Geometry of Black Holes Revised July 2018
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4. Kruskal Coordinates and Penrose Diagrams
4. Kruskal Coordinates and Penrose Diagrams. 4.1. Removing a coordinate Singularity at the Schwarzschild Radius. The Schwarzschild metric has a singularity at r = rS where g 00 → 0 and g11 → ∞ . However, we have already seen that a free falling observer acknowledges a smooth motion without any peculiarity when he passes the horizon. This suggests that the behaviour at the Schwarzschild radius is only a coordinate singularity which can be removed by using another more appropriate coordinate system. This is in GR always possible provided the transformation is smooth and differentiable, a consequence of the diffeomorphism of the spacetime manifold. Instead of the 4-dimensional Schwarzschild metric we study a 2-dimensional t,r-version. The spherical symmetry of the Schwarzschild BH guaranties that we do not loose generality. −1 ⎛ r ⎞ ⎛ r ⎞ ds 2 = ⎜1− S ⎟ dt 2 − ⎜1− S ⎟ dr 2 (4.1) ⎝ r ⎠ ⎝ r ⎠ To describe outgoing and ingoing null geodesics we divide through dλ2 and set ds 2 = 0. −1 ⎛ rS ⎞ 2 ⎛ rS ⎞ 2 ⎜1− ⎟t& − ⎜1− ⎟ r& = 0 (4.2) ⎝ r ⎠ ⎝ r ⎠ or rewritten 2 −2 ⎛ dt ⎞ ⎛ r ⎞ ⎜ ⎟ = ⎜1− S ⎟ (4.3) ⎝ dr ⎠ ⎝ r ⎠ Note that the angle of the light cone in t,r-coordinate.decreases when r approaches rS After integration the outgoing and ingoing null geodesics of Schwarzschild satisfy t = ± r * +const. (4.4) r * is called “tortoise coordinate” and defined by ⎛ r ⎞ ⎜ ⎟ r* = r + rS ln⎜ −1⎟ (4.5) ⎝ rS ⎠ −1 dr * ⎛ r ⎞ so that = ⎜1− S ⎟ . (4.6) dr ⎝ r ⎠ As r ranges from rS to ∞, r* goes from -∞ to +∞. We introduce the null coordinates u,υ which have the direction of null geodesics by υ = t + r * and u = t − r * (4.7) From (4.7) we obtain 1 dt = ()dυ + du (4.8) 2 and from (4.6) 28 ⎛ r ⎞ 1 ⎛ r ⎞ dr = ⎜1− S ⎟dr* = ⎜1− S ⎟()dυ − du (4.9) ⎝ r ⎠ 2 ⎝ r ⎠ Inserting (4.8) and (4.9) in (4.1) we find ⎛ r ⎞ 2 ⎜ S ⎟ ds = ⎜1− ⎟ dudυ (4.10) ⎝ r ⎠ Fig. -
3+1 Formalism and Bases of Numerical Relativity
3+1 Formalism and Bases of Numerical Relativity Lecture notes Eric´ Gourgoulhon Laboratoire Univers et Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, Universit´eParis 7 arXiv:gr-qc/0703035v1 6 Mar 2007 F-92195 Meudon Cedex, France [email protected] 6 March 2007 2 Contents 1 Introduction 11 2 Geometry of hypersurfaces 15 2.1 Introduction.................................... 15 2.2 Frameworkandnotations . .... 15 2.2.1 Spacetimeandtensorfields . 15 2.2.2 Scalar products and metric duality . ...... 16 2.2.3 Curvaturetensor ............................... 18 2.3 Hypersurfaceembeddedinspacetime . ........ 19 2.3.1 Definition .................................... 19 2.3.2 Normalvector ................................. 21 2.3.3 Intrinsiccurvature . 22 2.3.4 Extrinsiccurvature. 23 2.3.5 Examples: surfaces embedded in the Euclidean space R3 .......... 24 2.4 Spacelikehypersurface . ...... 28 2.4.1 Theorthogonalprojector . 29 2.4.2 Relation between K and n ......................... 31 ∇ 2.4.3 Links between the and D connections. .. .. .. .. .. 32 ∇ 2.5 Gauss-Codazzirelations . ...... 34 2.5.1 Gaussrelation ................................. 34 2.5.2 Codazzirelation ............................... 36 3 Geometry of foliations 39 3.1 Introduction.................................... 39 3.2 Globally hyperbolic spacetimes and foliations . ............. 39 3.2.1 Globally hyperbolic spacetimes . ...... 39 3.2.2 Definition of a foliation . 40 3.3 Foliationkinematics .. .. .. .. .. .. .. .. ..... 41 3.3.1 Lapsefunction ................................. 41 3.3.2 Normal evolution vector . 42 3.3.3 Eulerianobservers ............................. 42 3.3.4 Gradients of n and m ............................. 44 3.3.5 Evolution of the 3-metric . 45 4 CONTENTS 3.3.6 Evolution of the orthogonal projector . ....... 46 3.4 Last part of the 3+1 decomposition of the Riemann tensor . -
Part 3 Black Holes
