Black holes in superstring theory - thermodynamic stability of five dimensional black holes in heterotic brane-worlds

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(1)“Black Holes in Superstring Theory: Thermodynamic Stability of Five Dimensional Black Holes in Heterotic Brane-worlds.”. By: Andrés Garcı́a Escovar. Graduate Project Presented in Partial Fulfillment for the Title of M.Sc in Physics Director: Dr. José M. Rolando Roldán December/2010.

(2) Acknowledgements Le doy gracias, primero que todo, a DIOS porque me dió la oportunidad de tener salud y hogar durante el curso de la maestrı́a. Agradezco a mis padres, por todo su amor y apoyo incondicional que me han brindado durante estos largos años de estudio. También quiero agradecer a mi novia, Paola Pinilla, por todo el amor comprensión y apoyo en lo momentos tanto fáciles cómo difı́ciles; a Myriam, Germán y Pacho porque también me apoyaron y me ayudaron en momentos crı́ticos. Le quiero agradecer a toda mi familia, que siempre estuvo ahı́ cuando los necesité. A mi mejor amigo, Miguel Dı́az, porque siento cómo si fueramos hermanos, aunque ya no nos veamos mucho; a Alejandro Ospina, porque se que a la distancia estaremos siempre conectados, primo. También estoy agradecido con mi supervisor Dr. José M. Rolando Roldán, principalmente por su paciencia y el tiempo que dedicó a atender mis dudas. Adicionalmente, quiero agradecerle a todo el departamento de Fı́sica de la Universidad de los Andes, por la grandiosa oportunidad y el honor que ha sido haber trabajado estos dos años ahı́. Le quiero también agradecer al grupo de Altas Energı́as y al grupo del Superstring Seminar (SSS), que estoy seguro será un grupo de investigación en crecimiento y productividad. Le quiero dar también gracias a PhD Comics por sus caricaturas siempre acertadas de cómo es la vida de un estudiante de posgrado; si, PhD Comics me entiende. . . Finalmente, le quiero dar gracias a todos los que tuvieron que vivir mi odisea por Facebook, por aguantar la larga crónica que fue este trabajo de grado, y con sus mensajes de apoyo; las notificaciones de like siempre me dieron ánimos para seguir adelante. Si por alguna razón siente que usted debe estar incluido en estos agradecimientos, no dude en hacerlo. Andrés Garcı́a E. Noviembre,18/2010. i.

(3) ii. Abstract In Superstring Theory, analogous objects to classical (non-quantum) Black Holes may appear as the result of considering the superposition of massive D-branes. The similarity of these objects with classical Black Holes has lead to the definition of microscopic quantities that resemble entropy, area and other geometrical characteristics that these objects possess. Since Superstring Theories are presumed to have information on quantum gravitational phenomena, the analysis of these properties is crucial for understanding the microscopic structure of Black Holes. Like for classical Black Holes, for Superstring Theory Black Holes there exist several solutions depending on the conditions imposed on the brane-world action. The aim of this project is to analyze the thermal stability/instability conditions for Black Holes that appear in Heterotic Superstring Theory. That is, starting from the general free action, in ten dimensions, and considering the coupling to external fields (such as Maxwell Fields), get the solutions of the induced metric on the world-sheet and analyze the behavior of the solutions, for variations of parameters such as mass, charge, angular momentum, metric and other fields, thus reaching a conclusion as to which conditions give thermodynamic stability, if any..

(4) Introduction The development of Superstring Theory (ST) as a unified field theory of the four known fundamental interactions has been a key step towards a quantum gravitational theory. With ST, phenomena and questions that General Relativity alone can’t explain, are expected to be cleared; some of these include: Is there a link between General Relativity with statistical mechanics? What lies beyond the horizon of a Black Hole? Are naked singularities allowed? Can the Bekenstein-Hawking entropy be obtained using statistical methods? These questions hope to be answered by a quantum gravitational theory. One of the questions this work tries to address is the issue regarding black hole stability. The mathematical proof by Hawking that Black Holes (BHs) can radiate with a black body spectrum, in the quantum regime, has led to the conjecture that BHs can evaporate. If a BH can evaporate, it means that there is an instability. It is thought that BH stability/instability in lower dimensions is due to the stability/instability of BHs in higher dimensions. This work tries to address the problem regarding thermodynamic stability of BHs in five dimensional heterotic string theory compacted on a T 5 ; it is motivated by the fact that Hawking radiation makes the area of the horizon decrease, thus reducing its entropy, δSBH < 0. If a BH were stable, the entropy would tend to increase, as the laws for classical BHs dictate [1]; thus, a simpler criteria should exist to analyze BH stability. The purpose of this work is to make progress towards this solution by perturbing the physical parameters that define the entropy, rather than by the methods used by others, and comparing the results. The problem is approached in the context of ST given that it is the only theory that has found a statistical interpretation for BH entropy, and has given the expected classical Bekenstein-Hawking entropy; making it suitable for extensive research. The work is divided into three parts: (I) the first part basically introduces the reader to useful subjects; in chapter 1 the historical development of ST and black holes is presented, with comments on future or immediate perspectives the area offers; in chapter 2 a brief introduction to black holes is presented, along with some of the relevant topics for part II; in chapter 3, a quick review of Kaluza-Klein theories and compactification of the heterotic string are presented. (II) iii.

(5) iv. The second part is the calculation; chapter 4 shows in detail how to take the action from the String Frame to the Einstein Frame; in chapter 5, the existence of the perturbation to the metric and the charges is discussed, this, to justify the perturbation method used; in chapter 6, the perturbation to the metric is performed and the relations to be used are defined; in chapter 7 the numerical study and the results are shown, concluding if the BHs are stable or not. (III) Finally, the third part consists of an appendix with the conventions used, the list of tables, the list of figures and the bibliography. It is recommended that the conventions are looked at before reading the document..

(6) Contents Abstract. ii. Introduction I. iii. Preliminaries. 1 Some History 1.1 More Relevant History. . . . . . 1.2 The Planck Scale . . . . . . . . 1.3 ST Historical Development. . . in 1.3.1 Including Fermions . . . 1.4 A Comment on Supersymmetry. 1 . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 2 2 3 4 6 7. 2 Black Holes 2.1 The Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . 2.1.1 The Real Nature of r = 2M . . . . . . . . . . . . . . . . 2.2 Other Black Holes in General Relativity . . . . . . . . . . . . . 2.2.1 Related Line Elements . . . . . . . . . . . . . . . . . . . 2.3 Black Hole Thermodynamics and The Four Laws of Black Holes 2.4 Black Holes in ST . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Maldacena or AdS/CFT Correspondence . . . . . . 2.4.2 The Entropy Function Formalism . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 8 8 10 10 10 12 13 14 14. 3 Compactification and the Heterotic String 3.1 Restrictions and the Einstein Equations . . 3.2 Common Compactification Surfaces . . . . 3.2.1 The Torus . . . . . . . . . . . . . . 3.2.2 Calabi-Yau Manifolds . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 15 16 17 17 18. II. . . . . . . Brief . . . . . .. . . . . .. The Calculation. . . . . .. . . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . .. 19. 4 The Einstein Frame 20 4.1 Changes in the Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 v.

(7) 4.1.1 The Christoffel Symbols . . . . . . . . 4.1.2 The Riemann Tensor . . . . . . . . . . 4.1.3 The Ricci Tensor . . . . . . . . . . . . 4.1.4 The Curvature Scalar . . . . . . . . . . 4.2 Changes in the Determinant of the Metric and 4.3 The Action in the Einstein Frame . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Fields . . . . . . . .. 5 The Metric and Perturbations to Einstein’s Equations 5.1 The Metric and the Christoffel Symbols . . . . . . . . . . . . 5.1.1 The Riemann Tensor . . . . . . . . . . . . . . . . . . 5.1.2 The Ricci Tensor and the Curvature Scalar . . . . . . 5.2 The Electromagnetic Field Tensor . . . . . . . . . . . . . . . 5.3 The Perturbation to the Equations: General Discussion . . . 5.3.1 A Comment on Checking Stability Using this Method. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 21 21 22 23 23 25. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 26 26 29 30 31 34 36. 6 Introducing a Generalized Perturbation and Consequences 6.1 Changes in the Fields and Scalar Curvature . . . . . . . . . . 6.1.1 The Christoffel Symbols . . . . . . . . . . . . . . . . . 6.1.2 The Riemann Tensor . . . . . . . . . . . . . . . . . . . 6.1.3 The Ricci Tensor . . . . . . . . . . . . . . . . . . . . . 6.1.4 The Scalar Curvature . . . . . . . . . . . . . . . . . . . 6.1.5 Changes in the Fields . . . . . . . . . . . . . . . . . . . 6.1.6 The Action . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Einstein Equations . . . . . . . . . . . . . . . . . . . . . . 6.3 Defining the Perturbation to the Charges . . . . . . . . . . . . 6.4 The Special Case h ∝ g . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 38 39 39 40 41 41 42 43 43 44 46. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 47 48 49 49 50. 7 Numerical Study 7.1 Charge Expansion . . . . . . . . . . 7.2 Choosing the Numerical Parameters 7.2.1 The Graphs . . . . . . . . . 7.3 Analysis and Conclusions . . . . . . III. Appendixes. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 53. A Conventions 54 A.1 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 List of Tables. 57. List of Figures. 58. Bibliography. 60.

