DOCSLIB.ORG

Spacetime Penrose Inequality for Asymptotically Hyperbolic Spherical Symmetric Initial Data

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spacetime Penrose Inequality for Asymptotically Hyperbolic Spherical Symmetric Initial Data U.U.D.M. Project Report 2020:27 Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherical Symmetric Initial Data Mingyi Hou Examensarbete i matematik, 30 hp Handledare: Anna Sakovich Examinator: Julian Külshammer Augusti 2020 Department of Mathematics Uppsala University Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherically Symmetric Initial Data Mingyi Hou Abstract In 1973, R. Penrose conjectured that the total mass of a space-time containing black holes cannot be less than a certain function of the sum of the areas of the event horizons. In the language of differential geometry, this is a statement about an initial data set for the Einstein equations that contains apparent horizons. Roughly speaking, an initial data set for the Einstein equations is a mathematical object modelling a slice of a space-time and an apparent horizon is a certain generalization of a minimal surface. Two major breakthroughs concerning this conjecture were made in 2001 by Huisken and Ilmanen respectively Bray who proved the conjecture in the so-called asymptotically flat Riemannian case, that is when the slice of a space-time has no extrinsic curvature and its intrinsic geometry resembles that of Euclidean space. Ten years later, Bray and Khuri proposed an approach using the so-called generalized Jang equation which could potentially be employed to deal with the general asymptotically flat case by reducing it to the Riemannian case. Bray and Khuri have successfully implemented this strategy under the assumption that the initial data is spherically symmetric. In this thesis, we study a suitable modification of Bray and Khuri's approach in the case when the initial data is asymptotically hyperbolic (i.e. modelled on a hy- perboloidal slice of Minkowski spacetime) and spherically symmetric. In particular, we show that the Penrose conjecture for such initial data holds provided that the generalized Jang equation has a solution with certain asymptotic behaviour near infinity. Furthermore, we prove a result concerning the existence of such a solution. Contents 1 Introduction1 2 Preliminaries4 2.1 Apparent Horizons............................4 2.2 Asymptotically Flat Initial Data.....................5 2.3 Asymptotically Hyperbolic Initial Data.................5 2.4 Spherically Symmetric Asymptotically Hyperbolic Initial Data..........................6 3 The Generalized Jang Equation8 3.1 The Generalized Jang Equation Approach...............8 3.2 Spherical Symmetry...........................9 4 Motivation for Asymptotics and the Warping Factor 12 4.1 Behavior at Infinity............................ 12 4.2 Proof of the Inequality.......................... 15 5 Existence and Asymptotics 18 5.1 Existence................................. 18 5.2 Asymptotics................................ 19 Chapter 1 Introduction In the 1970s, physicist R. Penrose conjectured that the total mass of a spacetime containing black holes with event horizons of total area A should be at least pA=16π. In his argument, cosmic censorship, a famous conjecture saying that singularities arising in the solutions of the Einstein equations are typically hidden within event horizons, is one of the fundamental ingredients. In fact, a proof of a suitable version of the Penrose inequality would give indirect support to cosmic censorship and a counter example of the inequality would very likely involve a spacetime for which cosmic censorship fails to hold [Mar09]. Thus, finding a proof for the inequality has become an active area of research. An important case of the Penrose inequality is a statement about asymptotically flat initial data. Recall that a spacetime is a Lorentzian manifold (N ; gN ) satisfying the Einstein equations, 1 Ric − R g = 8πT (1.1) N 2 N N where T is the so-called stress-energy tensor, RicN is the Ricci curvature tensor and RN is the scalar curvature. We call a triple (M; g; K) an initial data set (or Cauchy data) if (M; g) is a Riemannian manifold and K is a symmetric 2-tensor, such that the so-called constraint equations ( 1 µ = (R − jKj2 + tr (K)2) 2 g g g (1.2) J = divg(K − trg(K)g) are satisfied. Here Rg is the scalar curvature and µ and J are physically energy and momentum density respectively. In the case when (M; g) is a space-like slice of a spacetime (N ; gN ), with unit normal n, we have µ = T (n; n) and J = T (n; ·). In what follows, we will assume that any given initial data satisfies the so-called dominant energy condition µ > jJj (1.3) which is known to be satisfied in all physically reasonable spacetimes. Note that if (1.2) is satisfied then (M; g; K) is indeed the initial data required to pose an initial value problem for the Einstein equations in the sense that the time evolution of the initial data is a spacetime (N ; gN ) satisfying (1.1) and (M; g) is a space-like hypersurface in the spacetime with second fundamental form K [BI04]. 