DOCSLIB.ORG

Conserved Charges in Extended Theories of Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conserved Charges in Extended Theories of Gravity Conserved Charges in Extended Theories of Gravity Hamed Adami∗ Department of Science, University of Kurdistan, Sanandaj, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran Mohammad Reza Setare† Department of Science, Campus of Bijar, University of Kurdistan, Bijar, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran Tahsin Çağrı Şişman‡ Department of Astronautical Engineering, University of Turkish Aeronautical Association, 06790 Ankara, Turkey Bayram Tekin§ Department of Physics, Middle East Technical University, 06800 Ankara, Turkey (Dated: September 13, 2019) We give a detailed review of construction of conserved quantities in extended theories of gravity for asymptotically maximally symmetric spacetimes and carry out explicit compu- tations for various solutions. Our construction is based on the Killing charge method, and a proper discussion of the conserved charges of extended gravity theories with this method requires studying the corresponding charges in Einstein’s theory with or without a cosmo- logical constant. Hence we study the ADM charges (in the asymptotically flat case but in generic viable coordinates), the AD charges (in generic Einstein spaces, including the anti-de Sitter spacetimes) and the ADT charges in anti-de Sitter spacetimes. We also discuss the conformal properties and the behavior of these charges under large gauge transformations as well as the linearization instability issue which explains the vanishing charge problem for some particular extended theories. We devote a long discussion to the quasi-local and off- shell generalization of conserved charges in the 2+1 dimensional Chern-Simons like theories and suggest their possible relevance to the entropy of black holes. Contents arXiv:1710.07252v3 [hep-th] 12 Sep 2019 I. Reading Guide and Conventions 4 I GLOBAL CONSERVED CHARGES II. Introduction 4 III. ADT Energy 6 A. AD and ADM Conserved Charges 9 B. Large diffeomorphisms (gauge transformations) 12 C. Schwarzschild-(A)dS Black Holes in n-dimensions 14 2 D. Kerr-AdS Black Holes in n-dimensions 14 E. Computation of the charges for the solitons 16 1. The AdS Soliton 16 2. Eguchi-Hanson Solitons 17 F. Energy of the Taub-NUT-Reissner-Nordstr¨om metric 17 IV. Charges of Quadratic Curvature Gravity 18 µν V. Charges of f Rρσ Gravity 21 A. Example 1: Charges of Born-Infeld Extension of New Massive Gravity (BINMG) 25 VI. Conserved Charges of Topologically Massive Gravity 26 1. Logarithmic solution of TMG at the chiral point 31 2. Null warped AdS3 32 3. Spacelike stretched black holes 32 VII. Charges in Scalar-Tensor Gravities 33 VIII. Conserved Charges in the First Order Formulation 33 IX. Vanishing Conserved Charges and Linearization Instability 35 II CONSERVED CHARGES FOR EXTENDED 3D GRAVITIES: QUASI-LOCAL APPROACH X. Introduction 38 XI. Chern-Simons-like theories of gravity 39 A. First order formalism of gravity theories 39 B. Lorentz-Lie derivative and total variation 42 C. Gravity in three dimensions 43 D. Chern-Simons-like theories of gravity as the general 45 XII. Some examples of Chern-Simons-like theories of gravity 46 A. Example.1: Einstein’s gravity with a negative cosmological constant 46 1. Banados-Teitelboim-Zanelli black hole 47 B. Example.2: Minimal Massive Gravity 47 C. Example.3: New Massive Gravity 48 1. Rotating Oliva-Tempo-Troncoso black hole as a solution of new massive gravity 49 D. Example.4: Generalized Massive Gravity 50 E. Example.5: Generalized Minimal Massive Gravity 50 1. Solutions of Einstein gravity with negative cosmological constant 51 2. Warped black holes 52 XIII. Extended off-shell ADT current 53 XIV. Off-shell extension of the covariant phase space method 56 XV. Quasi-local conserved charge 57 XVI. Black hole entropy in Chern-Simons-like theories of gravity 58 XVII. Asymptotically AdS3 spacetimes 61 A. Brown-Henneaux boundary conditions 61 B. General massive gravity 63 3 XVIII. Asymptotically spacelike warped anti-de Sitter spacetimes in General Minimal Massive Gravity 65 A. Asymptotically spacelike warped AdS3 spacetimes 65 B. Conserved charges of asymptotically spacelike warped AdS3 spacetimes in General Minimal Massive Gravity 67 C. The algebra of conserved charges 68 D. Mass, angular momentum and entropy of warped black hole solution of general minimal massive Gravity 70 XIX. Conserved charges of the