(8) Part I. Preliminaries. 1.

(9) Chapter 1. Some History Superstring theory (ST) is considered by many physicists today as one of the most important theories in physics. The goal of ST is to unify the four fundamental interactions into a single theory. As it is well known, ST had its origins as a model for hadron interaction, based on work by Italian physicist Gabriele Veneziano in 1968, this model was known as a dual resonance model. As time passed, not much was accomplished by the dual resonance model and it was superseded by the development of quantum chromodynamics (QCD), as the model for strong interactions. It was in the early 1970s that it was realized that the Veneziano model could be understood as a model of quantum relativistic vibrating strings. Some years later, in the mid 1970s, John Schwarz and Joel Sherk propose string theory as a quantum theory of gravity. While the former was happening, in 1973 − 1974, the supersymmetric model was developed by Julius Wess and Bruno Zumino. The first “superstring revolution” took place in 1984, when it was noticed that, due to symmetries, the theory could be interpreted as a unification theory, the main people involved were Michael Green, John Schwarz and Edward Witten. It was not until the second superstring revolution, ten years later, when Joseph Polchinski and Petr Horava introduced the concept of further extended objects called D-branes, that helped to solve the problem of Dirichlet boundary valued conditions [2].. 1.1. More Relevant History. . .. The attempts of unification can be traced back to J.C. Maxwell in the second half of the 19th century. Maxwell successfully “unified” the electric and magnetic theory into a single one, now called the electromagnetic theory. After Einstein proposed his general theory of relativity in 1915, he tried to unify it with electromagnetism, along with the efforts of Theodor Kaluza and Oskar Klein, the fathers of the modern approach to compactification. In 1916 the first exact solution to the Einstein equations was found by Karl Schwarzschild. The solutions were further studied by Arthur Eddington and Georges Lemaître; they noticed singularities and discarded the singularity encountered at the Schwarzschild radius as a real singularity, using a suitable set of coordinates. The properties of these objects and singularities were extensively studied, until the correct interpretation was given by Julius Robert Oppenheimer (1939) and further discussed by David Finkelstein (1958). In general, little progress was 2.

(10) 3. 1.2. THE PLANCK SCALE. made during the 1940s and 1950s, and it was not until the 1960s that the important discoveries began to appear. In 1963 Roy Kerr found the general solution for these objects if they were rotating; in 1967 John Archibald Wheeler is said to have introduced the term “Black Hole” (BH) for these objects. Extensive work by Jacob Beckenstein led to the idea that the BHs could radiate as a black body, however this was not proven until 1975, when Stephen Hawking, using semiclassical methods of quantum fields propagating in curved spacetime backgrounds, mathematically showed that BHs emitted radiation in the quantum regime, which leads to the evaporation of BHs [3]. It can be shown that the radiation has a Planckian spectrum [4]. As a quantum theory of gravity, ST has to be able to handle extreme gravitational effects. As an example, the Bekenstein-Hawking entropy should be able to be derived from it, counting the microstates in BHs; furthermore, the correct expression for the entropy, at least in the macroscopic limit, should come out naturally. Since the establishment of ST as a unification theory of all the interactions, there have been attempts to give descriptions of BHs using ST. Pioneering work by Andrew Strominger and Cumrun Vafa showed that the entropy of four and five dimensional extremal BHs in type II B STs gave the expected Bekenstein-Hawking entropy by counting soliton states in the D-branes [5], that are also known as Bogomol’nyiPrasad-Sommerfield (BPS) states. The same year (just before [5] was published) Juan Martı́n Maldacena published his doctoral dissertation on BHs and ST, where he tried to describe the nature of microscopic properties of BHs [6]; leading to the Maldacena or AdS/CFT correspondence in 1997, that stated the relation between an anti-de-Sitter space times a five sphere and a conformal field theory in four spacetime dimensions [2].. 1.2. The Planck Scale. It is believed that in nature there are quantities that remain constant, almost independent of the scale that is chosen. Since these quantities set fundamental scales, the idea is to use the nature of these constants, Table 1.1, and use dimensional analysis to build physical quantities out of these constants. These quantities were introduced by Max Planck, arguing that they were the same everywhere in the Universe; because of this, they are called Planck quantities. Expressions for these physical quantities in SI units are: r ~G = 1.62 × 10−35 m (Planck Length), lp = 3 c r ~c mp = = 2.18 × 10−8 kg (Planck Mass), G r ~G = 5.39 × 10−44 s (Planck Time), tp = 5 c s Tp =. ~c5 = 1.42 × 1032 K 2 GkB. (Planck Temperature).. Quantities such as linear momentum, energy and angular momentum, for example, are defined and derived in terms of these quantities; this using their quantum relativistic definition..

(11) 4. CHAPTER 1. SOME HISTORY. It is thought that when this scale is approached, quantum gravitational interactions become meaningful; also, since a fundamental theory of gravity requires spacetime to emerge from fundamental quantities, the two above mentioned quantities, the Planck length and the Planck time, are thought to be outstanding candidates for the fundamental length scales; thus, they are thought to be the “quanta” of spacetime i.e. length or time intervals smaller than these values are not allowed, and only integer multiples of these quantities are allowed.. 1.3. ST Historical Development. . . in Brief. ST began with the research on dual resonance models by Veneziano, however these models do not account for the modern approach in ST. The first proposed model for ST began with the Nambu-Goto action, written by Yoichiro Nambu and Tetsuo Goto. The aim of the Nambu-Goto action is to define an invariant area over the world sheet in which the string lives: r Z 2 ∂X ∂X 02 0 2 S = −T dσ dτ Ẋ X − Ẋ · X , Ẋ = , X0 = ; (1.1) ∂τ ∂σ. where T denotes the tension of the string and having in mind that X is in general a vector. The Nambu-Goto action is equivalent to the Polyakov action: Z √ T d2 σ −h hαβ ηµν ∂α X µ ∂β X ν , h = det (h) . (1.2) S=− 2 The advantage of writing the action in the Polyakov form, is that it considers the world sheet metric hαβ separately from the string metric ηµν . Simple calculations show that the Polyakov action is Weyl invariant and parametrization invariant; that is, it keeps the form of the action Symbol. Quantity. c. 3.00 × 108 ms. G. 6.67 × 10−11 N·m kg2. kB. J 1.38 × 10−23 K. h. 2. 6.63 × 10−34 J · s. Brief Description. . . Speed of light in vacuum. Believed to be the top speed of an object, only non-massive objects are allowed to reach this speed. The special theory of relativity is based on this assumption. Newtonian Gravitational Constant. Constant first measured by Henry Cavendish. It appears in Newton’s universal law of gravitation as the proportionality constant in the inverse square force law. It is thought not vary as a consequence of the cosmological principle. Boltzmann Constant. It is believed to set the limit between macroscopic and microscopic physics. It relates the energy of a system to its temperature. Planck Constant. Sets the fundamental energy scale in nature. Introduced by Max Planck in 1900 as a fitting constant that fixed the ultra-violet catastrophe problem.. Table 1.1: Brief description of some quantities that are thought to be fundamental constants. The reduced Planck constant, ~ = h/ (2π), is used more often than h alone..

