Notes on Lorentzian Causality ESI-EMS-IAMP Summer School on Mathematical Relativity
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Universal Thermodynamics in the Context of Dynamical Black Hole
universe Article Universal Thermodynamics in the Context of Dynamical Black Hole Sudipto Bhattacharjee and Subenoy Chakraborty * Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India; [email protected] * Correspondence: [email protected] Received: 27 April 2018; Accepted: 22 June 2018; Published: 1 July 2018 Abstract: The present work is a brief review of the development of dynamical black holes from the geometric point view. Furthermore, in this context, universal thermodynamics in the FLRW model has been analyzed using the notion of the Kodama vector. Finally, some general conclusions have been drawn. Keywords: dynamical black hole; trapped surfaces; universal thermodynamics; unified first law 1. Introduction A black hole is a region of space-time from which all future directed null geodesics fail to reach the future null infinity I+. More specifically, the black hole region B of the space-time manifold M is the set of all events P that do not belong to the causal past of future null infinity, i.e., B = M − J−(I+). (1) Here, J−(I+) denotes the causal past of I+, i.e., it is the set of all points that causally precede I+. The boundary of the black hole region is termed as the event horizon (H), H = ¶B = ¶(J−(I+)). (2) A cross-section of the horizon is a 2D surface H(S) obtained by intersecting the event horizon with a space-like hypersurface S. As event the horizon is a causal boundary, it must be a null hypersurface generated by null geodesics that have no future end points. In the black hole region, there are trapped surfaces that are closed 2-surfaces (S) such that both ingoing and outgoing congruences of null geodesics are orthogonal to S, and the expansion scalar is negative everywhere on S. -
Light Rays, Singularities, and All That
Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition . -
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. -
Black Hole Formation and Stability: a Mathematical Investigation
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 55, Number 1, January 2018, Pages 1–30 http://dx.doi.org/10.1090/bull/1592 Article electronically published on September 25, 2017 BLACK HOLE FORMATION AND STABILITY: A MATHEMATICAL INVESTIGATION LYDIA BIERI Abstract. The dynamics of the Einstein equations feature the formation of black holes. These are related to the presence of trapped surfaces in the spacetime manifold. The mathematical study of these phenomena has gained momentum since D. Christodoulou’s breakthrough result proving that, in the regime of pure general relativity, trapped surfaces form through the focusing of gravitational waves. (The latter were observed for the first time in 2015 by Advanced LIGO.) The proof combines new ideas from geometric analysis and nonlinear partial differential equations, and it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized by S. Klainerman and I. Rodnianski, and more recently by these authors and J. Luk. Here, we investigate the dynamics of the Einstein equations, focusing on these works. Finally, we address the question of stability of black holes and what has been known so far, involving recent works of many contributors. Contents 1. Introduction 1 2. Mathematical general relativity 3 3. Gravitational radiation 12 4. The formation of closed trapped surfaces 14 5. Stability of black holes 25 6. Cosmic censorship 26 Acknowledgments 27 About the author 27 References 27 1. Introduction The Einstein equations exhibit singularities that are hidden behind event hori- zons of black holes. A black hole is a region of spacetime that cannot be ob- served from infinity. -
Global Properties of Asymptotically De Sitter and Anti De Sitter Spacetimes
Global properties of asymptotically de Sitter and Anti de Sitter spacetimes Didier A. Solis May, 17, 2006 arXiv:1803.01171v1 [gr-qc] 3 Mar 2018 Ad Majorem Dei Gloriam. To all those who seek to fulfill God's will in their lives. iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to: • My advisor Dr. Gregory Galloway, for his constant support and help. This dissertation would not had seen the light of day without his continued encour- agement and guidance. • Dr. N. Saveliev, Dr. M. Cai and Dr. O. Alvarez for their valuable comments and thorough revision of this work. • The Faculty and staff at the Math Department for all the knowledge and affec- tion they shared with me. • CONACYT (Consejo Nacional de Ciencia y Tecnolog´ıa)for the financial support granted to the author. • My family: Douglas, Rosalinda, Douglas Jr, Rosil´uand Josu´efor all their prayers and unconditional love. They are indeed the greatest blessing in my life. • All my friends, especially Guille, for always being there for me to share the struggles and joys of life. • To the CSA gang, for being a living example of faith lived in love and hope. • Last but by no means least, to God, without whom there is no math, laughter or music. iv Contents 1 Introduction 1 1.1 Lorentz vector spaces . 1 1.2 Spacetimes . 6 1.2.1 The Levi-Civita connection . 7 1.2.2 Covariant derivative . 11 1.2.3 Geodesics . 12 1.3 Causal theory . 15 1.3.1 Definitions . 15 1.3.2 Global causality conditions . -