Rotating Oliva-Tempo-Troncoso black hole 72 XX. Explicit examples of the black hole entropy 73 A. Banados-Teitelboim-Zanelli black hole entropy 73 1. Minimal Massive Gravity 74 2. General Massive Gravity 74 3. General Minimal Massive Gravity 74 B. Warped black hole entropy from the gravity side 75 C. Entropy of the rotating Oliva-Tempo-Troncoso black hole 75 XXI. Near horizon symmetries of the non-extremal black hole solutions of the General Minimal Massive Gravity 76 A. Near horizon symmetries of non-extremal black holes 77 B. Near horizon conserved charges and their algebra for General Minimal Massive Gravity 78 XXII. Extended near horizon geometry 80 A. Application to the general minimal massive gravity 81 B. Soft hair and the soft hairy black hole entropy 83 XXIII. Horizon Fluffs In Generalized Minimal Massive Gravity 84 A. "Asymptotic" and "near horizon" conserved charges and their algebras 85 1. Asymptotic conserved charges 85 2. Near horizon conserved charges 86 B. Relation between the asymptotic and near horizon algebras 86 XXIV. Asymptotically flat spacetimes in General Minimal Massive Gravity 87 A. Asymptotically 2+1 Dimensional Flat Spacetimes 87 B. Conserved charges of asymptotically flat spacetimes in general minimal massive gravity 89 C. Thermodynamics 91 XXV. Conserved Charges of Minimal Massive Gravity Coupled to Scalar Field 93 A. Minimal massive gravity coupled to a scalar field 93 B. Field equations 94 C. Quasi-local conserved charges 96 D. General formula for entropy of black holes in minimal massive gravity coupled to a scalar field 98 E. Application for the BTZ black hole with ϕ = ϕ0 99 F. Virasoro algebra and the central extension 100 XXVI. Conclusions 102 XXVII. Acknowledgments 102 A. EQCA and Linearization of the Field Equations 102 B. Procedure of obtaining equations (331)-(336) 108 C. Technical proof of equation (338) 109 4 References 109 I. READING GUIDE AND CONVENTIONS This review is naturally divided into two parts: In Part I, we discuss global conserved charges for generic modified or extended gravity theories. Global here refers to the fact that the integrals defining the charges are on a spacelike surface on the boundary of the space. In Part II, we discuss quasi-local and off-shell conserved charges; particularly, for 2+1 dimensional gravity theories with Chern-Simons like actions. In what follows, the meaning of these concepts will be elaborated. Here, let us briefly denote our conventions of the signature of the metric and the Riemann curvature: We use the mostly plus signature with g = diag(−, +, ..., +) and take the Riemann tensor to be defined as ρ ρ σ [∇µ, ∇ν] V = Rµν σV , (1) and the Ricci tensor is defined as ρ Rµν = R µρν, (2) while the scalar curvature is µ R = R µ. (3) Part I GLOBAL CONSERVED CHARGES II. INTRODUCTION The first “casualty of gravity” is Noether’s theorem [1], well at least for the case of rigid spacetime symmetries and their corresponding conserved charges. The “Equivalence Principle” in any form is both a blessing and a curse: it makes gravity locally trivial (essentially a coordinate effect in the extreme limit of going to a point-like lab) but then it also, for the same reason, makes it hard to find local observables in gravity. In fact all the observables must be global. To heuristically understand this, as an example, consider the energy momentum density of the gravitational field: Being a bosonic (spin-2) field, its kinetic energy density is expected to be of the form K ∼ ∂g∂g which makes no covariant sense as it can be set to zero in a locally inertial frame (or in Riemann normal coordinates). This state of affairs of course affects both the classical gravity and the would-be quantum gravity. The first thing one wants to know and compute in any theory are the “observables” and it turns out generically, a spacetime has no observables and it would not make sense to construct classical and quantum theories for those spacetimes. For example, in a ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] 5 spacetime such as the de Sitter spacetime which does not have a global timelike Killing vector or an asymptotic spatial infinity, defining a global conserved positive energy is not possible [2]. In the absence of a tensorial quantity that can represent local gravitational energy-mass, mo- mentum etc, one necessarily resorts to new ideas; two of which are quasilocal expressions that try to capture gravitational quantities contained in a finite region of space or the global