Part 3 Black Holes Harvey Reall Part 3 Black Holes March 13, 2015 ii H.S. Reall Contents Preface vii 1 Spherical stars 1 1.1 Cold stars . .1 1.2 Spherical symmetry . .2 1.3 Time-independence . .3 1.4 Static, spherically symmetric, spacetimes . .4 1.5 Tolman-Oppenheimer-Volkoff equations . .5 1.6 Outside the star: the Schwarzschild solution . .6 1.7 The interior solution . .7 1.8 Maximum mass of a cold star . .8 2 The Schwarzschild black hole 11 2.1 Birkhoff's theorem . 11 2.2 Gravitational redshift . 12 2.3 Geodesics of the Schwarzschild solution . 13 2.4 Eddington-Finkelstein coordinates . 14 2.5 Finkelstein diagram . 17 2.6 Gravitational collapse . 18 2.7 Black hole region . 19 2.8 Detecting black holes . 21 2.9 Orbits around a black hole . 22 2.10 White holes . 24 2.11 The Kruskal extension . 25 2.12 Einstein-Rosen bridge . 28 2.13 Extendibility . 29 2.14 Singularities . 29 3 The initial value problem 33 3.1 Predictability . 33 3.2 The initial value problem in GR . 35 iii CONTENTS 3.3 Asymptotically flat initial data . 38 3.4 Strong cosmic censorship . 38 4 The singularity theorem 41 4.1 Null hypersurfaces . 41 4.2 Geodesic deviation . 43 4.3 Geodesic congruences . 44 4.4 Null geodesic congruences . 45 4.5 Expansion, rotation and shear . 46 4.6 Expansion and shear of a null hypersurface . 47 4.7 Trapped surfaces . 48 4.8 Raychaudhuri's equation . 50 4.9 Energy conditions . 51 4.10 Conjugate points . -
How the Quantum Black Hole Replies to Your Messages
Gerard 't Hooft How the Quantum Black hole Replies to Your Messages Centre for Extreme Matter and Emergent Phenomena, Science Faculty, Utrecht University, POBox 80.089, 3508 TB, Utrecht Qui Nonh, Viet Nam, July 24, 2017 1 / 45 Introduction { Einstein's theory of gravity, based on General Relativity, and { Quantum Mechanics, as it was developed early 20th century, are both known to be valid at high precision. But combining these into one theory still leads to problems today. Existing approaches: { Superstring theory, extended as M theory { Loop quantum gravity { Dynamical triangulation of space-time { Asymptotically safe quantum gravity are promising but not (yet) understood at the desired level. In particular when black holes are considered. 2 / 45 one encounters problems with: { information loss { incorrectly entangled states { firewalls We shall show that fundamental new ingredients in all these theories are called for: { the gravitational back reaction cannot be ignored, { one must expand the momentum distributions of in- and out-particles in spherical harmonics, and { one must apply antipodal identification in order to avoid double counting of pure quantum states. This we will explain. We do not claim that these theories are incorrect, but they are not fool-proof. The topology of space and time is not (yet) handled correctly in these theories. This is why 3 / 45 { the gravitational back reaction cannot be ignored, { one must expand the momentum distributions of in- and out-particles in spherical harmonics, and { one must apply antipodal identification in order to avoid double counting of pure quantum states. This we will explain. We do not claim that these theories are incorrect, but they are not fool-proof. -
Arxiv:Hep-Th/9209055V1 16 Sep 1992 Quantum Aspects of Black Holes
EFI-92-41 hep-th/9209055 Quantum Aspects of Black Holes Jeffrey A. Harvey† Enrico Fermi Institute University of Chicago 5640 Ellis Avenue, Chicago, IL 60637 Andrew Strominger∗ Department of Physics University of California Santa Barbara, CA 93106-9530 Abstract This review is based on lectures given at the 1992 Trieste Spring School on String arXiv:hep-th/9209055v1 16 Sep 1992 Theory and Quantum Gravity and at the 1992 TASI Summer School in Boulder, Colorado. 9/92 † Email address: [email protected] ∗ Email addresses: [email protected], [email protected]. 1. Introduction Nearly two decades ago, Hawking [1] observed that black holes are not black: quantum mechanical pair production in a gravitational field leads to black hole evaporation. With hindsight, this result is not really so surprising. It is simply the gravitational analog of Schwinger pair production in which one member of the pair escapes to infinity, while the other drops into the black hole. Hawking went on, however, to argue for a very surprising conclusion: eventually the black hole disappears completely, taking with it all the information carried in by the infalling matter which originally formed the black hole as well as that carried in by the infalling particles created over the course of the evaporation process. Thus, Hawking argued, it is impossible to predict a unique final quantum state for the system. This argument initiated a vigorous debate in the physics community which continues to this day. It is certainly striking that such a simple thought experiment, relying only on the basic concepts of general relativity and quantum mechanics, should apparently threaten the deterministic foundations of physics. -