(12) 1.3. ST HISTORICAL DEVELOPMENT. . . IN BRIEF. 5. when the coordinates ~σ → ~σ 0 , where ~σ 0 denotes a new coordinate system; it is also invariant under the scaling of h, that is h → h0 = Λ (~σ ) h [7]: Z Z √ √ T T 0 0 αβ µ ν 2 (1.3) d σ −h h ηµν ∂α X ∂β X = − d2 σ 0 −h0 h0αβ ηµν ∂α X µ ∂β X ν ; S=− 2 2. the above action is a pure bosonic action. Of course, when solving the differential equation or the equations of motion, with the imposed constraints based on the equivalence with the Nambu-Goto action (that is not the subject of discussion), one must apply boundary conditions; the boundary conditions determine the type of strings one is dealing with. Closed strings must obey: X µ (σ, τ ) = X µ (σ + π, τ ) ;. (1.4). the periodicity of π is just a convention. As for open strings, the boundary conditions can be either imposed in the partial derivatives with respect to σ or with respect to τ , in both cases, equal to zero. The conditions are called Neumann and Dirichlet boundary conditions respectively: ∂X µ ∂σ. = 0. (Neumann boundary conditions). ∂X µ ∂τ. = 0. (Dirichlet boundary conditions).. These conditions are applied at the extremes of the string, namely σ = 0 or σ = π; of course, different types of conditions may be considered at different boundaries; these conditions have different implications in the physical meaning of the solution [8]. The most relevant type of strings for us are the heterotic strings, that are closed strings, and have solutions of the form: X µ (σ, τ ) = XRµ (τ − σ) + XLµ (τ + σ) ,. (1.5). where XR (τ − σ) and XL (τ + σ) represent right moving modes and left moving modes respectively. As in ordinary quantum field theory (QFT), the solutions can be written in terms of the Fourier modes: X 1 i xµ 1 2 µ µ µ −2i(σ−τ ) , (1.6a) + `s p (σ − τ ) + `s α e XR (σ − τ ) = 2 2 2 n n n∈Z/{0} µ X 1 x 1 2 µ i µ µ −2i(σ+τ ) XL (σ + τ ) = . (1.6b) + `s p (σ + τ ) + `s α̃ e 2 2 2 n n n∈Z/{0}. `s is defined as the string length; this length is in the Planck scale. With further analysis and after defining the algebra and the generators (Virasoro generators), along with the vacuum, one should be able to get a massive spin two particle, that is a graviton like particle [8]. Of course, this would only be a bosonic string, without fermions. The modes αn,µ and α̃n,µ may be regarded † as creation and annihilation operators in their respective sectors, considering α−n,µ = αn,µ and † α̃−n,µ = α̃n,µ . Thus, the vacuum defined by the theory can be excited in a way such that: |1, 1; µ, νi = α−1,µ α̃−1,ν |0i,. (1.7). it is seen that this state is a massless spin-2 particle that is identified as the graviton or the boson that carries the gravitational interaction [9]..

(13) 6. CHAPTER 1. SOME HISTORY. 1.3.1. Including Fermions. Since an only bosonic model is not realistic, fermions must also be included. The coupling to fermions is accomplished by using the action in equation (1.1) and adding a non-interacting term of the form: Z µ S = −T d2 σ ∂α X µ ∂ α Xµ − iψ ρα ∂α ψµ , (1.8) where ψ is a Majorana two component spinor, the ρα s are two 2×2 Dirac matrices and ψ = ψ † ρ0 [8, 9]. The Majorana spinor is defined to be: ψ+ . (1.9) ψ= ψ− After imposing boundary conditions on the spinors, for the action to be “well-defined” [8] one has to set, at σ = 0, ψ+ = ±ψ− , the positive sign is used by convention; at σ = π one finds that one may impose two conditions: R : ψ+ (π, τ ) = ψ− (π, τ ) ,. (1.10a). NS : ψ+ (π, τ ) = −ψ− (π, τ ) .. (1.10b). The R conditions are called Ramond conditions, first proposed by Pierre Ramond, and the NS conditions, that are called Neveu-Schwarz conditions proposed by André Neveu and John Schwarz. The R conditions make reference to the fermionic sector of the theory and the NS conditions to the bosonic case; this can be achieved by expanding the functions in Fourier modes and analyzing the boundary conditions [8]. After analyzing the on-shell mass condition, a negative mass state called a Tachyon appears. This Tachyon is eliminated using a GliozziScherk-Olive (GSO) projection [8, 9], whose aim is to remove unwanted modes that account for anomalies. In the bosonic sector involving the spinors, the functions are written in terms of the Fourier modes as follows: 1 X µ −in(τ +σ) µ ψ+ =√ , dn e 2 1 n∈Z+ 2. 1 X µ −in(τ −σ) µ ψ− =√ . bn e 2 1. (1.11). n∈Z+ 2. For the fermionic sector the expansion is similar, however the index n runs in Z [8, 9]. In the bosonic sector, the vacuum excitations are written as [9]: |n, m; µ, νi =. YY n,µ m,ν. {α−n,µ } {b−m,ν } |0i,. (1.12). where the operators labeled by αn,µ are the same operators as in equation (1.7), this theory also contains the graviton. Of course, these are very basic concepts given that existing anomalies are canceled using more sophisticated techniques..

(14) 1.4. A COMMENT ON SUPERSYMMETRY. 7. Why the Name “Heterotic”?. Heterotic ST is named this way due to the property that the theory can be divided in left and right moving modes; [9] defines this property as, literally, “heterosis”, where the right and left moving modes are treated independently. The word heterosis is translated as: “hybrid vigor”.. 1.4. A Comment on Supersymmetry. Supersymmetry had its origins in the search for a group that contained both the Poincaré group and an internal symmetry group, in a nontrivial union. The characteristics of the group that was being searched for were: 1. unitarity, 2. and finite dimensional; so that it could be finitely represented, given that a finite number of particles and a discrete mass spectrum is needed; however, the Coleman-Mandula theorem hindered this from happening [9]. The path to avoid the Coleman-Mandula theorem restriction turned out to be a Z2 -graded group algebra or a superalgebra. The idea behind string theory was to relate the internal symmetries in the Lagrangian that couples bosons and fermions and give generators such that these preserve the symmetries, and that relate the two types of particles: Q|Bosoni = |Fermioni,. Q|Fermioni = |Bosoni,. (1.13). [10]. The resulting group algebra turned out to be a superalgebra; this was first noticed by Bunji Sakita and Jean-Loup Gervais, in 1971. However, it was not until 1973 that Julius Wess and Bruno Zumino introduced the first supersymmetrical model for four dimensional spacetime [2]. Even though today we do not have experimental evidence that string theory is right; and will not probably have for some decades; there might be indirect evidence that at least a part of it is right. Using the “Minimal Supersymmetric Standard Model” (MSSM) predictions, at the LHC and at Fermilab in Batavia-Illinois, experiments have been designed to determine if there are indeed supersymmetric particles, whose masses are presumed to be in the 100 GeV to, 10 TeV scale, approximately [10]. If supersymmetry is found, a big step towards a unified field theory could be at hand; if supersymmetric particles are not found, it can only be argued that the MSSM needs to be corrected and/or the search will have to go on at higher energies..

(15) Chapter 2. Black Holes Black Holes (BHs) appear hidden in the solutions to Einstein’s equations in the form of real singularities, that is, singularities that can not be removed using a coordinate transformation, even more, some singularities, called imaginary singularities, can be removed using a coordinate transformation. The imaginary singularities, even if they can be removed using coordinate changes, are usually seen as “no-return” boundaries. Due to the nature of these objects, BHs where initially thought to be insatiable matter devourers and whatever went in would never go out again; that is, BHs could not radiate, implying that these objects had zero temperature. These ideas remained for over 55 years. It was not until the 1970’s that Stephen Hawking, with ideas from Jacob Bekenstein, showed that BHs had entropy, furthermore, that the entropy was proportional to the area of the event horizon. In 1973, Hawking showed that the radiation emitted by the BHs was thermal i.e. had the characteristics of a black body [3]. In the same year, Hawking (along with Brandon Carter and Werner Israel) proved the no-hair theorem, this leading to the characterization of BHs by their mass, electric charge and angular momentum. These quantities must be measurable from outside the horizon, of course. From today’s astronomical observations, it is believed that BHs are real objects and can be detected using indirect phenomena near them. From direct observations of several spiral galaxies, at the galactic cores it is believed that there are enormous BHs where matter is orbiting around them.. 2.1. The Schwarzschild Black Hole. When Karl Schwarzschild obtained the first solution to Einstein’s equations, he encountered a solution for the metric that contained particular features. The solution is usually given in terms of the line element ds2 = gµν dxµ dxν , where gµν denotes the metric components. The line element Schwarzschild found was: −1 2M 2M 2 2 dr 2 + r 2 dΩ2 , (2.1) dt + 1 − ds = − 1 − r r 8.