expressions that are assigned to the totality of spacetime. Both of these involve integrations over some regions of space, and they are not tensorial. But, that does not mean that they are physically irrelevant. On the contrary, they are designed to answer physical questions such as what is the mass of a black hole or how much energy is radiated if two black holes merge as in the recent observations by the LIGO detectors, of course with the assumption that these systems can be isolated from the rest of the Universe? Especially, global expressions that we shall discuss, the Arnowitt-Deser- Misner (ADM) [3] and the Abbott-Deser (AD) conserved charges [4], and their generalizations to higher derivative models the Abbott-Deser-Tekin (ADT) conserved charges [5, 6] are well-defined for asymptotically flat and anti-de Sitter spaces, given the proper decay conditions on the metric and the extrinsic curvature of the Cauchy surface.
Load more

Recommended publications
  • Some Aspects in Cosmological Perturbation Theory and F (R) Gravity
    Some Aspects in Cosmological Perturbation Theory and f (R) Gravity Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Leonardo Castañeda C aus Tabio,Cundinamarca,Kolumbien Bonn, 2016 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Peter Schneider 2. Gutachter: Prof. Dr. Cristiano Porciani Tag der Promotion: 31.08.2016 Erscheinungsjahr: 2016 In memoriam: My father Ruperto and my sister Cecilia Abstract General Relativity, the currently accepted theory of gravity, has not been thoroughly tested on very large scales. Therefore, alternative or extended models provide a viable alternative to Einstein’s theory. In this thesis I present the results of my research projects together with the Grupo de Gravitación y Cosmología at Universidad Nacional de Colombia; such projects were motivated by my time at Bonn University. In the first part, we address the topics related with the metric f (R) gravity, including the study of the boundary term for the action in this theory. The Geodesic Deviation Equation (GDE) in metric f (R) gravity is also studied. Finally, the results are applied to the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime metric and some perspectives on use the of GDE as a cosmological tool are com- mented. The second part discusses a proposal of using second order cosmological perturbation theory to explore the evolution of cosmic magnetic fields. The main result is a dynamo-like cosmological equation for the evolution of the magnetic fields.
    [Show full text]
  • Modified Theories of Relativistic Gravity
    Modified Theories of Relativistic Gravity: Theoretical Foundations, Phenomenology, and Applications in Physical Cosmology by c David Wenjie Tian A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Theoretical Physics (Interdisciplinary Program) Faculty of Science Date of graduation: October 2016 Memorial University of Newfoundland March 2016 St. John’s Newfoundland and Labrador Abstract This thesis studies the theories and phenomenology of modified gravity, along with their applications in cosmology, astrophysics, and effective dark energy. This thesis is organized as follows. Chapter 1 reviews the fundamentals of relativistic gravity and cosmology, and Chapter 2 provides the required Co-authorship 2 2 Statement for Chapters 3 ∼ 6. Chapter 3 develops the L = f (R; Rc; Rm; Lm) class of modified gravity 2 µν 2 µανβ that allows for nonminimal matter-curvature couplings (Rc B RµνR , Rm B RµανβR ), derives the “co- herence condition” f 2 = f 2 = − f 2 =4 for the smooth limit to f (R; G; ) generalized Gauss-Bonnet R Rm Rc Lm gravity, and examines stress-energy-momentum conservation in more generic f (R; R1;:::; Rn; Lm) grav- ity. Chapter 4 proposes a unified formulation to derive the Friedmann equations from (non)equilibrium (eff) thermodynamics for modified gravities Rµν − Rgµν=2 = 8πGeffTµν , and applies this formulation to the Friedman-Robertson-Walker Universe governed by f (R), generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, f (R; G) generalized Gauss-Bonnet and dynamical Chern-Simons gravities. Chapter 5 systemati- cally restudies the thermodynamics of the Universe in ΛCDM and modified gravities by requiring its com- patibility with the holographic-style gravitational equations, where possible solutions to the long-standing confusions regarding the temperature of the cosmological apparent horizon and the failure of the second law of thermodynamics are proposed.