1 Roughly speaking, an initial data (M; g; K) is asymptotically flat if near infinity g approaches the Euclidean metric δ and K approaches 0. For an asymptotically flat initial data set there is a well known notion of mass [ADM08] that in a certain sense measures how fast the initial data set becomes “flat” at infinity. The argument which can be used to motivate the Penrose inequality roughly goes as follows [Mar09]. Assume a spacetime (N ; gN ) which is asymptotically flat, in the sense that it looks like the Minkowski spacetime [ONe83] at infinity. Intuitively, the event horizon H, is the boundary of the region around a black hole which not even the light can escape. Note that it is a null hypersurface which is at least Lipschitz continuous [Wal84]. Assume now, that the spacetime admits an asymptotically flat slice with total ADM mass MADM and which intersects H on a cut S. If H is a smooth hypersurface, then this cut is a smooth embedded surface which has a well-defined area jSj. Consider any other cut S1 to the future of S along the event horizon. The black hole area law [Haw71; Haw72] states jS1j > jSj, provided that the null energy condition, which is a certain strengthening of (1.3), holds. From physical considerations, the spacetime is expected to settle down to some equilibrium configuration. Assuming that all matter fields are swallowed by the black hole in the process, uniqueness theorems for stationary black holes (see e.g. [Heu96]) imply that the spacetime must approach the Kerr black hole spacetime, where the area of the section S of the event horizon is independent of the cut and takes the value p 2 2 2 2 AKerr = 8πM(M + M − L =M ) 6 16πM , where M is the total mass and L is the total angular momentum of the spacetime. In particular, M should be the asymptotic value of the Bondi mass along the future null infinity. Since gravitational waves carry positive energy, the Bondi mass cannot increase to the future [BVM62; SB62]. The Penrose inequality p MADM > jSj=16π follows provided that the Bondi mass approaches the ADM mass MADM which is known to be true under certain assumptions (see e.g. [Hay03; Kro03; AM79]). After a period of heuristic proofs and partial results, important breakthroughs have been made in the past decades. First, Huisken and Ilmanen [HI01] were able to prove the Penrose inequality in the so called time symmetric case, i.e. when the initial data has K = 0, using a geometric evolution equation known as the inverse mean curvature flow. Shortly afterwards, another proof using the so-called conformal flow was found by Bray. Roughly speaking, the result of [HI01] is for a single black hole, while that of [Bra01] allows for multiple ones. In summary, we have the following result in the time symmetric case: Theorem 1 (Riemann Penrose Inequality). Let (M; g) be a complete, smooth, asymptotically flat 3-manifold with nonnegative scalar curvature, total mass M, and with the boundary (possibly disconnected) which is an outermost minimal surface of total area A. Then r A M (1.4) > 16π with equality if and only if (M; g) is isometric to a Schwarzschild manifold 3 (R n f0g; gSchw). 2 The Penrose inequality is a strengthening of the famous positive mass theorem, saying that an asymptotically flat initial data (M; g; K) satisfying dominant energy condition has MADM > 0, with equality if and only if it is a slice of the Minkowski spacetime. This result was proven in [SY82] by reducing it to the time symmetric case K = 0 that was earlier settled in [SY79]. The proof uses a certain quasi-linear elliptic partial differential equation, the so-called Jang equation. It was suggested by Bray and Khuri (see [BK11; BK10]) that a generalization of the Jang equation could be a way to reduce the general Penrose inequality to the case of time symmetry. This idea has been successfully implemented in the asymptotically flat spherically symmetric case [BK10]. In this thesis, we provide some further support to the generalized Jang equation