(16) 2.1. THE SCHWARZSCHILD BLACK HOLE. 9. in spherical coordinates. As can be seen, the line element contains two singularities: When r = 0; that is a real singularity. When r = Rs = 2M; that is an imaginary singularity. The value of R at this point is called the Schwarzschild radius. A hint of what the Schwarzschild radius is:. Although the Schwarzschild radius, Rs , first appeared in general relativity, and the right way to calculate it is using general relativity, a simple calculation using Newton’s law of gravitation can give an insight as to the nature of Rs . Using the equivalence principle, it can be shown that light is also influenced by gravity, as it was predicted by Einstein and shown by Sir Arthur Eddington in the eclipse experiment in 1919. Thus, the aim is to apply Newton’s law of gravitation to massive bodies and then extend the argument to massless particles, such as light. There is a very simple notion in classical physics that relates the minimum velocity needed to “escape” a planets gravitational influence to the radius of the planet and its mass; this velocity is called the escape velocity. Suppose one sends a rocket of mass m with velocity v from a perfectly spherical shaped body of mass M, where m << M, to space and we wish to find the escape velocity from this body; the idea would be to write the total energy of the rocket when it starts its engines and the final total energy at infinity, i.e. the rocket barely escapes the body’s gravitational influence. This can written as: r mM 2M 1 2 , Ef = 0 ⇒ Ei = Ef ⇒ vs = G . (2.2) Ei = mv − G 2 R R Thus, if one had the escape velocity and M, one could find the radius of the body by solving for R: R=G. 2M . vs2. (2.3). Notice that this equation does not depend on the mass of the rocket, or the object launched. If one were curious, one could ask: what would be the radius for light not to escape?, this is achieved by setting vs = c: Rs = G. 2M , c2. (2.4). in natural units c = G = 1; that is the definition of the Schwarzschild radius! In principle this could be done for the reasons discussed above. Thus, if a body were to have enough mass and the mass was concentrated in a radius less than or equal to the Schwarzschild radius, nothing inside that region would ever come out again; of course, this is in the context of classical mechanics. A detailed analysis with general relativity will show this to be correct, furthermore, it will reveal that causality is directed towards the real singularity r = 0 i.e. geodesics inside the horizon end in r = 0; this implies that not only will a particle be trapped, but will be located at r = 0..

(17) 10. CHAPTER 2. BLACK HOLES. Real singularities can be found by using the square of the Riemann tensor, namely, the Kretschmann scalar [11]: K = Rµανβ Rµανβ ,. (2.5). where Rµανβ denotes the components of the Riemann tensor. 2.1.1. The Real Nature of r = 2M. It is not clear from just the line element in equation (2.1) what will happen if a particle were to cross the boundary r = 2M. A simple analysis, without going into the details of changing to an appropriate geometry and for the sake of brevity, reveals what happens. Consider a particle that falls radially inwards, dθ = dφ = 0. When r < 2M, the line element switches signs in the time and radial component; that is, the time component changes into a spacelike component and radial component into a timelike component. Since particles must move slower than the speed of light, their paths are restricted to be timelike; the only way to accomplish this for ∆r to be smaller than zero; thus, a particle that has crossed the boundary r = 2M from the outside, it will never go back, neither will light rays. If a signal were to be sent from within the region r < 2M, it would never reach the outside of the BH.. 2.2. Other Black Holes in General Relativity. There are just a few number of additional BHs that appear in general relativity, these are characterized by physical quantities, due to the no-hair theorem. The following table shows names of the possible solutions that contain BHs and the quantities that characterizes them: Name Schwarzschild Reissner-Nördstrom Kerr Kerr-Newman. Characterized by M Q L X X X X X X X X. Table 2.1: List of BHs that are known and physically viable in general relativity. The can be characterized by a combination of physical quantities such as mass, M , electric charge, Q, and angular momentum, L.. 2.2.1. Related Line Elements. The line element for these solutions, from which the metric can be determined, are given by: Schwarzschild: −1 2M 2M 2 2 ds = − 1 − dt + 1 − dr 2 + r 2 dΩ2 . r r. (2.6).

(18) 2.2. OTHER BLACK HOLES IN GENERAL RELATIVITY. 11. This is the line element corresponding to the solution inside or outside a spherically symmetric body. If the mass is concentrated under the limit r = 2M, a BH is formed; r = 2M is the location of the horizon. Reissner-Nördstrom: . Q2 2M + 2 ds = − 1 − r r 2. . −1 Q2 2M + 2 dt + 1 − dr 2 + r 2 dΩ2 , r r 2. (2.7). This solution belongs to an electric charged spherically symmetric body, this solution can be found using Birkhoff’s theorem [11]. It can be seen that the condition for the metric to have a singularity is, grr = ∞: p r = r± = 2M ± 2 M 2 − Q2 .. (2.8). This condition tells us that there are two possible horizons. The first one, r+ , is the event horizon; the second one, r− , is called a Cauchy horizon. When Q = M, the horizon is said to be degenerate and the BH is said to be extremal. Kerr-Newman: This is the most general situation that has been found in general relativity that contains a BH solution (without violating the physical laws); written in BoyerLindquist coordinates: ds2 = −. 2 ρ2 2 2 sin2 (θ) ! 2 ∆ 2 2 dφ − a dt dt − a sin (θ) dφ r + a + dr + ρ2 dθ2 , (2.9) + ρ2 ρ2 ∆. where ρ and ∆ are defined as: ρ2 ≡ r 2 + a2 cos2 (θ) , 2. 2. (2.10a) 2. ∆ ≡ r − 2Mr + a + Q , L , (Angular Momentum per unit mass) a≡ M. (2.10b) (2.10c). [11]. It can be seen that, again, there are two horizons and the condition for these BHs to exist is: M 2 ≥ Q2 + a2 .. (2.11). BHs where M 2 = Q2 + a2 are said to be extremal and the horizon degenerate. In the case where there is no angular momentum, L = 0 → a = 0, the solution is clearly a Reissner-Nördstrom one. When both the electric charge and angular momentum are zero, Q = 0 and L = 0, the solution is clearly a Schwarzschild type solution. When only Q = 0, it reduces to the Kerr metric; thus, this can be thought as the most general metric that contains BH solutions in general relativity..

(19) 12. CHAPTER 2. BLACK HOLES. Figure 2.1: The idea of an ergosphere in a Kerr BH [12]. Some Comments. The idea of a Schwarzschild BH, although theoretically attractive, is not realistic due to the fact there are few chances that during a collapse a body will not have charge and/or, moreover, angular momentum i.e. the body’s rotation along an axis passing through it. Thus, the most realistic model would be the one describing a Kerr-Newman spacetime, considering axial symmetry. The Kerr BH introduces the feature of frame dragging, that is, a region of spacetime where a particle is confined to move in, but moves in an orbit without ever falling into the BH, for certain values of the energy [11]. This region is called the ergosphere and is located between r+ and r− (see Figure 2.1), where r+ and r− are given by the solution to ∆ = 0 in equation (2.10b), with Q = 0. For these BHs, there is an extremal solution when M 2 = a2 .. 2.3. Black Hole Thermodynamics and The Four Laws of Black Holes. When BHs were discovered, it was thought that these objects were completely black; that is, because of their classical properties, it was thought that once something went in, it would never come out again. Extensive work by Demetrius Christodoulou with reversible and irreversible transformations on BHs led to the idea that BHs could, in some way, behave like a thermodynamic ensemble [13]; furthermore, ideas by Bardeen, Carter and Hawking led to the formulation of the four laws of BHs [1]. At the same time methods to define quantities that resembled classical thermodynamic quantities for BHs were being developed; this was achieved by Jacob Bekenstein in the early 1970’s [14]. In a series of publications by Hawking [3, 15], he discovered that BHs could radiate and a temperature for this objects could be defined. Starting from the second law of thermodynamics: dE = T dS − P dV,. (2.12). the relation found by Bekenstein in [14] were: ~ · dL ~ + Φ dQ, dM = Θ dα + Ω where α is the rationalized area of the horizon: ~ A L 2 α= = r+ − a2 , ~a = ; 4π M. (2.13). (2.14).

