whose form

is unknown yet, due to the fact that the junction conditions only allow the calculation of the rest mass

and pressure and not the electric potential.

In order to determine the form of Φ, first note that the temperature plays a role of an integration

factor, which implies that there will be integrability conditions that must be specified in order to guarantee

the existence of an expression for the entropy, i.e. that the differential dS is exact. Using the definition

35

β ≡ 1/T , these integrability conditions are(∂β

∂A

)M,Q

=

(∂βp

∂M

)A,Q

(6.19)(∂β

∂Q

)M,A

= −(∂βΦ

∂M

)A,Q

(6.20)(∂βp

∂Q

)M,A

= −(∂βΦ

∂A

)M,Q

. (6.21)

This system of equations leads to the differential equation(∂p

∂Q

)M,R

+1

8πR

(∂Φ

∂R

)r+,r−

+ Φ

(∂p

∂M

)R,Q

= 0 (6.22)

where the second term has been expressed in the variables (r+, r−, R) and the other terms in the variables

(M,Q,R) for simplicity. With this choice of variables, it is possible to show that the solution for the

differential equation is

Φ(R, r+, r−) =φ(r+, r−)−

√r+r−√GR

k(6.23)

where φ(r+, r−) ≡ Φ(∞, r+, r−) is an arbitrary function that corresponds physically to the electric

potential of the shell if it was located at infinity. On the other hand, equation (6.19) by itself leads to

specific form for β. Changing the variables (R,M,Q) to (R, r+, r−), that equation becomes(∂β

∂R

)r+,r−

= βR(r+ + r−)− 2r+r−

2R3k2(6.24)

which has the analytic solution

β(R, r+, r−) = b(r+, r−)k (6.25)

where b(r+, r−) ≡ β(∞, r+, r−) is the inverse of the temperature of the shell if its radius is infinite. Like

it happened in Chapter 5, we recover again Tolman’s formula for the temperature of a body in curved

spacetime [66]. The arbitrariness of this function is due to the fact that the matter fields the shell made

of are not specified. Inserting the results (6.23) and (6.25) in the first law (6.18), the differential of the

entropy is found to be

dS = b(r+, r−)1− c(r+, r−)r−

2Gdr+ + b(r+, r−)

1− c(r+, r−)r+

2Gdr− (6.26)

where c(r+, r−) ≡ φ(r+, r−)/Q. Altough it might seem at first that to integrate (6.26) two functions (c

and b) need to be given as input, that is not actually the case. In fact, Eq. (6.26) has its own integrability

condition if dS is to be an exact differential. Indeed, it must satisfy the equation

∂b

∂r−(1− r−c)−

∂b

∂r+(1− r+c) =

∂c

∂r−br− −

∂c

∂r+br+. (6.27)

This shows that only one function, b or c, needs to be specified a priori since the other remaining function

can be obtained by solving the differential equation (6.27) with respect to that function. We will choose

to always specify the function b first and from it derive the expression for c, although it wouldn’t make

any difference if was the function c to be specified and b to be derived from it.

From the moment we have the functions b and c specified, one can perform a thermodynamic stability

analysis by using the stability conditions for a charged system(∂2S

∂M2

)A,Q

≤ 0 (6.28)

36

(∂2S

∂A2

)M,Q

≤ 0 (6.29)

(∂2S

∂Q2

)M,A

≤ 0 (6.30)

(∂2S

∂M2

)(∂2S

∂A2

)−(

∂2S

∂M∂A

)2

≥ 0 (6.31)

(∂2S

∂A2

)(∂2S

∂Q2

)−(

∂2S

∂A∂Q

)2

≥ 0 (6.32)

(∂2S

∂M2

)(∂2S

∂Q2

)−(

∂2S

∂M∂Q

)2

≥ 0 (6.33)

(∂2S

∂M2

)(∂2S

∂Q∂A

)−(

∂2S

∂M∂A

)(∂2S

∂M∂Q

)≥ 0 , (6.34)

whose derivation is closely similar to the one in [30] for the case where there is no charge. The full

computations are presented in Appendix B.


To obtain an explicit expression for the entropy, we must first specify an adequate thermal equation

of state for b. The most simple possible choice is a power law of the form

b(r+, r−) = 2Ga(r+ + r−)α (6.35)

where a and α are free coefficients. The integrability equation (6.27) automatically fixes the function c

to be of the form

c(r+, r−) = 2Gd(r+r−)δ

(r+ + r−)α. (6.36)

Inserting these results in Eq. (6.26) and integrating, gives the entropy

S(M,R,Q) = a

[(r+ + r−)α+1

α+ 1− d (r+r−)δ+1

δ + 1

]+ S0 (6.37)

where we have considered only α > 0, a > 0 and d > 0 for the sake of simplicity of the upcoming

stability analysis. Although this choice somewhat narrows down the range of cases to which the analysis

is applicable, it only rules out the cases where −1 < α < 0, since for values α ≤ −1 it would give a

diverging entropy in the limit r+ → 0 and r− → 0, something which is not physically acceptable. Indeed,

in such limit we would expect the entropy to be zero, which not only requires α > −1 but also S0 = 0.