Singularities, Black Holes, and Cosmic Censorship: a Tribute to Roger Penrose
Foundations of Physics (2021) 51:42 https://doi.org/10.1007/s10701-021-00432-1 INVITED REVIEW Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose Klaas Landsman1 Received: 8 January 2021 / Accepted: 25 January 2021 © The Author(s) 2021 Abstract In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose’s work on general rela- tivity. His 1965 singularity theorem (for which he got the prize) does not in fact imply the existence of black holes (even if its assumptions are met). Similarly, his versatile defnition of a singular space–time does not match the generally accepted defnition of a black hole (derived from his concept of null infnity). To overcome this, Penrose launched his cosmic censorship conjecture(s), whose evolution we discuss. In particular, we review both his own (mature) formulation and its later, inequivalent reformulation in the PDE literature. As a compromise, one might say that in “generic” or “physically reasonable” space–times, weak cosmic censorship postulates the appearance and stability of event horizons, whereas strong cosmic censorship asks for the instability and ensuing disappearance of Cauchy horizons. As an encore, an “Appendix” by Erik Curiel reviews the early history of the defni- tion of a black hole. Keywords General relativity · Roger Penrose · Black holes · Ccosmic censorship * Klaas Landsman [email protected] 1 Department of Mathematics, Radboud University, Nijmegen, The Netherlands Vol.:(0123456789)1 3 42 Page 2 of 38 Foundations of Physics (2021) 51:42 Conformal diagram [146, p. 208, Fig. -
Spacetime Continuity and Quantum Information Loss
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Nazarbayev University Repository universe Article Spacetime Continuity and Quantum Information Loss Michael R. R. Good School of Science and Technology, Nazarbayev University, Astana 010000, Kazakhstan; [email protected] Received: 30 September 2018; Accepted: 8 November 2018; Published: 9 November 2018 Abstract: Continuity across the shock wave of two regions in the metric during the formation of a black hole can be relaxed in order to achieve information preservation. A Planck scale sized spacetime discontinuity leads to unitarity (a constant asymptotic entanglement entropy) by restricting the origin of coordinates (moving mirror) to be timelike. Moreover, thermal equilibration occurs and total evaporation energy emitted is finite. Keywords: black hole evaporation; information loss; remnants 1. Introduction In this note, the role of continuity and information loss (via the tortoise coordinate, r∗), in understanding the crux of the phenomena of particle creation from black holes [1], is explored. In particular, the relationship to the simplified model of the moving mirror [2–4] is investigated because it has identical Bogolubov coefficients [5]. Uncompromising continuity is relaxed in favor of unitarity by a single additional parameter generalization of the r∗ coordinate. An understanding of the correspondence between the black hole and the moving mirror in this new context is initiated [6–8]. We prioritize information preservation, and find finite evaporation energy, thermal equilibrium, and analytical beta coefficients. Moreover, we determine and assess a left-over remnant (e.g., [9,10]). Broadly motivating, delving into the ramifications of external effects on quantum fields have led to interesting results on a wide variety of phenomena all the way from, e.g., relativistic superfluidity [11,12] to the creation of quantum vortexes in analog spacetimes [13,14]. -
Penrose Diagrams