(20) 13. 2.4. BLACK HOLES IN ST. and r+ denotes the outer horizon. The other terms are defined as: θ=. (r+ − r− ) , 4α. ~ = ~a , Ω α. Φ=. Qr+ ; α. (2.15). these quantities can be identified with temperature, angular frequency of rotation and the electric potential, respectively [14]. This definition of the second law of BHs is particularly useful if one is working in the classical regime.. 2.4. Black Holes in ST. When working with general relativity, one can obtain the field equations from Hamilton’s principle, using as a coordinate invariant Lagrangian the scalar curvature multiplied by the negative determinant of the metric: Z √ 1 −g R dx4 , (2.16) S= 16π treating the metric as the field1 . When extremizing the action with respect to the metric one gets the Einstein equations: 1 Rµν − Rg µν := 8πT µν , | {z2 }. g = det (g) ,. (2.17). Gµν. without considering a cosmological constant. As superstring theory is formulated using Hamilton’s principle, one defines a coordinate independent Lagrangian, that accounts for the energy of the fields present in the theory. It must be noticed that the effective action contains several fields that can be associated with the electromagnetic field tensor or the angular momentum tensor in higher dimensions. When extremizing the action in ST, assuming one starts with the vacuum equations (that is, the Lagrangian contains only the scalar curvature), with respect to the metric field, one gets equations that are analogous to Einstein’s equations, however, in several dimensions. The form of the equations may resemble a Schwarzschild, Reissner-Nördstrom or Kerr type of solution, with the so called “stringy corrections”, that are basically corrections induced by dimensional compactification. As there are more dimensions, there is a wider variety of BH solutions that will appear.. 1. This was noticed by Hilbert an computed by Emmy Noether, Hilbert’s student at the time. Einstein did the same some days after Noether, however, Hilbert finally gave the credit to Einstein..

(21) 14. 2.4.1. CHAPTER 2. BLACK HOLES. The Maldacena or AdS/CFT Correspondence. One of the most important advances in ST in the area of BHs is the AdS/CFT correspondence. In 1996 Juan Maldacena wrote his PhD thesis on superstring theory and BHs; in his thesis [6], Maldacena finds the correspondence of maximally symmetric BHs, that have an anti-de Sitter near horizon type metric, along with a duality transformation, resembles a supergravity theory in 1 + 1 dimensions. The supergravity theories are scale invariant and are often called conformal field theories or CFT for short. The first correspondence was initially found for AdS2 , however, this idea can be extended to various AdSn spacetimes as shown in [16]. A simple example regarding AdS2 can be found in [17]. 2.4.2. The Entropy Function Formalism. If a theory exhibits a continuous symmetry, a conserved quantity can be found; this is just the statement of Noether’s theorem. With the aid of the Noethers theorem, one can find a continuity equation that gives a conserved current. With this conserved current, one can find conserved quantities that are called Noether charges. In the early 1990’s, Robert M. Wald proposes that entropy can be viewed as a collection of Noether charges that are represented by Killing fields [18]. Wald managed to reproduce the form of the entropy proposed in [14], assuming a static Killing field; that was later extended to a proposal for dynamical entropy, carried by a dynamical Killing field [18]. Killing Field Definition. A Killing field is a field F such that the Lie derivative of the metric with respect to its vectors vanish: LF g = 0,. (2.18). or in terms of the vectors in the field F : Dµ xν + Dν xµ = 0;. (2.19). where Dµ denotes the covariant derivative. In [18], various assumptions limit the theories in which this is valid. One of the most important assumptions, is that the BHs treated are not extremal [18, 19]. The entropy function formalism developed in [19], allows for the entropy of extremal BHs, that are near-horizon AdS2 , to be computed using the Wald formula, both in the classical and quantum regimes [17, 19]..

(22) Chapter 3. Compactification and the Heterotic String One of the main features of ST is the appearance of extra dimensions for the theory to be consistent, that is, the theory should not have anomalies. These anomalies first appear in the Virasoro algebra when trying to stablish algebra satisfied by the generators; the algebra that is satisfied is: c ! 3 m − m δm+n,0 , (3.1) [Lm , Ln ] = (m − n) Lm+n + 12. [8, 9]. The term that contains the δm+n,0 is called a conformal anomaly and c is called the “central charge”. It turns out that c is the number of spacetime dimensions, d. When quantizing the theory using the Becchi-Rouet-Stora-Tyutin (BRST) formalism, one finds that the Fadeev-Popov ghosts cancel the anomaly term when d = 26 [8]. The 26 dimensions are interpreted in several ways; one of the interpretations is that these dimensions correspond to physical dimensions that are so small that can’t be perceived, or to extra degrees of freedom for which there are geometric analogs. The extra dimensions carry corrections to the action and to the equations of motion, this feature plays an important role in BH physics.. The idea of Kaluza-Klein theories developed in 1919, when Theodor Kaluza wrote to Einstein about the possibility of unifying the electromagnetic and gravitational field into a single theory [9]. After Einstein published his theory on general relativity, one of the first attempts to unify the electromagnetic theory and gravitation were performed by Theodor Kaluza and Oskar Klein. What Kaluza and Klein proposed, was to add an extra dimension to the general relativity metric: ĝµν Aµ4 gAB = , µ, ν = {0, 1, 2, 3} , A, B = {0, 1, 2, 3, 4} ; (3.2) A4ν φ where A4ν is proportional to the row four vector potential, Aµ4 the transpose of A4ν and φ an extra field. Based on this, one can impose certain restrictions on A4ν and ĝµν to get the appropriate equations. The strongest assumption is that the 5th dimension is curled up in a circle with such a small radius, that it can only be perceived at very high energies. 15.

(23) 16. CHAPTER 3. COMPACTIFICATION AND THE HETEROTIC STRING. 3.1. Restrictions and the Einstein Equations. The restrictions imposed on the metric elements of equation (3.2) are focused on reproducing the classical Einstein equations at low energies, that is, just taking into account the coupling to the massless Maxwell field Aµ , i.e. the photon. The restriction on the metric elements are defined as follows: ĝµν =gµν + κ2 Aµ Aν ,. (3.3a). A4ν =ATν4 = κAν ;. (3.3b). where T denotes the transposition operation, and κ a coupling constant. The idea of the 5th dimension being curled up into a small circle adds a restriction to the position coordinate: x4 = x4 + 2πR,. (3.4). that is, the 5th dimension should be periodic; the constant denoted by R is called the compactification radius. Of course, this is the simple case where the compactification is made on a circle of radius R. If one says that the theory was living in a five dimensional manifold M5 , with an almost1 arbitrary structure, the manifold is now represented as the product of two manifolds: M5 = M4 × S 1 ,. (3.5). where M4 denotes an Einstein manifold and S 1 a circle of radius R. The consequence of R being small can be interpreted as ∂5 → 0 [9]. One can calculate the Christoffel symbols using the definition (A.3); based on this, one can calculate the Ricci tensor and the scalar curvature: R=−. 1 1 Rµν gµν − Fµν Fµν + corrections, 2 2κ 4. thus, the action can be written as: Z κ2 1 µν 4√ ; S ≈ − 2 dx −g R + Fµν F 2κ 2. (3.6). (3.7). that are the Einstein equations coupled to the electromagnetic field [9]. This dimensional reduction technique can be applied to an arbitrary number of dimensions. The idea is to start with a theory of D dimensions, that we want to reduce to p dimensions, p < D, that is, we want to compactify D − p dimensions. This is achieved by assuming a manifold that can be expressed as the product of two manifolds: D M Rp × |{z} = |{z}. Initial Manifold 1. Final Manifold. D−p |S {z }. .. Compactification Manifold. This almost implies that restrictions concerning compactness, smoothness and other properties apply.. (3.8).

(24) 17. 3.2. COMMON COMPACTIFICATION SURFACES. 3.2. Common Compactification Surfaces. The surface on which one can compactify the dimensions is up to some point arbitrary, as long as the theory can be reduced to the number of required dimensions. The process of compactification may be difficult or physically unclear, however, some compactification surfaces for some theories have shown to give results in the expected path. 3.2.1. The Torus. The most common and most simple surface in which one dimension can be compacted is on a circle. The condition for dimensional reduction would be: MD = MD−1 × S 1 ,. (3.9). that is a dimension compacted on a circle. If one wanted to compact a second dimension, one could choose again another circle: MD = MD−2 × S 1 × S 1 ;. (3.10). the S 1 × S 1 surface is a torus T 2 . In general, if one wanted to compact p dimensions on circles one would get the product of the final manifold times a p−torus: MD = MD−p × T p ,. 1 1 Tp = S × . . . × S}1 . | × S {z. (3.11). p times. Compactification on a Circle. As a simple example, one could take the 26− dimensional bosonic string, and compact the action on a circle. The compactification process on a circle of radius R is quite simple and the implications of this can be further extended to tori. The idea is to start with a set of coordinates {X µ } and take a look at the string action: Z 1 S=− dσ 2 ∂α Xµ ∂ α X µ . (3.12) 2π As has been stated in chapter 1, X µ can be written as a solution in terms of the Fourier modes, as in equation (1.6), with X µ = X µ (σ, τ ); the idea is to make the coordinate periodic in the σ coordinate with dimension length 2πR, in mathematical terms: X µ (σ + π, τ ) = X µ (σ, τ ) + 2πRW ;. (3.13). where W denotes the times the dimensions winds the surface. Since the dimension is periodic, momentum becomes quantized in that dimension: pµ =. K , R. K ≡ Kaluza-Klein momentum,. K ∈ Z.. (3.14). Using arbitrary labeling of the spatial dimenions and because it is a 26−dimensional theory, the dimension chosen to be compactified will be the 25th dimension, i.e. µ = 25; thus, the.