Also note that, unlike the case of Chapter 4 and 5, we don’t need the multiplicative parameters a and d

to be dimensionless because there is no pair of values (α, δ) which lead to the charged black hole limit,

thus making it irrelevant to explicitly show the dimensions since there is no way of comparing Eq. (6.37)

to the Bekenstein-Hawking entropy of a charged black hole.

37

Proceeding to the thermodynamic stability treatment, we start with Eq. (6.28), which can be shown

to be equivalent to

GQ2 − 2R2k2α+ (1− k2)R2 ≥ 0. (6.38)

Solving for k, this leads to the restriction

k ≤

√1

2α+ 1

(1 +

GQ2

R2

). (6.39)

Going now to Eq. (6.29), it gives

[GQ2 − (1− k)2R2

] [α(Q2 − (1− k)2R2) + 3(Q2 + (1− k2)R2)

]≤ 0. (6.40)

Since the second multiplicative term on the left must be positive, one can solve for k and obtain the set

of values which satisfy the inequality:

α

α+ 3−

√9

(α+ 3)2+GQ2

R2≤ k ≤ α

α+ 3+

√9

(α+ 3)2+GQ2

R2. (6.41)

As for Eq. (6.30), it reduces to

dR(2δ + 1)(GQ2)δ(GQ2

R + (1− k2)R)α ≥ R2(1− k2) + (2α+ 1)GQ2

R2(1− k2) +GQ2. (6.42)

Although one cannot conclude anything directly from the above inequality, it is nonetheless worth to

note that the right side is greater than zero, and so δ must obey the condition

δ ≥ −1

2. (6.43)

Regarding Eq. (6.31), it is possible to show that it implies the condition

G2Q4(α+ 3)− 2GQ2R2(2k2α+ 2k2 − k + α− 1) + (1− k)2R4(3k2α+ k2 + 2αk + α− 1) ≤ 0, (6.44)

which does not provide any information on itself since it is an polynomial of order 4 in the variable k.

Nonetheless, it does need to be satisfied once an region of allowed values for k is known, which will be

ascertained in a few moments.

Concerning Eq. (6.32), we are led to


R + (1− k2)R)α ≤

G2Q4(3α+ 1) + 2(1− k)GQ2R2(2α(k − 1) + 2k − 1)− (1− k)3R4(k(α+ 3)− α+ 3)

[(1− k)2R2 −GQ2] [(k − 1)R2(k(α+ 3)− α+ 3)], (6.45)

which does not contain any new information. On the other hand, when Eq. (6.33) is simplified to


R + (1− k2)R)α ≥ R2(1− k2) + (2α+ 1)GQ2 − 2R2k2α

R2(1− k2) +GQ2 − 2R2k2α(6.46)

and one notices that the numerator on the right side must be positive, another constraint on k naturally

appears, namely

k ≤√

1

2α+ 1+GQ2

R2. (6.47)

38

Finally, the last condition (6.34) gives the inequality

GQ2(α+ 1)−R2[(α+ 1)k2 + α− 1

]≥ 0 (6.48)

which constricts the values of k to be within the interval

k ≤√−α− 1

α+ 1+GQ2

R2. (6.49)

The definitive region of permitted values for k is the intersection of the conditions (6.39), (6.41), (6.47)

and (6.49). It is possible to show that such intersection gives the range

α

α+ 3−

√9

(α+ 3)2+GQ2

R2≤ k ≤

√−α− 1

α+ 1+GQ2

R2(6.50)

where α must be restricted to

α ≥1 + GQ2

R2

1− GQ2

R2

. (6.51)

Returning to Eq. (6.44), it is now possible to verify if the interval (6.50) satisfies said condition, which

indeed it does.

6.5 Thin shell entropy in the limit of a black hole

Since the result for the differential of the entropy of a thin shell is valid for any radius R, we can take

the limit of a black hole R→ r+ and see if the resulting entropy is consistent with (2.6).

Let us consider then the case of a charged thin shell, for which the differential of the entropy has been

deduced to be (6.26). Taking the limit R → r+ doesn’t change the functional form of the differential

since the dependence in R has been naturally eliminated, but it does imply an infinite pressure as the

shell reaches that limit, according to (6.17), so all the derived results have no physical meaning for radii

R < r+. Thus, we need only to give as input the inverse of the temperature of the shell at infinity

b(r+, r−).