    [Show full text]
  • Extended General Relativity for a Curved Universe
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4 Extended General Relativity for a Curved Universe Mohammed. B. Al-Fadhli1* 1College of Science, University of Lincoln, Lincoln, LN6 7TS, UK. Abstract: The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB). Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses the positive curvature of the early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with quantum mechanics and the necessity of dark matter and dark energy might indicate its incompleteness. By utilising the Einstein–Hilbert action, this study presents extended field equations considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which could remove singularities from the theory and facilitate its quantisation. The extended equations have been utilised to derive the evolution of the Universe with reference to the scale factor of the early Universe and its radius of curvature. Keywords: Astrometry, General Relativity, Boundary Term. 1. INTRODUCTION Considerable efforts have been devoted to Recent evidence by Planck Legacy recent release modifying gravity in form of Conformal Gravity, (PL18) indicated the presence of an enhanced lensing Loop Quantum Gravity, MOND, ADS-CFT, String amplitude in the cosmic microwave background theory, 퐹(푅) Gravity, etc.
    [Show full text]
  • Fundamental Equations for Hypersurfaces
    Appendix A Fundamental Equations for Hypersurfaces In this appendix we consider submanifolds with codimension 1 of a pseudo- Riemannian manifold and derive, among other things, the important equations of Gauss and Codazzi–Mainardi. These have many useful applications, in particular for 3 + 1 splittings of Einstein’s equations (see Sect. 3.9). For definiteness, we consider a spacelike hypersurface Σ of a Lorentz manifold (M, g). Other situations, e.g., timelike hypersurfaces can be treated in the same way, with sign changes here and there. The formulation can also be generalized to sub- manifolds of codimension larger than 1 of arbitrary pseudo-Riemannian manifolds (see, for instance, [51]), a situation that occurs for instance in string theory. Let ι : Σ −→ M denote the embedding and let g¯ = ι∗g be the induced metric on Σ. The pair (Σ, g)¯ is a three-dimensional Riemannian manifold. Other quanti- ties which belong to Σ will often be indicated by a bar. The basic results of this appendix are most efficiently derived with the help of adapted moving frames and the structure equations of Cartan. So let {eμ} be an orthonormal frame on an open region of M, with the property that the ei ,fori = 1, 2, 3, at points of Σ are tangent μ to Σ. The dual basis of {eμ} is denoted by {θ }.OnΣ the {ei} can be regarded as a triad of Σ that is dual to the restrictions θ¯i := θ j |TΣ(i.e., the restrictions to tangent vectors of Σ). Indices for objects on Σ always refer to this pair of dual basis.
    [Show full text]
  • Algebraic Structures
    Appendix A Algebraic Structures An algebraic structure is a set X with one or more operations satisfying certain axioms. In this appendix, we will briefly review groups and vector spaces. A.1 Groups Group.Agroup is a set G equipped with an operation m : G × G → G such that: 1. ∀x, y, z ∈ G, m(x, m(y, z)) = m(m(x, y), z) (associativity); 2. ∀x ∈ G, there exists an element u ∈ G such that m(u, x) = m(x, u) = x (existence of the identity element); 3. ∀x ∈ G, there exists x−1 such that m(x, x−1) = m(x−1, x) = u (existence of the inverse element). If m(x, y) = m(y, x) ∀x, y ∈ G then G is an abelian group. Note that in every group the identity element is unique. If we assume there are two identity elements, say u and u, we should have m(u, u) = u and m(u, u) = u, and therefore u = u. Even the inverse element must be unique. If x had two inverse elements, say x−1 and x−1, then x−1 = m(u, x−1) = m(m(x−1, x), x−1) = m(x−1, m(x, x−1)) = m(x−1, u) = x−1 . (A.1) The set of real numbers R with the common operation of sum is an abelian group with the identity element 0; the inverse of the element x is denoted −x.ThesetR/{0} with the common operation of product is an abelian group with the identity element 1; the inverse of the element x is denoted 1/x.