(25) 18. CHAPTER 3. COMPACTIFICATION AND THE HETEROTIC STRING. mode expansion for the right and left movers can be expressed as, following [8, 20], and using equation (1.6): X 1 1 25 i 25 0K µ −2i(σ−τ ) XR (τ − σ) = x + α − W R (τ − σ) + √ `s αn e , (3.15a) 2 R 2 n∈Z/{0} n X 1 1 25 i 25 0K µ −2i(σ+τ ) XL (τ + σ) = x + α . (3.15b) + W R (τ + σ) + √ `s α̃ e 2 R n n 2 n∈Z/{0}. It must be noted that this α0 is the Regee parameter of the theory and is written in terms of `s as: α0 =. `2s ; 2. (3.16). [20]. The modification of the momentum changes the effective mass; To calculate the mass contribution due to this compactification requires much more background, so the result be given instead of calculated. The contribution to the mass, in terms of the mass operator will be [20]: 2. M =. M02. + 4 (NR + NL − 2) ,. NR =. ∞ X. m=1. µ α−m αµm ,. NL =. ∞ X. µ α̃−m α̃µm ;. (3.17). m=1. µ µ µ = α̃m , that are the exitacion operators for each mode, in their where α−m = αµ†m and α̃−m respective sector. The important thing to notice in this case is that the R sector and L sector both carry different field strengths due to compactification and mass correction terms; this will be useful later in chapter 6. †. 3.2.2. Calabi-Yau Manifolds. The most popular surfaces in which to compact are Calabi-Yau manifolds; Calabi-Yau manifolds are special manifolds that are defined as: “Kähler manifolds having n complex dimensions and vanishing first Chern class” [20]. Kähler manifolds can be considered as a product of a Riemann space, a complex manifold with U (n) structure and a symplectic manifold. Having this definition, the importance of these manifolds can inmediately be seen: these manifolds can shed information on symmetry breaking for gravity (i.e. the Riemann manifold) and Yang-Mills theories (i.e. SU (n), subgroup of U (n)), among others. Even thought the properties of these manifolds are not going to be used explicitly, it is worth mentioning them because these manifolds are the ones used nowadays to compactify the heterotic string, and are the most promising to reproduce the MSSM2 [8, 9, 20].. 2. See section 1.4 for comments..

(26) Part II. The Calculation. 19.

(27) Chapter 4. The Einstein Frame Passing to the Einstein Frame in ST is important given that it involves a transformation of the metric that decouples the scalar curvature from the other fields, turning the action into an Einstein-Hilbert action type, and an action defined in terms of what can be cosidered as a matter Lagrangian. One usually starts with an action of the type: Z 2 1 1 p 1 D ~ dx̃ (4.1) − H̃2 − F̃2 , −g̃ exp (−2φ) R̃ + 4 ∇φ S= 16π V 4 4 g̃. where the tilde, ∼, denotes we are working with the metric g̃. For D = 5, this is an action for N = 4 supersymmetry, in type II B theories compacted on a K3 × S 1 manifold (a Calabi-Yau manifold), or a heterotic theory compacted on a T 5 manifold [5]. We shall remember that, at this point, the square of the gradient of the dilatonic field is calculated using the metric g̃: 2 ~ ∇φ = g̃µν (∂µ φ)(∂ν φ). (4.2) g̃. The transformation we should use is: 4 φ gµν ; g̃µν = exp D−2. (4.3). also called a Weyl transformation [7]. We must check that this is a valid transformation. Using the definition of the metric (in component notation) and equation (4.3): −4 µν g̃ = exp φ gµν . (4.4) D−2 Thus, µσ. g̃ g̃σν. −4 4 µσ = exp φ g φ gσν = gµσ gσν exp D−2 D−2 = δνµ .. (4.5) 20.

(28) 21. 4.1. CHANGES IN THE SCALAR CURVATURE. Given that the Christoffel symbols are written in terms of the metric and its inverse, if the dilatonic field is not constant, the Riemann tensor will be modified and will induce changes in the action. We shall now look at the induced changes in the Riemann tensor once one passes to the Einstein frame, even more, how the Riemann tensor in the String Frame is written in terms of the Riemann tensor in the Einstein Frame, given the transformation in equation (4.3). With the changes in the Riemann tensor, one can find how the Ricci tensor and scalar curvature, in the String Frame, are written in terms of the Ricci tensor and scalar curvature in the Einstein Frame.. 4.1. Changes in the Scalar Curvature. We shall now write the Christoffel symbols in the String Frame in terms of the Christoffel symbols in the Einstein Frame, given the transformation in equation (4.3), and then examine how this affects the Riemann tensor, the Ricci tensor and scalar curvature. 4.1.1. The Christoffel Symbols. The Christoffel symbols are written in terms of the metric: 1 Γ̃αµν = g̃αβ (g̃βµ,ν + g̃βν,µ − g̃µν,β ) . 2. (4.6). Using the transformation in equation (4.3), and considering g̃µν,α = ∂α (g̃µν ):. Γ̃αµν. 4 4 −4 αβ ∂ν exp φ g φ gβµ + ∂µ exp φ gβν exp D−2 D−2 D−2 4 φ gµν −∂β exp D−2 ! αβ 2 1 g gβν φ,µ + gαβ gβµ φ,ν − gαβ gµν φ,β . (4.7) = gαβ (gβµ,ν + gβν,µ − gµν,β ) + 2 (D − 2). 1 = 2. One can now write the Christoffel symbols of the String Frame in terms of the Christoffel symbols of the Einstein Frame and corrections induced by the Weyl transformation: Γ̃αµν = Γαµν + 4.1.2. ! α 2 δν φ,µ + δµα φ,ν − gαβ gµν φ,β . (D − 2). (4.8). The Riemann Tensor. Now that we have the Christoffel symbols in the String Frame written in terms of the Christoffel symbols in the Einstein Frame, we can go ahead and write the Riemman tensor in the String Frame and see how this is written in terms of the Christoffel symbols in the Einstein Frame. One begins by writing the Riemann tensor in the String Frame: α R̃βµν = Γ̃ανβ,µ − Γ̃αµβ,ν + Γ̃ασµ Γ̃σνβ − Γ̃ασν Γ̃σµβ ,. symbol Γ̃ριγ ≡ inChristoffel the String Frame.. (4.9).

(29) 22. CHAPTER 4. THE EINSTEIN FRAME. We will concentrate in the third and fourth term of equation (4.9), these terms will give the main corrections for the scalar curvature once the Riemann tensor is contracted twice. We shall see first how the third term transforms and generalize the expression for the fourth term: The third term transforms as: ! α ! σ 2 2 α αλ α σ α δσ φ,µ + δµ φ,σ − g gσµ φ,λ δ φ,ν + δνσ φ,β − Γσνβ + Γ̃σµ Γ̃νβ = Γσµ + (D − 2) (D − 2) β σγ g gβν φ,γ ) 2 4 α σ α α αγ α α ~ =Γσµ Γνβ + ∇φ δ φ φ + δ φ φ − g g φ φ + 2δ φ φ − δ g ,µ ,ν ,µ ,β βν ,µ ,γ ,β ,ν βν β ν µ µ (D − 2)2 (4.10) −gαλ gβµ φ,λφ,ν − gαλ gµν φ,λ φ,β + gαλ gβν φ,λ φ,µ + corrections.. Thus,. Γ̃ασµ Γ̃σνβ. −. Γ̃ασν Γ̃σβµ. 2 4 α α α ~ + − g φ φ − δ φ φ − δ δ µ βν ∇φ ν ,β ,µ µ ,β ,ν 2 (D − 2) 2 ,α ,α α ~ (4.11) + gβν φ φ,µ − gβµ φ φ,ν + corrections. +δν gβµ ∇φ. =Γασµ Γσνβ. Γασν Γσβµ. Thus, the Riemann tensor in the String Frame will be written as, in terms of the Riemann tensor in the Einstein Frame: 2 2 4 α α α α α α ~ ~ R̃βµν = Rβµν + + gβν φ,αφµ g − δ ∇φ g φ φ + δ φ φ − δ δ µ βν ∇φ ν βµ ν ,β ,µ µ ,β ,ν 2 (D − 2) −gβµ φ,α φν ] + corrections 4.1.3. (4.12). The Ricci Tensor. We now make use of the definition of the Ricci tensor and obtain it by contracting the α index with the µ index in equation (4.12):. α α R̃βν = δαµ R̃βµν = R̃βαν. 2 2 4 α α α α ~ ~ ∇φ + ∇φ − δ g δ φ φ − δ φ φ + δ g = + βν ,β ,ν ,β ,α βα α α ν ν (D − 2)2 2 ,α ~ gβν ∇φ − gβα φ φ,ν + corrections α Rβαν. = Rβν. 2 4 ~ + (D − 2) φ,β φ,ν − (D − 2) gβν ∇φ + corrections. (D − 2)2. (4.13).