At this point one must note that, as the shell approaches its gravitational radius, quantum fields are

inevitably present and their back-reaction will diverge unless we choose the Hawking temperature (2.5)

for the temperature of the shell at infinity, i.e.

b(r+, r−) =1

TBH=

4π

~r2+

r+ − r−. (6.52)

For a charged shell we must also specify the function c(r+, r−), whose form is fixed by the differential

equation (6.27) upon substitution of the function (6.52). Such equation can be analytically solved, yielding

the family of solutions

c(r+, r−) =a(r+, r−)(r+ − r−) + r−

r2+

(6.53)

where a(r+, r−) is an arbitrary constant of integration, which we are free to fix as a(r+, r−) = 1. Thus,

we have

c(r+, r−) =1

r+. (6.54)

39

Inserting this last result along with the function (6.52) in the differential (6.26) and integrating, we obtain

the entropy of the shell

S(M,Q) =πr2

+

l2p=

1

4

Ashell

l2p(6.55)

which is equal to the entropy of a charged black hole (2.6) since Ashell = ABH. Therefore, we are able to

reproduce the behaviour of a charged black hole by taking the shell to its gravitational radius.

We also note that the similarities between the thin shell approach and the black hole mechanics

approach are evident if we express the differential of the entropy of the charged shell (6.26) in terms of

the black hole mass m and charge Q, given in terms of the variables (r+, r−) by

m =r+ − r−

2G, Q =

√r+r−G

. (6.56)

Such differential becomes

T0dS = dm− c(r+, r−)QdQ (6.57)

where we have defined T0 ≡ 1/b(r+, r−) which is the temperature the shell would possess if located at

infinity. As we have seen, when we take the shell to its gravitational radius, we must fix T0 = TBH and

consequently c(r+, r−) = 1/r+, so the conservation of energy of the shell is expressed as

TBHdS = dm− Q

r+dQ. (6.58)

We see that the first law of thermodynamics for the shell is quite similar to the energy conservation of the

black hole (2.2) if we take the Bekenstein formula (2.4). Indeed, Eq. (6.58) suggests that Q/r+ should

play the role of the black hole electric potential ΦBH, which in fact is true, as shown in [25], for example.

6.6 Conclusions

This chapter introduced a new thermodynamic variable into the study of self-gravitating systems by

considering the case of an electrically charged thin shell. As was the case in the previous chapters, the rest

mass and pressure were obtained using the thin shell formalism and the junction conditions which, upon

substitution in the first law of thermodynamics, led to a differential for the entropy. This time, however,

there were two undetermined functions instead of just the usual thermal equation of state, where the new

arbitrary function corresponded to the electric potential at infinity. This situation is naturally associated

to the fact that the first law of thermodynamics (6.18) has a new term involving the electric potential

of the shell, and so the presence of three different differentials leads to two more integrability conditions

apart from (6.19) which was the only one to appear in Chapters 4 and 5.

The fact that the differential for the entropy in its final form, i.e. with the rest mass and pressure

equations inserted, had two independent differentials implied that it also had a new integrability condition,

which related the two undetermined functions for the temperature and electric potential of the shell.

Therefore, choosing a specific form for one of the functions automatically fixed the other one. With this

in mind, the most simple ansatz was chosen for the thermal equation of state, resulting in the expression

(6.37) for the entropy of the shell.

40

Despite the increase in complexity in the thermodynamic stability analysis due to the existence of

four new stability equations, it was possible to obtain the unique range (6.50) for k, as well as the regions

of allowed values (6.51) and (6.43) for the parameters α and δ, respectively.

As for the case where the thermal equation of state was equal to the inverse of the Hawking tem-

perature, it was shown that the resulting entropy was equal to the Bekenstein-Hawking entropy of a

non-extremal charged black hole.

41

7Entropy of an extremal black hole

through thin matter shells: asolution to the debate

7.1 Introduction

The entropy of an extremal black hole is a subject of wide debate. On one hand, such a black

hole has zero temperature, according to the Hawking temperature formula, and so it should have zero

entropy according to one of the formulations of the third law of thermodynamics [30]. Hawking [69] and

Teitelboim [70] have also given some topological arguments which point to the same conclusion. On the

other hand, there is no convincing reason why the Bekenstein-Hawking entropy formula should not be

valid in the extremal case.

This chapter is dedicated to the study of the limit of an extremal black hole formed by taking a

charged shell to its gravitational radius. There are two ways it can be done: by taking a non-extremal

shell to its horizon and imposing the extremal condition; and to take an a priori extremal shell to the

black hole limit. There will be section dedicated to each one of these limits, culminating in a possible

solution for the existing controversy.

7.2 Entropy of an extremal black hole through a non-extremalshell

The first way we are going to employ to calculate the entropy of an extremal black hole is through a

non-extremal shell. Taking the shell to its gravitational radius, the mechanical equations of the shell are

43

obtained by inserting the limit R→ r+ in Eqs. (6.15)-(6.17), giving a rest mass

M =r+

G(7.1)

and a pressure

p =∞. (7.2)

The differential of the entropy remains of the form (6.26) since it is functionally independent of the radius

R, so the direct integration of the differential will follow the same steps as in Chapter 6, resulting in the

entropy

S(M,Q) =πr2

+

l2p=

1

4

Ashell

l2p. (7.3)

Imposing now the extremal limit Gm2 = Q2, or equivalently r+ = r−, the form of the above expression

remains intact. We arrive at the conclusion that the entropy of an extremal black hole should be given

by the usual Bekenstein-Hawking entropy.