Penrose Diagrams R. L. Herman March 4, 2015 1 Schwarzschild Geometry The line element for empty space outside a spherically symmetric source of curvature is given by the Schwarzschild line element, 2M 2M −1 ds2 = − 1 − dt2 + 1 − dr2 + r2 dθ2 + sin2 θ dφ2 : (1) r r We have used geometrized units (c = 1 and G = 1). We want to investigate the geometry by looking at its causal structure. Namely, what do the light cones look like? We will consider radial null curves. Radial null curves are curves followed by light rays (ds2 = 0) for which θ and φ are constant. Thus, 2M 2M −1 − 1 − dt2 + 1 − dr2 = 0: r r Therefore, the slope of the light cones in r − t space is given by dt 2M −1 = ± 1 − : dr r dt We note that for large r; dr ! ±1: This indicates that for large r light rays dt travel as if in flat spacetime. As light rays approach r = 2M; dr ! ±∞: Thus, the light cones have infinite slope and close, not allowing any causal structure. This can be seen from the solution t(r) = ±r ± 2M ln jr − 2Mj + constant. In Figure 1 we show the radial light rays for the Schwarzschild geometry. Note how the solutions on either side of r = 2M suggest that no information can cross the event horizon. This is a result of the coordinate singularity at r = 2M: We will explore other coordinate systems in order to see how light rays outside the event horizon, r = 2M; are connected to those inside. 1 t r r = 2M Figure 1: Radial light rays for the Schwarzschild geometry given by t(r) = ±r ± 2M ln jr − 2Mj + constant.: 1.1 Eddington-Finkelstein Coordinates We begin with the Schwarzschild line element in geometrized units (c = 1 and G = 1), 2M 2M −1 ds2 = − 1 − dt2 + 1 − dr2 + r2 dθ2 + sin2 θ dφ2 : r r We will first transform this system using different sets of coordinates. -
Catalogue of Spacetimes
Catalogue of Spacetimes e2 e1 x2 = 2 x = 2 q ∂x2 1 x2 = 1 ∂x1 x1 = 1 x1 = 0 x2 = 0 M Authors: Thomas Müller Visualisierungsinstitut der Universität Stuttgart (VISUS) Allmandring 19, 70569 Stuttgart, Germany [email protected] Frank Grave formerly, Universität Stuttgart, Institut für Theoretische Physik 1 (ITP1) Pfaffenwaldring 57 //IV, 70550 Stuttgart, Germany [email protected] URL: http://go.visus.uni-stuttgart.de/CoS Date: 21. Mai 2014 Co-authors Andreas Lemmer, formerly, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Alcubierre Warp Sebastian Boblest, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart deSitter, Friedmann-Robertson-Walker Felix Beslmeisl, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Petrov-Type D Heiko Munz, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Bessel and plane wave Andreas Wünsch, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Majumdar-Papapetrou, extreme Reissner-Nordstrøm dihole, energy momentum tensor Many thanks to all that have reported bug fixes or added metric descriptions. Contents 1 Introduction and Notation1 1.1 Notation...............................................1 1.2 General remarks...........................................1 1.3 Basic objects of a metric......................................2 1.4 Natural local tetrad and initial conditions for geodesics....................3 1.4.1 Orthonormality condition.................................3 1.4.2 Tetrad transformations...................................4 -
An Introduction to the Theory of Rotating Relativistic Stars
An introduction to the theory of rotating relativistic stars Lectures at the COMPSTAR 2010 School GANIL, Caen, France, 8-16 February 2010 Éric Gourgoulhon Laboratoire Univers et Théories, UMR 8102 du CNRS, Observatoire de Paris, Université Paris Diderot - Paris 7 92190 Meudon, France [email protected] arXiv:1003.5015v2 [gr-qc] 20 Sep 2011 20 September 2011 2 Contents Preface 5 1 General relativity in brief 7 1.1 Geometrical framework . ..... 7 1.1.1 The spacetime of general relativity . ....... 7 1.1.2 Linearforms .................................. 9 1.1.3 Tensors ..................................... 10 1.1.4 Metrictensor .................................. 10 1.1.5 Covariant derivative . 11 1.2 Einsteinequation................................ .... 13 1.3 3+1formalism .................................... 14 1.3.1 Foliation of spacetime . 15 1.3.2 Eulerian observer or ZAMO . 16 1.3.3 Adapted coordinates and shift vector . ....... 17 1.3.4 Extrinsic curvature . 18 1.3.5 3+1 Einstein equations . 