approach of Bray and Khuri by implementing it in the case when the initial data set is asymptotically hyperbolic spherically symmetric. This setting is of great interest for both geometrical and physical reasons, in particular, it is important for the study of gravitational radiation [Fra04]. The thesis is organized as follows: In Chapter 2, we introduce terminology re- quired in this thesis. In Chapter 3, we briefly review the generalized Jang equation and rewrite it as an ordinary differential equation under the assumption of spherical symmetry. In Chapter 4, we prove a theorem stating that the Penrose inequality for asymptotically hyperbolic spherically symmetric initial data holds provided that a solution to the generalized Jang equation exists and has certain asymptotic behavior at infinity. Finally, in Chapter 5, we prove the existence of a solution to the gener- alized Jang equation and obtain a (non-optimal) result regarding its asymptotics.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Dynamics of the Four Kinds of Trapping Horizons & Existence of Hawking
    Dynamics of the four kinds of Trapping Horizons & Existence of Hawking Radiation Alexis Helou1 AstroParticule et Cosmologie, Universit´eParis Diderot, CNRS, CEA, Observatoire de Paris, Sorbonne Paris Cit´e Bˆatiment Condorcet, 10, rue Alice Domon et L´eonieDuquet, F-75205 Paris Cedex 13, France Abstract We work with the notion of apparent/trapping horizons for spherically symmetric, dynamical spacetimes: these are quasi-locally defined, simply based on the behaviour of congruence of light rays. We show that the sign of the dynamical Hayward-Kodama surface gravity is dictated by the in- ner/outer nature of the horizon. Using the tunneling method to compute Hawking Radiation, this surface gravity is then linked to a notion of temper- ature, up to a sign that is dictated by the future/past nature of the horizon. Therefore two sign effects are conspiring to give a positive temperature for the black hole case and the expanding cosmology, whereas the same quantity is negative for white holes and contracting cosmologies. This is consistent with the fact that, in the latter cases, the horizon does not act as a separating membrane, and Hawking emission should not occur. arXiv:1505.07371v1 [gr-qc] 27 May 2015 [email protected] Contents 1 Introduction 1 2 Foreword 2 3 Past Horizons: Retarded Eddington-Finkelstein metric 4 4 Future Horizons: Advanced Eddington-Finkelstein metric 10 5 Hawking Radiation from Tunneling 12 6 The four kinds of apparent/trapping horizons, and feasibility of Hawking radiation 14 6.1 Future-outer trapping horizon: black holes . 15 6.2 Past-inner trapping horizon: expanding cosmology .
    [Show full text]
  • Stephen Hawking: 'There Are No Black Holes' Notion of an 'Event Horizon', from Which Nothing Can Escape, Is Incompatible with Quantum Theory, Physicist Claims
    NATURE | NEWS Stephen Hawking: 'There are no black holes' Notion of an 'event horizon', from which nothing can escape, is incompatible with quantum theory, physicist claims. Zeeya Merali 24 January 2014 Artist's impression VICTOR HABBICK VISIONS/SPL/Getty The defining characteristic of a black hole may have to give, if the two pillars of modern physics — general relativity and quantum theory — are both correct. Most physicists foolhardy enough to write a paper claiming that “there are no black holes” — at least not in the sense we usually imagine — would probably be dismissed as cranks. But when the call to redefine these cosmic crunchers comes from Stephen Hawking, it’s worth taking notice. In a paper posted online, the physicist, based at the University of Cambridge, UK, and one of the creators of modern black-hole theory, does away with the notion of an event horizon, the invisible boundary thought to shroud every black hole, beyond which nothing, not even light, can escape. In its stead, Hawking’s radical proposal is a much more benign “apparent horizon”, “There is no escape from which only temporarily holds matter and energy prisoner before eventually a black hole in classical releasing them, albeit in a more garbled form. theory, but quantum theory enables energy “There is no escape from a black hole in classical theory,” Hawking told Nature. Peter van den Berg/Photoshot and information to Quantum theory, however, “enables energy and information to escape from a escape.” black hole”. A full explanation of the process, the physicist admits, would require a theory that successfully merges gravity with the other fundamental forces of nature.