7.3 Entropy of an extremal black hole through an extremal shell

The second way of obtaining the entropy of an extremal black hole is to consider the extremal condition

before taking the black hole limit, i.e., to consider a shell whose matter obeys the condition r+ = r−.

An extremal shell has the mechanical equations

M =r+

G(7.4)

and

p = 0, (7.5)

as can be seen from Eqs. (6.15)-(6.17). The entropy differential of such a shell is obtained by taking the

extremal limit in Eq. (6.26), leading to

dS = b(r+)1− c(r+)r+

Gdr+ ≡ f(r+)dr+ (7.6)

where b(r+) ≡ b(r+, r+), c(r+) ≡ c(r+, r+) and the function f is defined as

f(r+) ≡ b(r+)1− c(r+)r+

G. (7.7)

If we now take the shell to its gravitational radius, we must again follow the logic of Chapter 6 and choose

b(r+) = 1/TBH due to back-reaction effects. However, since the entropy differential depends only on r+,

there is no longer an integrability condition that relates the functions b and c. Consequently, although

the function b is fixed, the function c is not, and so the function f is arbitrary. Thus we conclude that

the entropy of the extremal shell in the black hole limit is given by

S(M,Q) =

∫f(r+) dr+. (7.8)

This suggests that the entropy of an extremal black hole can assume any value, including the Bekenstein-

Hawking entropy, depending on the constitution of the matter that collapsed to form the black hole.

44

7.4 Conclusions

We have shown our solution for the ongoing debate concerning the entropy of an extremal black hole

obtained with the formalism of thin mater shells. Although it would necessary a full quantum theory

of gravity to fully understand the result obtained, it is nonetheless interesting to see that the use of

the junction conditions leads inevitably to the suggestion that extremal black holes are a different class

of objects than non-extremal black holes, due to the fact that their entropy seems to depend on the

particularities of the matter distribution which originated the black hole.

The result obtained here goes in line with the works of Lemos and Zaslavskii [28, 29] using quasi-black

holes.

45

8Conclusions

In this thesis we sought out to study the thermodynamic properties of black holes by using thin

matter shells and the junction conditions. The same procedure was applied to a variety of different

spacetimes, which consisted in imposing the junction conditions on the thin shell such that the interior

and exterior spacetimes to the shell formed together a single solution of the Einstein equations. This

led to the specific mass and pressure necessary for the shell to remain static. By inserting those in the

first law of thermodynamics, we were able to obtain the entropy differential in each situation, were the

thermal equation of state remained an arbitrary function of the gravitational radius of the system. An

ansatz was then given for this undetermined function, thus allowing the calculation of a specific entropy

as well as an analysis of the intrinsic thermodynamic stability of the shell. In the cases where the black

holes could exist in the given spacetime, the shells were taken to their gravitational radius, leading to

the Bekenstein-Hawking entropy. This result is by no means trivial, since there is no reason a priori for a

two dimensional system to have the Bekenstein-Hawking entropy once it was assumed to be at a constant

Hawking temperature throughout its distribution. In fact, since the shell is exactly at the event horizon

of the black hole when it is taken to its gravitational radius and in that limit the usual black hole entropy

is recovered, then we are strongly inclined to believe that this is evidence that the degrees of freedom of

a black hole are situated at its event horizon.

The particular case of a charged shell stood out for having an interesting application when the shell

was at its event horizon, more precisely when the shell was made of extremal matter. The entropy derived

from a non-extremal shell was shown not to change when the extremal condition was imposed if the shell

was already at the extremal limit. However, if the shell satisfies the extremal condition before being

taken to the black hole limit, then the result is entirely distinct, where it was proven that the shell gives

47

an entropy which depends on its constitution. Since every shell reproduced the black hole entropy when

considered at their gravitational radius, then this inevitably leads us to conclude that the entropy of an

extremal black hole shares the exact same dependence, thus suggesting that extremal black holes belong

to a different class of objects distinct from the non-extremal black holes.

48

AThin Shell Formalism

A.1 Introductory definitions

Consider a d−1 dimensional hypersurface Σ that partitions a d dimensional spacetime into two regions

V + and V −. Each region is covered by a coordinate patch xα±, where the plus or minus signs correspond

to the regions V + or V −, respectively. The problem we are interested in is the following: what conditions

must the metric satisfy in order for both regions to be smoothly joined at Σ?

To address this question, first assume that the hypersurface is parametrized by a coordinate system ya

which is the same on both sides of the hypersurface1. Suppose as well that a third continuous coordinate

system xα overlaps with xα+ and xα− in open regions of V + and V −. We will make all the calculations in

the xα coordinates but they are merely temporary since the final results will not depend on them. Now,

if

xα = xα(ya) (A.1)

are the parametric equations that describe the hypersurface, then differentiating (A.1) with respect to

ya yields the vectors

eαa =∂xα

∂ya, (A.2)

which are tangent to the lines of constant ya on Σ. Perpendicular to (A.2) are the normal vectors nα,

which we choose to point from V − to V +. To find such normal field, we start by piercing Σ orthogonally

with a congruence of geodesics and parametrize their proper distance l such that l > 0 in V +, l < 0 in

V − and l = 0 at Σ. This implies that a displacement away from the hypersurface along any geodesic

1Henceforth, greek letters will be used for indexes of the spacetime and roman letters will represent the indexes on thehypersurface.