Communications-Mathematics and Applied Mathematics/Download/8110
A Mathematician's Journey to the Edge of the Universe "The only true wisdom is in knowing you know nothing." ― Socrates Manjunath.R #16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore560010, Karnataka, India *Corresponding Author Email: [email protected] *Website: http://www.myw3schools.com/ A Mathematician's Journey to the Edge of the Universe What’s the Ultimate Question? Since the dawn of the history of science from Copernicus (who took the details of Ptolemy, and found a way to look at the same construction from a slightly different perspective and discover that the Earth is not the center of the universe) and Galileo to the present, we (a hoard of talking monkeys who's consciousness is from a collection of connected neurons − hammering away on typewriters and by pure chance eventually ranging the values for the (fundamental) numbers that would allow the development of any form of intelligent life) have gazed at the stars and attempted to chart the heavens and still discovering the fundamental laws of nature often get asked: What is Dark Matter? ... What is Dark Energy? ... What Came Before the Big Bang? ... What's Inside a Black Hole? ... Will the universe continue expanding? Will it just stop or even begin to contract? Are We Alone? Beginning at Stonehenge and ending with the current crisis in String Theory, the story of this eternal question to uncover the mysteries of the universe describes a narrative that includes some of the greatest discoveries of all time and leading personalities, including Aristotle, Johannes Kepler, and Isaac Newton, and the rise to the modern era of Einstein, Eddington, and Hawking. -
Global Hyperbolicity in Space-Time Manifold
Munich Personal RePEc Archive Global Hyperbolicity in Space-Time Manifold Mohajan, Haradhan Assistant Professor, Premier University, Chittagong, Bangladesh. 10 February 2016 Online at https://mpra.ub.uni-muenchen.de/83036/ MPRA Paper No. 83036, posted 01 Dec 2017 09:38 UTC International Journal of Professional Studies (IJPS) 2016, Vol. No. 1, Issue 1, Jan-Jun GLOBAL HYPERBOLICITY IN SPACE-TIME MANIFOLD Haradhan Kumar Mohajan Assistant Professor, Faculty of Business Studies, Premier University, Chittagong, Bangladesh _________________________________________________________________________ ABSTRACT Global hyperbolicity is the most important condition on causal structure space-time, which is involved in problems as cosmic censorship, predictability etc. An open set O is said to be globally hyperbolic if, i) for every pair of points x and y in O the intersection of the future of x and the past of y has compact closure i.e., a space-time M, g is said to be globally hyperbolic if the sets J x J y are compact for all , yx M (i.e., no naked singularity can exist in space-time topology), and ii) strong causality holds on O i.e., there are no closed or almost closed time like curves contained in O. Here J x is causal future and J x is the causal past of an event x. If a space-time is timelike or null geodesically incomplete but cannot be embedded in a larger space-time then we say that it has a singularity. An attempt is taken here to discuss global hyperbolicity and space-time singularity by introducing definitions, propositions and displaying diagrams appropriately. Keywords: Cauchy surface, causality, global hyperbolicity, space-time manifold, space-time singularities. -
Energy Conditions in General Relativity and Quantum Field Theory Arxiv:2003.01815V2 [Gr-Qc] 5 Jun 2020
Energy conditions in general relativity and quantum field theory Eleni-Alexandra Kontou∗ Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom Department of Physics, College of the Holy Cross, Worcester, Massachusetts 01610, USA Ko Sanders y Dublin City University, School of Mathematical Sciences and Centre for Astrophysics and Relativity, Glasnevin, Dublin 9, Ireland June 8, 2020 Abstract This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on mat- ter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in math- ematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are sys- tematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy arXiv:2003.01815v2 [gr-qc] 5 Jun 2020 condition, has recently received increased attention. -
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 . -
Gravitational Waves and Their Memory in General Relativity