    [Show full text]
  • Lectures on Gravitation
    IFUP–TH/2017 LECTURES ON GRAVITATION Pietro Menotti Dipartimento di Fisica, Universit`adi Pisa, Italy March 2017 arXiv:1703.05155v1 [gr-qc] 15 Mar 2017 2 3 INDEX Foreword 7 Acknowledgments 7 Chapter 1: The Lorentz group 9 1. The group of inertial transformations: the three solutions 2. Lorentz transformations 3. SL(2,C) is connected 4. SL(2,C) is the double covering of the restricted Lorentz group 5. The restricted Lorentz group is simple 6. SL(2,C) is simply connected 7. Little groups 8. There are no non-trivial unitary finite dimensional representation of the Lorentz group 9. The conjugate representation 10. Relativistically invariant field equations 11. The Klein-Gordon equation 12.The Weyl equation 13. The Majorana mass 14. The Dirac equation 15. The Rarita- Schwinger equation 16. Transformation properties of quantum fields Chapter 2: Equivalence principle and the path to general relativity 35 1. Introduction 2. Motion of a particle in a gravitational field 3. Hamiltonian of a particle in a gravitational field Chapter 3: Manifolds 41 1. Introduction 2. Mappings, pull-back and push-forward 3. Vector fields and diffeomorphisms 4. The Lie derivative 5. Components of the Lie derivative 4 6. Commuting vector fields 7. Stokes’s theorem Chapter 4: The covariant derivative 49 1. Introduction 2. Transformation properties of the connection 3. Parallel transport 4. Geodesics 5. Curvature 6. Torsion 7. The structure equations 8. Non-abelian gauge fields on differential manifolds 9. Non compact electrodynamics 10. Meaning of torsion 11. Vanishing of the torsion 12. Metric structure 13. Isometries 14. Metric compatibility 15.
    [Show full text]
  • Variational Structures and Cosmological Dynamics in Higher Order Gravity
    VARIATIONAL STRUCTURES AND COSMOLOGICAL DYNAMICS IN HIGHER ORDER GRAVITY IOANNIS MIRITZIS July 1997 Contents 1 Introduction 1 I Variational issues in higher order gravity theo- ries 11 2 Constrained variations and nonlinear Lagrangians 13 2.1 Variational principles in General Relativity . 14 2.1.1 Hilbert variation . 15 2.1.2 Palatini variation . 18 2.2 Palatini variation for general Lagrangians . 21 2.2.1 f(R)Lagrangians.................. 22 ab 2.2.2 f(R(ab)R )Lagrangians .............. 25 abcd 2.2.3 f RabcdR Lagrangians. .. 29 2.2.4 Comments on the Palatini variation . 29 2.3 Constrained Palatini Variation . 31 2.4 ConformalstructureandWeylgeometry . 37 2.5 Summary .......................... 39 II Cosmological dynamics 41 3 Cosmic no-hair theorems 43 3.1 The inflationary scenario and de Sitter spacetime . 43 3.2 Cosmicno-hairconjecture . 47 3.3 Proof of the CNHT in a curvature squared theory for all Bianchi-type cosmologies . 51 3 4 Recollapse problems 61 4.1 The k =1 FRW universe in the conformal frame . 62 4.2 Conditions that ensure the existence of a maximal hyper- surface............................ 63 4.3 Summary .......................... 67 5 Conclusion 69 A Raychaudhuri’s equation 71 A.1 Extrinsiccurvature . .. .. 72 B Homogeneous cosmologies 75 C Energy conditions 79 D Conformal transformations 81 This thesis is submitted to the Department of Mathematics, University of the Aegean, in fulfilment of the requirements for the degree of PhD. Στoν Nικoλαo´ Xατζησαββα,´ φ´ιλo και δασκαλo.´ Abstract In the first part of this thesis we analyse the variational structure of arbitrary nonlinear Langrangian theories of gravity.