49

will be of the form dxα = nαdl where dl is the infinitesimal proper distance from Σ to a point P along a

geodesic. We also have that

nα = ε∂l

∂xα(A.3)

where ε = nαnα. The only values that ε can have are +1 or −1, in which case the hypersurface is said

to be respectively timelike or spacelike. It will also prove useful to introduce the notation for the jump

of a quantity across Σ, that is

[A] ≡ A(V +) ∣∣

Σ−A

(V −) ∣∣

Σ(A.4)

where A is some function of spacetime. Since both xα, ya and l are continuous across Σ, we arrive at the

result

[nα] = [eαa ] = 0. (A.5)

It will be essential as well to use the concept of induced metric which arises when one wants to know

the metric on an hypersurface alone instead of on the whole spacetime. Its definition comes out naturally

using a path on the hypersurface. Let xα(λ) be a path in Σ, where λ is the curve parameter. Then an

infinitesimal line segment along this path is given by

dxα =dxα

dλdλ = eαady

a (A.6)

which can be substituted in the line element ds2 of the space time metric, giving

ds2 = gαβdxαdxβ = habdy

adyb (A.7)

where hab are the components of the so-called induced metric, given explicitly by

hab = gαβeαaeβb . (A.8)

The quantity (A.8) is invariant under a xα → xα′

transformation and transforms like a tensor for ya → ya′

transformations. Such a quantity is called a three-tensor. These tensors will play an important part in

defining a formalism independent of the coordinates xα. The induced metric, in particular, is used to raise

or lower indexes of three-tensors on the hypersurface which will be done frequently in the calculations

throughout this work.

A.2 First junction condition

The entire spacetime metric can be written using the language of distributions. More precisely, it can

be expressed as

gαβ = Θ(l)g+αβ + Θ(−l)g−αβ (A.9)

where g±αβ are the metrics in the regions V + and V − expressed2 in the coordinate system xα and Θ(l) is

the Heaviside distribution, defined as

Θ(l) =

+1, if l > 0

−1, if l < 0

indeterminate, if l = 0

. (A.10)

2From now on, all quantities with a + or - sign are to be interpreted as seen from V + or V −, respectively.

50

The distribution (A.10) has some important properties that will often be used in the calculations to

follow, namely

Θ2(l) = Θ(l), Θ(l)Θ(−l) = 0,d

dlΘ(l) = δ(l) (A.11)

where δ(l) is the Dirac distribution. The question of whether or not the spacetimes in each side of Σ

join smoothly is then equivalent to asking if the metric of the whole spacetime is a valid solution of the

Einstein equations

Rαβ −1

2gαβR = 8πGdTαβ (A.12)

where Rαβ is the Ricci curvature tensor, R ≡ gαβRαβ is the Ricci scalar curvature and Gd is Newton’s

gravitational constant in d dimensions. The natural units c = 1 were also used. The Ricci tensor is

obtained from the Riemann tensor through Rαβ = Rµαµβ , where the Riemann tensor has the explicit

form

Rαβγδ = Γαδβ,γ − Γαγβ,δ + ΓαγλΓλδβ − ΓαδλΓλγβ (A.13)

where the notation f,µ = ∂f∂xµ was used and Γ are the Christoffel symbols given by

Γαβδ =1

2gαλ (gλβ,δ + gλδ,β − gβδ,λ) . (A.14)

If (A.9) is to be a valid solution, then the geometrical quantities of which (A.12) is made of must be

correctly defined as distributions. The first concerning quantity that arises is the derivative of the metric

when calculating (A.14). Using equations (A.3) and (A.11), a simple calculation shows that

gαβ,γ = Θ(l)g+αβ,γ + Θ(−l)g−αβ,γ + εδ(l) [gαβ ]nγ . (A.15)

The first two terms are well behaved but the last one will induce terms of the form Θ(l)δ(l) when the

Christoffel symbols are calculated and terms like those are not defined as distributions. The only way to

avoid this is if

[gαβ ] = 0 (A.16)

but this holds only in the xα coordinates since from the beginning they are assumed to be continuous

across Σ, i.e. the condition [xα] = 0 is satisfied a priori and therefore so is (A.16). However, this doesn’t

need to be true in other coordinate systems, thus revealing the need for a relation which is independent

of the coordinates xα. This can be achieved by doing

[gαβ ] eαaeβb =

[gαβe

αaeβb

]= 0 (A.17)

where equation (A.5) was used. Since the quantity inside the square brackets is the induced metric (A.8),

which is a three-tensor, we arrive at the coordinate xα independent relation

[hab] = 0 (A.18)

also called the first junction condition. In other words, (A.18) states that the metric in Σ must be the

same viewed from either side of it. This condition must always be satisfied if the hypersurface is to have

a well defined geometry. It also implies the relation (A.16) while keeping the coordinate independence

which can also be seen by the fact that (A.18) produces only six conditions while (A.16) produces ten:

the difference corresponds to the four equations arising from the necessary additional condition [xα] = 0

when there is no coordinate independence.