Surveys in Differential Geometry XX Gravitational waves and their memory in general relativity Lydia Bieri, David Garfinkle, and Shing-Tung Yau Abstract. General relativity explains gravitational radiation from bi- nary black hole or neutron star mergers, from core-collapse supernovae and even from the inflation period in cosmology. These waves exhibit a unique effect called memory or Christodoulou effect, which in a detector like LIGO or LISA shows as a permanent displacement of test masses and in radio telescopes like NANOGrav as a change in the frequency of pulsars’ pulses. It was shown that electromagnetic fields and neutrino radiation enlarge the memory. Recently it has been understood that the two types of memory addressed in the literature as ‘linear’ and ‘non- linear’ are in fact two different phenomena. The former is due to fields that do not and the latter is due to fields that do reach null infinity. 1. Introduction Gravitational waves occur in spacetimes with interesting structures from the points of view of mathematics and physics. These waves are fluctuations of the spacetime curvature. Recent experiments [1] claim to have found imprints in the cosmic microwave background of gravitational waves produced during the inflation period in the early Universe, though it has been argued [28] that foreground emission could have produced the experimental signal. Gravitational radiation from non-cosmological sources such as mergers of binary black holes or binary neutron stars as well as core-collapse supernovae are yet to be directly detected. We live on the verge of detection of these waves and thus a new era of astrophysics 2010 Mathematics Subject Classification. -
The Location of Trapped Surfaces
The location of trapped surfaces José M M Senovilla Department of Theoretical Physics and History of Science University of the Basque Country, Bilbao, Spain Mathematical Relativity in Lisbon, Lisboa, 18th June 2009 Outline 1 Trapped surfaces. Formal definitions 2 Some simple questions on black holes 3 The future-trapped region T and its boundary B 4 B = AH 6 5 Closed trapped surfaces are clairvoyant ! 6 Fundamental general results 7 Application to Robertson-Walker spacetimes 8 Application to Vaidya: the hypersurface Σ 9 Where is B in Vaidya? 10 Conclusions and outlook References I Bengtsson and JMMS, Note on trapped surfaces in the Vaidya solution, Phys. Rev. D 79 024027 (2009) [arXiv:0809.2213] I Bengtsson and JMMS, The boundary of the region with trapped surfaces in spherical symmetry, Preprint JMMS, On the boundary of the region containing trapped surfaces, AIP Conf. Proc. 1122 72 (2009) [arXiv:0812.2767] Φ: S xµ = Φµ(λA): −! V Thus, the tangent vectors (seen on ) are: V µ µ @Φ ~eA Φ0(@ A ) e = ≡ λ () A @λA First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. Then, x S 8 2 Tx = TxS TxS? V ⊕ called tangent and normal parts. The trapped surface fauna: Definitions Let ( ; g) be a 4-dimensional Lorentzian manifold with metric V tensor g of signature ( ; +; +; +). − First fundamental form: µ ν γAB(λ) = g S(~eA;~eB) = gµν(Φ)e e j A B is positive-definite. So, S is assumed to be SPACELIKE. -
Spacetime Singularities and a Novel Formulation of Indeterminism
Spacetime singularities and a novel formulation of indeterminism Feraz Azhar∗ Department of Philosophy, University of Notre Dame, Notre Dame, IN, 46556, USA & Black Hole Initiative, Harvard University, Cambridge, MA, 02138, USA Mohammad Hossein Namjooy School of Astronomy, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (Dated: January 26, 2021) Spacetime singularities in general relativity are commonly thought to be problematic, in that they signal a breakdown in the theory. We address the question of how to interpret this breakdown, restricting our attention to classical considerations (though our work has ramifications for more general classical metric theories of gravity, as well). In particular, we argue for a new claim: spacetime singularities in general relativity signal indeterminism. The usual manner in which indeterminism is formulated for physical theories can be traced back to Laplace. This formulation is based on the non-uniqueness of future (or past) states of a physical system|as understood in the context of a physical theory|as a result of the specification of an antecedent state (or, respectively, of a subsequent state). We contend that for physical theories generally, this formulation does not comprehensively capture the relevant sense of a lack of determination. And, in particular, it does not com- prehensively capture the sense(s) in which a lack of determination (and so, indeterminism) arises due to spacetime singularities in general relativity. We thus present a novel, broader formulation, in which indeterminism in the context of some physical theory arises whenever one of the three following conditions holds: future (and/or past) states are (i) not unique|as for Laplacian notions of indeterminism; (ii) not specified; or (iii) `incoherent'|that is, they fail to satisfy certain desiderata that are internal to the theory and/or imposed from outside the theory.