(30) 23. 4.2. CHANGES IN THE DETERMINANT OF THE METRIC AND OTHER FIELDS. Thus, the Ricci tensor in the String Frame can be expressed as the Ricci tensor in the Einstein Frame: 2 4 ~ R̃βν = Rβν + φ,β φ,ν − gβν ∇φ + corrections. (4.14) (D − 2) 4.1.4. The Curvature Scalar. Now that we have the Ricci tensor in the String Frame expressed in terms of the Ricci tensor in the Einstein Frame, equation (4.14), one can contract the Ricci tensor to get the curvature scalar:. 4φ µν R̃µν g R̃ = g̃ R̃µν = exp − D−2 2 4 4φ µν ~ + corrections g φ,µ φ,ν − gµν ∇φ Rµν + = exp − D−2 (D − 2) 2 2 4 4φ ~ ~ + corrections. ∇φ − D ∇φ R+ = exp − D−2 (D − 2) µν. . . (4.15). Thus, the curvature scalar in the String Frame may be written as the curvature scalar in the Einstein Frame and some corrections: 4 (D − 1) ~ 2 4φ R− ∇φ + corrections. (4.16) R̃ = exp − D−2 (D − 2). 4.2. Changes in the Determinant of the Metric and Other Fields. The determinant of the metric in the String Frame and the fields, when contracting the indexes, are also affected. To finish passing from the String Frame to the Einstein Frame, one has to know how these are affected by the Weyl transformation. Change in The Metric’s Determinant. √ D The action considered is defined to have √ an invariant hypervolume −g̃ dV , where dV = dx is the regular coordinate volume and −g̃ is the square root of the negative determinant of the metric tensor g̃ in the String Frame: g̃ = det (g̃) .. (4.17). One knows from basic linear algebra that if one has an n dimensional square matrix, one multiplies the matrix by a non-zero or non-singular scalar and one takes the determinant; the determinant will be that of the matrix without multiplying it, times the multiplication factor to the nth power: det (A) = A. ⇒. det (bA) = bn A.. (4.18).

(31) 24. CHAPTER 4. THE EINSTEIN FRAME. In this case, because the metric allows an inverse and has D × D entries, and using the Weyl transformation, equation (4.3):. det (−g̃) = det − exp = exp. . 4 φ g D−2. 4D φ det (−g) . D−2. One gets that the volume element transforms as: p √ 2D D φ −g̃ dx̃ = exp −g dxD . D−2. (4.19). (4.20). Change in The Fields. As the action is a scalar quantity, the two fields that appear in the previous action, appear contracted as: F2 = Fµν Fµν ,. H̃2 = Hµν Hµν ,. (4.21). where the contraction is made using the metric in the String Frame. Thus, if one were to make the contractions using the metric in the Einstein Frame: For example, if one contracts the indexes for field F̃: F̃2 = Fµν Fµν = g̃µρ g̃νλ Fµν Fρλ . 8φ = exp − gµρ gνλ Fµν Fρλ D−2. (4.22). The result follows for the field H̃ . The square of the fields F̃ and H̃ are written in the Einstein Frame as: 8φ 8φ µν 2 2 Fµν F , H̃ = exp − Hµν Hµν . F̃ = exp − D−2 D−2. (4.23). One must also be careful with the square of the gradient of the dilatonic field that was left outside the calculation because the square of the gradient is still taken with the metric in the String Frame (hence the metric used to contract the indexes is g̃). The coefficient to be taken into account is:.

(32) 4.3. THE ACTION IN THE EINSTEIN FRAME. Thus, . 4.3. ~ ∇φ. ~ ∇φ. 2. 2. g̃. g̃. = g̃ φ,µ φ,ν = exp − µν. = exp −. 4 φ gµν φ,µφ,ν . D−2. 2 4 ~ φ ∇φ . D−2 g. 25. (4.24). (4.25). The Action in the Einstein Frame. Finally, one can write the action in the Einstein Frame. Using equations (4.1), (4.16), (4.20), (4.23) and (4.25): Z √ 1 2D 4 (D − 1) ~ 2 4φ D S= −g exp ∇φ dx R− φ exp (−2φ) exp − 16π V D−2 D−2 (D − 2) 2 4φ 1 1 8φ 8φ 2 ~ +4 exp − ∇φ − exp − H − exp − F2 . D−2 4 D−2 4 D−2 (4.26) That can be simplified to: Z 4φ 4 ~ 2 1 1 D√ dx −g R − ∇φ − exp − H2 S= 16π V D−2 4 D−2 1 2D − 8 − exp φ F2 . 4 D−2. (4.27). This way, the action can be written as: Stot = SEH + Seff ,. (4.28). R√ 1 −g R and an where Stot is the total action, SEH is the well known Einstein-Hilbert action 16π effective action Seff that may be viewed as the energy-stress tensor introduced by the charged1 fields. One can see that if D = 5, the action in equation (4.27) reduces to the action in the Einstein Frame in [5]. As a final remark: one can see that the calculation, although not trivial, is not very hard to perform. The importance of this calculation lies in the fact that we can find a transformation that decouples the scalar curvature from the dilatonic field and hence obtain equations that allow us to perturb the differential equations into simpler parts. The equations to be analyzed are those for the metric with an electromagnetic stress-energy tensor. The equations are to be perturbed according on how the solutions to the Einstein equations are chosen; that will be the discussion of the next chapter. 1. The use of this word, charged, is in the context in which physical quantities such as electric/magnetic charge, mass and angular momentum are considered as the physical parameters that describe totally (?) a black hole..

(33) Chapter 5. The Metric and Perturbations to Einstein’s Equations The following calculations concern the perturbation of the Einstein equations with an electromagnetic field; this perturbation is only used to determine the dynamic stability of static BHs [21, 22]. The existence of this perturbation will allow to justify the perturbation induced to the metric in chapter 6, leading to the equations to be studied to check stability, and perhaps generalize the method to analyze thermodynamic properties, rather than dynamic properties. This calculation is based on [22], that besides considering the Einstein equations with an electromagnetic tensor, it introduces a cosmological constant.. 5.1. The Metric and the Christoffel Symbols. A metric is assumed, such that it is maximally symmetric. The line element taken into account is of the form: ds2 = gab (y) dy a dy b + r 2 (y) γij (z) dz i dz j ,. (5.1). meaning that the spaces in which y and z live are independent. Furthermore, the y coordinates that can be identified with the radial and time coordinates of the system, the z coordinates make reference to the “angular” components of the system, that have their own metric γij (z). From equation (5.1), we recognize that the indexes a and b span the y coordinates, {y}, and the indexes i and j those of the angular coordinates, {z}.Thus, if one were to represent the metric as a matrix, it would look something like: gab 0 gµν = . (5.2) 0 gij This can be seen as the product of manifolds N 2 ×Kn ; given the properties of the metric and the independence of the coordinates, the convention for this section will be that the Latin indexes a − g will all span the coordinates of the N 2 manifold whereas the Latin indexes h − p will all span the coordinates over the Kn manifold1 ; the Greek indexes will span all the coordinates. 1. The Kn manifold is said to be an Einstein manifold. An Einstein manifold is defined as a manifold in which the Ricci tensor is proportional to the metric.. 26.