51

A.3 Second junction condition

The result (A.18) from last section implies that the Christoffel symbols will not contain any prob-

lematic terms. Indeed, calculating (A.14) using (A.9) and the first junction condition, one is lead to the

result

Γαβγ = Θ(l)Γ+αβγ + Θ(−l)Γ−αβγ . (A.19)

Only the derivatives of this last expression are need to construct (A.13), which are straightforwardly

shown to be

Γαβγ,δ = Θ(l)Γ+αβγ,δ + Θ(−l)Γ−αβγ,δ + εδ(l) [Γαβγ ]nδ (A.20)

where the Dirac distribution shows up again, posing no mathematical issues. Using this, the Riemann

tensor is readily calculated, giving

Rαβγδ = Θ(l)R+αβγδ + Θ(−l)R−αβγδ + δ(l)Aαβγδ (A.21)

where

Aαβγδ = ε ([Γαβδ]nγ − [Γαβγ ]nδ) . (A.22)

Looking at (A.21), we see that there’s still a δ(l) term. Again, this term is not problematic per se since

it is well defined as a distribution but it does represent a curvature singularity at Σ. The term Aαβγδ is

even a tensor since the difference between two Christoffel symbols transforms like one. Thus, it is worth

to study this term by finding out the specific form for the δ(l) part of the Einstein equations.

We begin by noting that gαβ is continuous across Σ, thus its tangential derivatives must also be

continuous and so gαβ,γ can only have a discontinuity along the normal vector nα. Mathematically, this

means that there must exist a tensor καβ such that

[gαβ,γ ] = καβnγ (A.23)

which solved for καβ gives

καβ = ε [gαβ,γ ]nγ . (A.24)

Equation (A.23) can be inserted in the calculation of [Γαβγ ], resulting in

[Γαβγ ] =1

2(καβnγ + καγnβ − κβγnα) . (A.25)

By making use of this last expression, we arrive at the explicit form for the δ(l) part of the Riemann

tensor

Aαβγδ =ε

2(καδnβnγ − καγnβnδ − κβδnαnγ + κβγn

αnδ) . (A.26)

Contracting the first and third indexes, one obtains the singular part of the Ricci tensor

Aαβ ≡ Aνβνδ =ε

2(κναn

νnβ + κνβnνnα − κnαnβ − εκαβ) (A.27)

where κ ≡ καα. Contracting the remaining indexes results in the singular part of the Ricci scalar

curvature

A ≡ Aαα = ε(καβn

αnβ − εκ). (A.28)

52

We can now construct the δ part of the Einstein equations, which is simply

Aαβ −1

2gαβA ≡ 8πGdSαβ (A.29)

where Sαβ is the associated stress-energy tensor. This expression appears in the total stress-energy tensor

among two others terms:

Tαβ = Θ(l)T+αβ + Θ(−l)T−αβ + δ(l)Sαβ . (A.30)

The first two terms are associated to the regions V ± so the δ(l) term must be associated to Σ, that is, it

is the surface stress-energy tensor of the hypersurface. This implies that when such stress-energy tensor

is non-null there must exist a distribution of energy where the hypersurface is located, also called a thin

shell. Writing out the terms explicitly, we get

16πεGdSαβ = κνβnνnα + κναn

νnβ − κnαnβ − εκαβ − (κµνnµnν − εκ) gαβ . (A.31)

Notice, however, that Sαβnβ = 0, or in other words, this stress-energy tensor is tangent to Σ and therefore

can be completely written in terms of tangent vectors to that hypersurface, like eαa . This means that

Sαβ = Sabeαaeβb , or conversely Sab = Sαβe

αaeβb , where Sab is a symmetric three-tensor. Decomposing the

metric in its normal and tangential components with respect to Σ

gαβ = habeαaeβb + nαnβ (A.32)

and multiplying both sides of (A.31) by eαaeβb , one can arrive at an expression for Sab, namely

16πGdSab = −καβeαaeβb + hmnκµνe

µme

νnhab. (A.33)

Note that even though καβ allows the calculation of explicit formulas, it would be desirable to express

those formulas as functions of more usual geometric quantities that characterize hypersurfaces. One such

important quantity is the extrinsic curvature, whose tensor components Kab are defined as

Kab = nα;βeαaeβb . (A.34)

Indeed, this quantity appears quite naturally in (A.33). To see this, we start by calculating the jump in

the covariant derivative of the normal vector components

[nα;β ] = − [Γγαβ ]nγ =1

2(εκαβ − κγαnβnγ − κγβnαnγ) (A.35)

where it was used the fact that [nα,β ] = 0 which in turn follows from (A.5). From this it is possible to

calculate the jump in extrinsic curvature, which is

[Kab] = [nα;β ] eαaeβb =

ε

2καβe

αaeβb . (A.36)