    [Show full text]
  • The Negative Energy in Generalized Vaidya Spacetime
    universe Communication The Negative Energy in Generalized Vaidya Spacetime Vitalii Vertogradov 1,2 1 Department of the Theoretical Physics and Astronomy, Herzen State Pedagogical University, 191186 St. Petersburg, Russia; [email protected] 2 The SAO RAS, Pulkovskoe Shosse 65, 196140 St. Petersburg, Russia Received: 17 August 2020; Accepted: 17 September 2020; Published: 22 September 2020 Abstract: In this paper we consider the negative energy problem in generalized Vaidya spacetime. We consider several models where we have the naked singularity as a result of the gravitational collapse. In these models we investigate the geodesics for particles with negative energy when the II type of the matter field satisfies the equation of the state P = ar (a 2 [0 , 1]). Keywords: generalized Vaidya spacetime; naked singularity; black hole; negative energy 1. Introduction In 1969 R. Penrose theoretically predicted the effect of the negative energy formation in the Kerr metric during the process of the decay or the collision. Furthermore, the nature of the geodesics for particles with negative energy was investigated [1,2]. It was shown that in the ergosphere of a rotating black hole closed orbits for such particles are absent. This geodesics must appear from the region inside the gravitational radius. Moreover, there was research devoted to the particles with negative energy in Schwarzschild spacetime which was conducted by A. A. Grib and Yu. V. Pavlov [3]. They showed that the particles with negative energy can exist only in the region inside of the event horizon. However, the Schwarzschild black hole is the eternal one and we must consider the gravitational collapse to speak about the past of the geodesics for particles with negative energy.
    [Show full text]
  • 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 .
    [Show full text]
  • M-Theory Solutions and Intersecting D-Brane Systems
    M-Theory Solutions and Intersecting D-Brane Systems A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in the Department of Physics and Engineering Physics University of Saskatchewan Saskatoon By Rahim Oraji ©Rahim Oraji, December/2011. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgrad- uate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Physics and Engineering Physics 116 Science Place University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E2 i Abstract It is believed that fundamental M-theory in the low-energy limit can be described effectively by D=11 supergravity.
    [Show full text]
  • The Conformal Flow of Metrics and the General Penrose Inequality 3
    THE CONFORMAL FLOW OF METRICS AND THE GENERAL PENROSE INEQUALITY QING HAN AND MARCUS KHURI Abstract. The conformal flow of metrics [2] has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for general initial data sets of the Einstein equations. The Penrose conjecture without the assumption of time symmetry is then reduced to solving a system of PDE with desirable properties. 1. Introduction Let (M,g,k) be an initial data set for the Einstein equations with a single asymptotically flat end. This triple consists of a 3-manifold M, on which a Riemannian metric g and symmetric 2-tensor k are defined and satisfy the constraint equations (1.1) 16πµ = R + (T rk)2 − |k|2, 8πJ = div(k + (T rk)g). The quantities µ and J represent the energy and momentum densities of the matter fields, respec- tively, whereas R denotes the scalar curvature of g. The dominant energy condition will be assumed µ ≥ |J|. This asserts that all measured energy densities are nonnegative, and implies that matter cannot travel faster than the speed of light. Null expansions measure the strength of the gravitational field around a hypersurface S ⊂ M and are given by (1.2) θ± := HS ± T rSk, where HS denotes the mean curvature with respect to the unit normal pointing towards spatial infinity.