(34) 5.1. THE METRIC AND THE CHRISTOFFEL SYMBOLS. 27. The metric γij (z) is symmetric, γij (z) = γji (z) and not necessarily diagonal γij (z) 6= 0, i 6= j; in general.. The Christoffel Symbols. To have the form of the Riemann tensor, the first thing is to obtain the Christoffel symbols from the most general form of the metric gµν , where, according to equation (5.1), the components gai = gia = 0. The Christoffel symbols are obtained using the definition: 1 Γµαβ = gµρ (gαρ,β + gβρ,α − gαβ,ρ ) . 2. (5.3). Given that gab and γij are both symmetric, the Christoffel symbols will be symmetric in their lower indexes. Thus, the Christoffel symbols are given by:. Γabc : 1 Γabc = gaµ (gbµ,c + gcµ,b − gbc,µ ) = 2Γabc , 2. (5.4). by definition.. Γijk : 1 Γijk = giµ (gjµ,k + gkµ,j − gkj,µ) 2 1 1 = gia (gja,k + gka,j − gkj,a) + gil (gjl,k + gkl,j − gkj,l ) {z } 2 |2 0. = Γ̂ijk (z) ,. (5.5). that are the symbols with respect to γij (z). We notice that the Christoffel symbols take this form because the radius, r (y), is not affected by the z derivatives..

(35) 28. CHAPTER 5. THE METRIC AND PERTURBATIONS TO EINSTEIN’S EQUATIONS. Γaij : 1 Γajk = gaµ (gjµ,k + gkµ,j − gkj,µ) 2 1 1 = gab (gjb,k + gkb,j − gkj,b) + gal (gjl,k + gkl,j − gkj,l) 2 {z } |2 0. . . 1 = gab gjb,k + gkb,j −gkj,b |{z} |{z} 2 0. 0. 1 = − gab gkj,b , 2 2 using gkj = r (y) γkj (z) ⇒ gkj,b = 2r (y) γkj (z) ∂b r (y), such that,. (5.6). Γajk = −r (y) γjk (z) ∂ a r (y) = −r (y) γjk (z) D a r (y) .. (5.7). We notice that the above equation may be written with a standard partial derivative ∂ a or a covariant derivative D a , given that the covariant derivative of a scalar is only the standard partial derivative. The reason we can do this is because a scalar does not have a direction and the covariant derivative makes reference to a specific direction.. Γija : Using the reasons from the previous deductions: 1 Γija = giµ (gjµ,a + gaµ,j − gaj,µ ) 2 1 1 = gil (gjl,a + gal,j − gaj,l ) + gib (gjb,a + gab,j − gja,b) 2 {z } |2 0. . . 1 = gil gjl,a + gal,j − gaj,l  , |{z} |{z} 2 0. 0. using gil = r 2 (y) γil (z) ⇒ gil,a = 2r (y) γil (z) ∂a r (y) and gil = Γija =. (5.8). δji δji ∂a r (y) = Da r (y) . r (y) r (y). γ il (z) , r 2 (y). thus: (5.9).

(36) 29. 5.1. THE METRIC AND THE CHRISTOFFEL SYMBOLS. Γabj : 1 Γabj = gaµ (gjµ,b + gbµ,j − gbj,µ ) 2 . . 1 1 = gal (gjl,b + gbl,j − gbj,l ) + gac  gjc,b +gbc,j − gbj,c  |{z} |{z} {z } 2 |2 0 0 0. but gbc,j. 1 = gac gbc,j , 2 = ∂j (gbc (y)) = 0, thus:. Γabj = 0.. (5.10). (5.11). Thus, the non-zero Christoffel symbols take the form: Γcab =2 Γcab ,. (5.12a). Γijk = Γ̂ijk (z) ,. (5.12b). Γaij = −r (y) γij (z) D a r (y) ,. (5.12c). Γija =. δji Da r (y) . r (y). (5.12d). Indeed, one must be careful with the mixed Christoffel symbols. It should be noticed that: Γija = Γiaj = 5.1.1. δji Da r (y) . r (y). (5.13). The Riemann Tensor. The next step in order to get Einstein’s equations is the Riemann tensor. For the sake of brevity, the calculation of some of the non-zero components of the Riemann tensor will be shown. The other components can be obtained by similar procedures and symmetries in the indexes. For this, we use the definition of the Riemann tensor as in equation (A.4) and expand the summation in the third and fourth summation terms of the definition, such that the summations are split in the indexes corresponding to the N 2 manifold and the Kn manifold: Rαβµν = Γαβν,µ − Γαβµ,ν + Γαµa Γaβν − Γανa Γaβµ + Γαµi Γiβν − Γανi Γiβµ .. (5.14).

(37) 30. CHAPTER 5. THE METRIC AND PERTURBATIONS TO EINSTEIN’S EQUATIONS. The calculation of the Riemann tensor over the N 2 manifold is really simple: Rabcd = Γabd,c − Γabc,d + Γaec Γebd − Γaed Γebc + Γaic Γibd − Γaid Γibc ,. Γif g = 0. = Γabd,c − Γabc,d + Γaec Γebd − Γaed Γebc =2Rabcd .. (5.15). The calculation of the Riemann tensor over the Kn manifold is a little more complex: Rijkl = Γijl,k − Γijk,l + Γika Γajl − Γila Γajk + Γikh Γhjl − Γilh Γhjk = R̂ijkl + Γika Γajl − Γila Γajk .. Using the expressions for the Christoffel symbols in equations (5.12a)-(5.12d), i δk i a Γka Γjl = Da r (−r (y) γjl (z) D a r) = − (Dr (y))2 δki γjl (z) . r (y). (5.16). (5.17). Finally: Rijkl = R̂ijkl − (Dr (y))2 δki γjl (z) − δli γjk (z) .. (5.18). The same calculations follow for the rest of the possible combinations of indexes. The components of the Riemann tensor will be given by: a Rbcd = 2Rabcd ,. i i Rjkl = R̂jkl − (Dr (y))2 δki γjl (z) − δli γjk (z) , a =− Ribj. D a Db r γij (z) . r. (5.19a) (5.19b). (5.19c). The totally covariant form of the Riemann tensor is as usual: σ Rαβµν = gασ Rβµν .. 5.1.2. (5.20). The Ricci Tensor and the Curvature Scalar. The Ricci tensor if defined as in equation (A.5); the ab components of the Ricci tensor are chosen such that it coincides with the Einstein equations:. Rab = Rσ aσb = Rc acb + Ri aib = 2R gab + Ri aib .. (5.21).

(38) 31. 5.2. THE ELECTROMAGNETIC FIELD TENSOR. We shall define: R̂ij = (n − 1) Kγij (z) ,. (5.22). where K is the sectional curvature and the trace of the metric’s angular components: γii (z) = n. One finds that, after a somewhat long algebra: Rab =. 12 Da Db r R gab − n , 2 r. Rai = 0,. (5.23a) (5.23b). r K − (Dr)2 + (n − 1) )γij (z) . (5.23c) r r2 The scalar curvature can be found by taking the trace of the Ricci tensor, Rµµ = gµν Rµν : Rij = (−. R =2R − 2n. 5.2. r K − (Dr)2 + n (n − 1) ; r r2. (5.24). The Electromagnetic Field Tensor. The perturbation to be used is a two-form electromagnetic field tensor that is defined as: 1 1 (5.25) F = E0 εab dxa ∧ dxb + fij dxi ∧ dxj , 2 2 where εab is the two dimensional Levi-Civita symbol. It is easy to prove that dF = 0, however one condition must be imposed for this to be possible. As a consequence E0 = E0 (y) and fij = fij (z), given that the spaces are disjoint. Before performing the calculation, it is necessary to remember that: G = aα dω α. ⇒. dG = ∂µ aα dω µ ∧ dω α .. (5.26). Given that the wedge product is antisymmetric, one must be careful with the order of the indexes. Thus, to have a consistent electromagnetic field, we require it to obey Maxwell type equations, dF = 0: 1 1 dF = ∂µ E0 εab dxµ ∧ dxa ∧ dxb + ∂µ fij dxµ ∧ dxi ∧ dxj 2 2 ⇒ dF =∂c E0 εab dxc ∧ dxa ∧ dxb + ∂i E0 εab dxi ∧ dxa ∧ dxb + ∂c fij dxc ∧ dxi ∧ dxj + ∂k fij dxk ∧ dxi ∧ dxj = 0.. (5.27). It should be verified that: dF = ∂c E0 εab dxc ∧ dxa ∧ dxb + ∂k fij dxk ∧ dxi ∧ dxj = 0.. (5.28).