18 2 Stationary and axisymmetric spacetimes 21 2.1 Stationary and axisymmetric spacetimes . .......... 21 2.1.1 Definitions ................................... 21 2.1.2 Stationarity and axisymmetry . ..... 25 2.2 Circular stationary and axisymmetric spacetimes . ............. 27 2.2.1 Orthogonal transitivity . ..... 27 2.2.2 Quasi-isotropic coordinates . ...... 29 2.2.3 Link with the 3+1 formalism . 31 3 Einstein equations for rotating stars 33 3.1 Generalframework ................................ 33 3.2 Einstein equations in QI coordinates . ......... 34 3.2.1 Derivation.................................... 34 3.2.2 Boundary conditions . 36 3.2.3 Caseofaperfectfluid ............................ 37 3.2.4 Newtonianlimit ................................ 39 3.2.5 Historicalnote ................................ 39 3.3 Spherical symmetry limit . ..... 40 3.3.1 Takingthelimit ............................... -
Black Holes from a to Z
Black Holes from A to Z Andrew Strominger Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA Last updated: July 15, 2015 Abstract These are the lecture notes from Professor Andrew Strominger's Physics 211r: Black Holes from A to Z course given in Spring 2015, at Harvard University. It is the first half of a survey of black holes focusing on the deep puzzles they present concerning the relations between general relativity, quantum mechanics and ther- modynamics. Topics include: causal structure, event horizons, Penrose diagrams, the Kerr geometry, the laws of black hole thermodynamics, Hawking radiation, the Bekenstein-Hawking entropy/area law, the information puzzle, microstate counting and holography. Parallel issues for cosmological and de Sitter event horizons are also discussed. These notes are prepared by Yichen Shi, Prahar Mitra, and Alex Lupsasca, with all illustrations by Prahar Mitra. 1 Contents 1 Introduction 3 2 Causal structure, event horizons and Penrose diagrams 4 2.1 Minkowski space . .4 2.2 de Sitter space . .6 2.3 Anti-de Sitter space . .9 3 Schwarzschild black holes 11 3.1 Near horizon limit . 11 3.2 Causal structure . 12 3.3 Vaidya metric . 15 4 Reissner-Nordstr¨omblack holes 18 4.1 m2 < Q2: a naked singularity . 18 4.2 m2 = Q2: the extremal case . 19 4.3 m2 > Q2: a regular black hole with two horizons . 22 5 Kerr and Kerr-Newman black holes 23 5.1 Kerr metric . 23 5.2 Singularity structure . 24 5.3 Ergosphere . 25 5.4 Near horizon extremal Kerr . 27 5.5 Penrose process . -
Astrophysical Wormholes
universe Article Astrophysical Wormholes Cosimo Bambi 1,* and Dejan Stojkovic 2 1 Center for Field Theory and Particle Physics and Department of Physics, Fudan University, Shanghai 200438, China 2 Department of Physics, State University of New York (SUNY) at Buffalo, Buffalo, NY 14260-1500, USA; [email protected] * Correspondence: [email protected] Abstract: Wormholes are hypothetical topologically-non-trivial structures of spacetime. From the theoretical point of view, the possibility of their existence is challenging but cannot be ruled out. This article is a compact and non-exhaustive review of past and current efforts to search for astro- physical wormholes in the Universe. Keywords: wormholes; black holes; spacetime topology; gravity 1. Introduction Wormholes are hypothetical spacetime structures with non-trivial topologies capable of connecting either two distant regions of the same universe or two different universes, as illustrated in Figure1. The entrances of a wormhole are called the “mouths” of the wormhole and the spacetime region connecting the mouths is called the “throat” of the wormhole. The simplest wormhole configuration has two mouths connected by a throat, but more complex structures are also possible [1]. Strictly speaking, wormholes are not a prediction of general relativity or of other theories of gravity. They are spacetime structures Citation: Bambi, C.; Stojkovic, D. that can potentially exist in curved spacetimes, so they exist in a very wide class of gravity Astrophysical Wormholes. Universe models. The formation mechanisms and stability of these spacetime structures depend on 2021, 7, 136. https://doi.org/ 10.3390/universe7050136 the specific gravity theory and often present problems, so the existence of wormholes in the Universe is challenging.