    [Show full text]
  • Richard Arnowitt, Bhaskar Dutta Texas A&M University Dark Matter And
    Dark Matter and Supersymmetry Models Richard Arnowitt, Bhaskar Dutta Texas A&M University 1 Outline . Understand dark matter in the context of particle physics models . Consider models with grand unification motivated by string theory . Check the cosmological connections of these well motivated models at direct, indirect detection and collider experiments 2 Dark Matter: Thermal Production of thermal non-relativistic DM: 1 Dark Matter content: ~ DM v m freeze out T ~ DM f 20 3 m/T 26 cm v 310 Y becomes constant for T>T s f 2 a ~O(10-2) with m ~ O(100) GeV Assuming : v ~ c c f 2 leads to the correct relic abundance m 3 Anatomy of sann Co-annihilation Process ~ 0 1 Griest, Seckel ’91 2 ΔM M~ M ~0 1 1 + ~ ΔM / kT f 1 e Arnowitt, Dutta, Santoso, Nucl.Phys. B606 (2001) 59, A near degeneracy occurs naturally for light stau in mSUGRA. 4 Models 5 mSUGRA Parameter space Focus point Resonance Narrow blue line is the dark matter allowed region Coannihilation Region 1.3 TeV squark bound from the LHC Arnowitt, Dutta, Santoso, Phys.Rev. D64 (2001) 113010 , Arnowitt, Dutta, Hu, Santoso, Phys.Lett. B505 (2001) 177 6 mSUGRA Parameter space Arnowitt, ICHEP’00, Arnowitt, Dutta, Santoso, Nucl.Phys. B606 (2001) 59, 7 Small DM at the LHC (or l+l-, t+t) High PT jet [mass difference is large] DM The pT of jets and leptons depend on the sparticle masses which are given by Colored particles are models produced and they decay finally to the weakly interacting stable particle DM R-parity conserving (or l+l-, t+t) High PT jet The signal : jets + leptons+ t’s +W’s+Z’s+H’s + missing ET 8 Small DM via cascade Typical decay chain and final states at the LHC g~ Jets + t’s+ missing energy ~ u uL Low energy taus characterize s the small mass gap e s s a However, one needs to measure M the model parameters to Y S predict the dark matter 0 u U ~ χ content in this scenario S 2 Arnowitt, Dutta, Kamon, Kolev, Toback 0 ~ ~ 1 Phys.Lett.
    [Show full text]
  • Roland E. Allen
    Roland E. Allen Department of Physics and Astronomy Texas A&M University College Station, Texas 77843-4242 [email protected] , http://people.physics.tamu.edu/allen/ Education: B.A., Physics, Rice University, 1963 Ph.D., Physics, University of Texas at Austin, 1968 Research: Theoretical Physics Positions: Research Associate, University of Texas at Austin, 1969 - 1970 Resident Associate, Argonne National Laboratory, summers of 1967 - 69 Assistant Professor of Physics, Texas A&M University, 1970 - 1976 Associate Professor of Physics, Texas A&M University, 1976 - 1983 Sabbatical Scientist, Solar Energy Research Institute, 1979 - 1980 Visiting Associate Professor of Physics, University of Illinois, 1980 - 1981 Professor of Physics, Texas A&M University, 1983 – Present Honors and Research Activities Honors Program Teacher/Scholar Award, 2005 University Teaching Award, 2004 College of Science Teaching Award, 2003 Deputy Editor, Physica Scripta (published for Royal Swedish Academy of Sciences), 2013-2016 Service on NSF, DOE, and university program evaluation panels, 1988-present Co-organizer of Richard Arnowitt Symposium (September 19-20, 2014) Organizer of Second Mitchell Symposium on Astronomy, Cosmology, and Fundamental Physics (April 10-14, 2006) Organizing Committee, Fifth Conference on Dark Matter in Astroparticle Physics (October 3-9, 2004) Organizer of Mitchell Symposium on Observational Cosmology (April 11-16, 2004) Organizer of Institute for Quantum Studies Research for Undergraduates Program (Summer, 2003) Organizer of Richard Arnowitt
    [Show full text]
  • Black Holes and Black Hole Thermodynamics Without Event Horizons
    General Relativity and Gravitation (2011) DOI 10.1007/s10714-008-0739-9 RESEARCHARTICLE Alex B. Nielsen Black holes and black hole thermodynamics without event horizons Received: 18 June 2008 / Accepted: 22 November 2008 c Springer Science+Business Media, LLC 2009 Abstract We investigate whether black holes can be defined without using event horizons. In particular we focus on the thermodynamic properties of event hori- zons and the alternative, locally defined horizons. We discuss the assumptions and limitations of the proofs of the zeroth, first and second laws of black hole mechan- ics for both event horizons and trapping horizons. This leads to the possibility that black holes may be more usefully defined in terms of trapping horizons. We also review how Hawking radiation may be seen to arise from trapping horizons and discuss which horizon area should be associated with the gravitational entropy. Keywords Black holes, Black hole thermodynamics, Hawking radiation, Trapping horizons Contents 1 Introduction . 2 2 Event horizons . 4 3 Local horizons . 8 4 Thermodynamics of black holes . 14 5 Area increase law . 17 6 Gravitational entropy . 19 7 The zeroth law . 22 8 The first law . 25 9 Hawking radiation for trapping horizons . 34 10 Fluid flow analogies . 36 11 Uniqueness . 37 12 Conclusion . 39 A. B. Nielsen Center for Theoretical Physics and School of Physics College of Natural Sciences, Seoul National University Seoul 151-742, Korea [email protected] 2 A. B. Nielsen 1 Introduction Black holes play a central role in physics. In astrophysics, they represent the end point of stellar collapse for sufficiently large stars.