Defining K ≡ habKab and inserting (A.36) in (A.33), we obtain

Sab = − ε

8πGd([Kab]− [K]hab) , (A.37)

which gives a relation between the surface stress energy-tensor and the jump in extrinsic curvature. Now,

a smooth transition across Σ is the same as saying that there can be no δ(l) part in the Einstein equations

which can only happen if Sab = 0. From (A.37), it is immediately seen that this can only happen if

[Kab] = 0 (A.38)

53

The above relation is called the second junction condition and since Kab is a three-tensor, this condition is

also independent of the xα coordinates. Together with (A.38), they form the set of necessary conditions

that must be satisfied in order for the spacetimes in each side of Σ to connect smoothly. However,

although (A.18) must always be satisfied, equation (A.38) needs not. If that’s the case, there is also a

physical interpretation to it: the smoothness in the transition across Σ no longer exists because there is

a thin matter shell present at the hypersurface, with stress-energy tensor given by (A.37).

Finally, notice that because of (A.5) the indexes in both junction conditions can be raised or lowered

freely without changing anything. In fact, the most direct and clean way of obtaining the surface stress-

energy tensor of the shell is by using one contravariant and one covariant index, i.e.

Sab = − ε

8πGd([Ka

b]− [K]hab) . (A.39)

54

BEquations of thermodynamic

stability for an electrically chargedsystem

In this appendix we shall show the derivation of the equations of thermodynamic stability for an

electrically charged system using the same approach as was used in [30] to derive Eqs. (3.45)-(3.47).

We start by considering two identical subsystems, each with an entropy S = S(M,A,Q), where M is

the internal energy of the system (equivalent to the rest mass), A is its area and Q its electric charge.

The usual state variables of a thermodynamic system are the internal energy U , volume V and other

conserved quantities N , like the number of particles, for example. However, the system we wish to study

is a thin shell, and thus it is more natural to use the variables (M,A,Q). Thermodynamic stability is

guaranteed if dS = 0 and d2S < 0 are both satisfied, or in other words, if the entropy is an extremum

and a maximum respectively.

Now suppose we keep A and Q constant and remove a positive amount of internal energy ∆M from

one subsystem to the other. The total entropy of the two subsystems goes from the value 2S(M,A,Q)

to S(M + ∆M,A,Q) +S(M −∆M,A,Q). If the initial entropy S(M,A,Q) is a maximum, then the sum

of initial entropies must be greater or equal to the sum of final entropies, i.e.

S(M + ∆M,A,Q) + S(M −∆M,A,Q) ≤ 2S(M,A,Q). (B.1)

Expanding S(M + ∆M,A,Q) and S(M −∆M,A,Q) in a Taylor series to second order in ∆M , we see

that Eq. (B.1) becomes (∂2S

∂M2

)A,Q

≤ 0 (B.2)

55

in the limit ∆M → 0. The same reasoning applies if we fix M and Q instead and apply a positive change

of area ∆A, so we must have

S(M,A+ ∆A,Q) + S(M,A−∆A,Q) ≤ 2S(M,A,Q). (B.3)

which in the limit ∆A→ 0 gives (∂2S

∂A2

)M,Q

≤ 0. (B.4)

If we fix M and A and make a positive change ∆Q on the charge, we have

S(M,A,Q+ ∆Q) + S(M,A,Q−∆Q) ≤ 2S(M,A,Q). (B.5)

and so it follows that (∂2S

∂Q2

)M,A

≤ 0. (B.6)

However, if we keep only one quantity fixed, like Q for example, we must also have a final sum of entropies

smaller than the initial sum if we apply a simultaneous change of area and internal energy rather than

separately, i.e.

S(M + ∆M,A+ ∆A,Q) + S(M −∆M,A−∆A,Q) ≤ 2S(M,A,Q). (B.7)

This inequality is satisfied by Eq. (B.2) and Eq. (B.4), but it also implies a new requirement. If we

expand the left side in a Taylor series to second order in ∆M and ∆A, and use the abbreviated notation

Sij = ∂2S/∂i∂j, we get

SMM (∆M)2 + 2SMA∆M∆A+ SAA(∆A)2 ≤ 0. (B.8)

Multiplying both sides by SMM and adding and subtracting S2MA(∆A)2 to the left side, allows the last

inequality to be written in the form

(SMM∆M + SMA∆A)2 + (SMMSAA − S2MA)(∆A)2 ≥ 0. (B.9)

Since the first term in the left side is always greater than zero, we see that it is sufficient to have(∂2S

∂M2

)(∂2S

∂A2

)−(

∂2S

∂M∂A

)2

≥ 0. (B.10)

This concludes the derivation of Eqs. (3.45)-(3.47). However, we can repeat the same calculations but

fixing M and A in turns. It is now straightforward to see that, when fixing M , we must have