    [Show full text]
  • Differential Geometry and Einstein's Field Equations Differential
    Differential geometry and Einstein's Field Equations Otis Saint - 10584344 Centre for Mathematical Sciences, University of Plymouth Manifolds, Vectors, and Tensors Curvature Tensors • General Relativity is founded upon the idea that space-time is a 4-dimensional manifold, • On a curved manifold, parallel transport of a vector around a closed loop causes the which can curve in the presence of matter. vector to change direction when it is returned to its original starting point. • (Simple definition) A manifold is a type of topological space for which local regions • The change in a vector's components after parallel transport around an infinitesimal around points, "look like" a subset of Rn. closed loop can be approximated as I ν • (Hard definition) A manifold is a Hausdorff topological space, M, such that each point β 1 β λ αdx 0 ∆Vk = RλναV x 0 dt (5) m 2 M is contained in some open subset Uk ⊂ M, called a coordinate patch, for which 2 dt n n there exists a homeomorphism φk : Uk ! ⊂ . β Rk R • Rλνα is the Riemann curvature tensor, and is defined to be • The tuple (Uk; φk), is called a coordinate chart on M, and allows us to define the notion β β β β τ β τ R = @νΓ − @αΓ + Γ Γ − Γ Γ (6) of coordinates on M, which we often denote (xµ). λνα λα λν τν λα τα λν n • By contracting the upper index with one of lower indices, we obtain a new tensor of type • One of the simplest examples of a manifold is the n-sphere, S .
    [Show full text]
  • Differential Geometry
    Di®erential Geometry Michael E. Taylor 1 2 Contents The Basic Object 1. Surfaces, Riemannian metrics, and geodesics Vector Fields and Di®erential Forms 2. Flows and vector ¯elds 3. Lie brackets 4. Integration on Riemannian manifolds 5. Di®erential forms 6. Products and exterior derivatives of forms 7. The general Stokes formula 8. The classical Gauss, Green, and Stokes formulas 8B. Symbols, and a more general Green-Stokes formula 9. Topological applications of di®erential forms 10. Critical points and index of a vector ¯eld Geodesics, Covariant Derivatives, and Curvature 11. Geodesics on Riemannian manifolds 12. The covariant derivative and divergence of tensor ¯elds 13. Covariant derivatives and curvature on general vector bundles 14. Second covariant derivatives and covariant-exterior derivatives 15. The curvature tensor of a Riemannian manifold 16. Geometry of submanifolds and subbundles The Gauss-Bonnet Theorem and Characteristic Classes 17. The Gauss-Bonnet theorem for surfaces 18. The principal bundle picture 19. The Chern-Weil construction 20. The Chern-Gauss-Bonnet theorem Hodge Theory 21. The Hodge Laplacian on k-forms 22. The Hodge decomposition and harmonic forms 23. The Hodge decomposition on manifolds with boundary 24. The Mayer-Vietoris sequence for deRham cohomology 3 Spinors 25. Operators of Dirac type 26. Cli®ord algebras 27. Spinors 28. Weitzenbock formulas Minimal Surfaces 29. Minimal surfaces 30. Second variation of area 31. The Minimal surface equation Appendices A. Metric spaces, compactness, and all that B. Topological spaces C. The derivative D. Inverse function and implicit function theorem E. Manifolds F. Vector bundles G. Fundamental existence theorem for ODE H.
    [Show full text]
  • General Relativity MATTHIAS BARTELMANN
    Lecture Notes Physik LN General Relativity MATTHIAS BARTELMANN HEIDELBERG UNIVERSITY PUBLISHING GENERAL RELATIVITY Lecture Notes Physik LN GENERAL RELATIVITY Matthias Bartelmann Institut für Theoretische Astrophysik Universität Heidelberg About the Author Matthias Bartelmann is professor for theoretical astrophysics at Heidelberg University. He mostly addresses cosmological questions, concerning in particular the formation and evolution of cosmic structures. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. This book is published under the Creative Commons Attribution 4.0 License (CC BY-SA 4.0). The cover is subject to the Creative Commons License CC BY-ND 4.0. The electronic open access version of this work is permanently available on Heidelberg University Publishing’s website: https://heiup.uni-heidelberg.de urn: urn:nbn:de:bsz:16-heiup-book-534-9 doi: https://doi.org/10.17885/heiup.534 Text © 2019, Matthias Bartelmann ISSN 2566-4816 (PDF) ISSN 2512-4455 (Print) ISBN 978-3-947732-59-3 (PDF) ISBN 978-3-947732-60-9 (Softcover) v Instead of a preface Before we begin, some words are in order on the purpose and the limits of these notes, on the notation used, and on some of the many people I am indebted to. Each of the chapters of these notes is meant for a week of two lectures of two hours each. Much more could be said about all of their topics in all directions, in terms of mathematics, physics, and experimental tests of general relativity.