SAA(∆A)2 + 2SAQ∆A∆Q+ SQQ(∆Q)2 ≤ 0, (B.11)

which is satisfied by (∂2S

∂A2

)(∂2S

∂Q2

)−(

∂2S

∂A∂Q

)2

≥ 0. (B.12)

Finally, by fixing A follows the inequality

SMM (∆M)2 + 2SMQ∆M∆Q+ SQQ(∆Q)2 ≤ 0 (B.13)

which implies the sufficient condition(∂2S

∂M2

)(∂2S

∂Q2

)−(

∂2S

∂M∂Q

)2

≥ 0. (B.14)

56

The last case left consists of doing a simultaneous change in all the state variables of the system, i.e.,

S(M + ∆M,A+ ∆A,Q+ ∆Q) + S(M −∆M,A−∆A,Q−∆Q) ≤ 2S(M,A,Q). (B.15)

To investigate the sufficient differential condition that this inequality implies, one must first expand

S(M + ∆M,A+ ∆A,Q+ ∆Q) and S(M −∆M,A−∆A,Q−∆Q) in a Taylor series to second order in

∆M , ∆A and ∆Q, which can be shown to lead to

SMM (∆M)2 + SAA(∆A)2 + SQQ(∆Q)2 + 2SMA∆M∆A+ 2SMQ∆M∆Q+ 2SQA∆A∆Q ≤ 0. (B.16)

Multiplying the above relation by SMM , noting that

(SMM∆M+SMA∆A+ SMQ∆Q)2 = S2MM (∆M)2 + S2

MA(∆A)2 + S2MQ(∆Q)2+

+ 2SMMSMA∆M∆A+ 2SMMSMQ∆M∆Q+ 2SMASMQ∆A∆Q (B.17)

and inserting this on Eq. (B.16), gives

(SMM∆M+SMA∆A+ SMQ∆Q)2 + (SMMSAA − S2MA)(∆A)2 + (SMMSQQ − S2

MQ)(∆Q)2+

+ 2(SMMSQA − SMASMQ)∆A∆Q ≥ 0. (B.18)

Recalling Eq. (B.10) and Eq. (B.14), and noting that the first term in the above inequality is always

positive, we conclude that the condition(∂2S

∂M2

)(∂2S

∂Q∂A

)−(

∂2S

∂M∂A

)(∂2S

∂M∂Q

)≥ 0 (B.19)

is sufficient to satisfy Eq. (B.15).

57

CNewton’s gravitational constant in d

dimensions

The generalization of Einstein’s equations to higher dimensions is automatic, since it is encoded in the

tensor indixes. However, Newton’s constant has to be generalized as well. In four dimensions, Einstein’s

equation is

Gµν = 8πGTµν (C.1)

where Gµν and Tµν are the Einstein and stress-energy tensors, respectively, and where we have used the

units c = 1. The most simple generalization of (C.1), and the one we use throughout this work, is

Gµν = 8πGdTµν (C.2)

where the indexes now run from 1, . . . , d instead of 1, . . . , 4 and Gd is the definition of Newton’s constant

in d dimensions which leaves Einstein’s equation in the form (C.2). To obtain such definition, we consider

the context of Newtonian gravity and recall the Poisson equation for the gravitational field

∇2φ = kρm, (C.3)

where φ is the gravitational potential, k is a constant and ρm is the density of matter. This equation

is the same for d dimensions. Integrating this equation over the space volume V and using the Gauss

theorem, we have ∫V

∇2φdd−1x =

∮Sd−2

(∇iφ)ni dSd−2 = k

∫V

ρm dd−1x = kM, (C.4)

59

where Sd−2 is the boundary surface surrounding the volume V , ni is the unit normal to the surface Sd−2

and M is the mass contained inside the volume. On the other hand, assuming spherical symmetry, i.e.,

(∇iφ)ni = −gr, (C.5)

where gr is the radial component of the gravitational field, one is led to∮Sd−2

(∇iφ)ni dSd−2 = −grSd−2 rd−2. (C.6)

Comparing Eq. (C.4) and Eq. (C.6), yields

gr = − k

Sd−2

M

rd−2. (C.7)

It is clear that k is Newton’s gravitational constant in d dimensions apart from some proportionality

constant. Simplicity would suggest the choice

k = GdSd−2 (C.8)

for k, as done in [59], since it gives

gr = −GdM

rd−2(C.9)

which is a straight generalization of Newton’s law of gravitation to d spacetime dimensions. However,

such choice implies that Einstein’s equation must be written as

Gµν =d− 2

d− 3Sd−2GdTµν (C.10)

which is not the desirable form we are interested in. Instead, the choice for k must be the one done in

[71], which is

k = 8πGdd− 3

d− 2(C.11)

because, despite leading to the slightly awkward gravitational force law

gr = −8πGdSd−2

d− 3

d− 2

M

rd−2, (C.12)

gives the correct form (C.2) for the Einstein equation.

This derivation of Newton’s constant in a d-dimensional spacetime closely follows that of [72].

60

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