    [Show full text]
  • Study of the Thermodynamics of Charged Rotating Black Holes in Five-Dimensional Anti-De Sitter Spacetimes
    American University in Cairo AUC Knowledge Fountain Theses and Dissertations Fall 7-26-2020 Study of the thermodynamics of charged rotating black holes in five-dimensional anti-de sitter spacetimes Hassan El-Sayed The American University in Cairo Follow this and additional works at: https://fount.aucegypt.edu/etds Recommended Citation APA Citation El-Sayed, H. (2020).Study of the thermodynamics of charged rotating black holes in five-dimensional anti- de sitter spacetimes [Master’s thesis, the American University in Cairo]. AUC Knowledge Fountain. https://fount.aucegypt.edu/etds/1475 MLA Citation El-Sayed, Hassan. Study of the thermodynamics of charged rotating black holes in five-dimensional anti- de sitter spacetimes. 2020. American University in Cairo, Master's thesis. AUC Knowledge Fountain. https://fount.aucegypt.edu/etds/1475 This Master's Thesis is brought to you for free and open access by AUC Knowledge Fountain. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AUC Knowledge Fountain. For more information, please contact [email protected]. The American University in Cairo School of Sciences and Engineering Study of the Thermodynamics of Charged Rotating Black Holes in Five-Dimensional Anti-de Sitter Spacetimes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science in Physics Author: Advisor: Hassan El-Sayed Dr. Adel Awad June 2020 This thesis was presented to the following thesis committee members: Dr. Adel Awad Professor Physics Department, School of Sciences and Engineering The American University in Cairo Thesis Advisor Dr. Mohammad AlFiky Assistant Professor Physics Department, School of Sciences and Engineering The American University in Cairo Internal Examiner Dr.
    [Show full text]
  • Vectors and Vector Analysis
    Appendix A Vectors and Vector Analysis A.1 Vector Algebra A.1.1. Let ai , i = 1, 2, 3 be the components of a vector a in the orthonormal basis ui of an Euclidean three-dimensional space. Using Einstein’s summation convention, the analytical expression of a is a = ai ui . (A.1) The analytical expression of the radius-vector is then r = xi ui . (A.2) A.1.2. Let a and b be two arbitrary vectors. Skipping addition and subtraction, one can define 1. The scalar product or dot product of the two vectors: a · b = (ai ui ) · (bk uk ) = ai bi = ab cos(a, b), (A.3) since ui · uk = δik. (A.4) 2. The vector product,orcross product of the two vectors: a × b =−b × a = ijka j bk ui , |a × b|=ab sin(a, b), (A.5) as well as 1 u × u = u , u = u × u , (A.6) i j ijk k s 2 sij i j © Springer-Verlag Berlin Heidelberg 2016 591 M. Chaichian et al., Electrodynamics, DOI 10.1007/978-3-642-17381-3 592 Appendix A: Vectors and Vector Analysis where ⎧ ⎨ +1, if i, j, k are an even permutation of 1, 2, 3, = − , , , , , , ijk ⎩ 1 if i j k are an odd permutation of 1 2 3 (A.7) 0, if any two indices are equal, is the Levi-Civita permutation symbol (see Sect. A.5). One can easily verify the property δil δim δin ijklmn = δ jl δ jm δ jn . (A.8) δkl δkm δkn For i = l,(A.8) becomes ijkimn = δ jmδkn − δ jnδkm, (A.9) while for i = l, j = m it becomes ijkijn = 2δkn.
    [Show full text]