    [Show full text]
  • Verlinde's Emergent Gravity and Whitehead's Actual Entities
    The Founding of an Event-Ontology: Verlinde's Emergent Gravity and Whitehead's Actual Entities by Jesse Sterling Bettinger A Dissertation submitted to the Faculty of Claremont Graduate University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate Faculty of Religion and Economics Claremont, California 2015 Approved by: ____________________________ ____________________________ © Copyright by Jesse S. Bettinger 2015 All Rights Reserved Abstract of the Dissertation The Founding of an Event-Ontology: Verlinde's Emergent Gravity and Whitehead's Actual Entities by Jesse Sterling Bettinger Claremont Graduate University: 2015 Whitehead’s 1929 categoreal framework of actual entities (AE’s) are hypothesized to provide an accurate foundation for a revised theory of gravity to arise compatible with Verlinde’s 2010 emergent gravity (EG) model, not as a fundamental force, but as the result of an entropic force. By the end of this study we should be in position to claim that the EG effect can in fact be seen as an integral sub-sequence of the AE process. To substantiate this claim, this study elaborates the conceptual architecture driving Verlinde’s emergent gravity hypothesis in concert with the corresponding structural dynamics of Whitehead’s philosophical/scientific logic comprising actual entities. This proceeds to the extent that both are shown to mutually integrate under the event-based covering logic of a generative process underwriting experience and physical ontology. In comparing the components of both frameworks across the epistemic modalities of pure philosophy, string theory, and cosmology/relativity physics, this study utilizes a geomodal convention as a pre-linguistic, neutral observation language—like an augur between the two theories—wherein a visual event-logic is progressively enunciated in concert with the specific details of both models, leading to a cross-pollinized language of concepts shown to mutually inform each other.
    [Show full text]
  • Particles Meet Cosmology and Strings in Boston
    PASCOS 2004 Particles meet cosmology and strings in Boston PASCOS 2004 is the latest in the symposium series that brings together disciplines from the frontier areas of modern physics. Participants at PASCOS 2004 and the Pran Nath Fest, which were held at Northeastern University, Boston. They include Howard Baer - front row sixth from left - then, moving right, Alfred Bartl, Michael Dine, Bruno Zumino, Pran Nath, Steven Weinberg, Paul Frampton, Mariano Quiros, Richard Arnowitt, MaryKGaillard, Peter Nilles and Michael Vaughn (chair, local organizing committee). The Tenth International Symposium on Particles, Strings and Cos­ redshift surveys suggests that the critical matter density of the uni­ mology took place at Northeastern University, Boston, on 16-22 verse is Qm ~ 0.3, direct dynamical measurements combined with August 2004. Two days of the symposium, 18-19 August, were the estimates of the luminosity density indicate Qm = 0.1-0.2. She devoted to the Pran Nath Fest in celebration of the 65th birthday of suggested that the apparent discrepancy may result from variations Matthews University Distinguished Professor Pran Nath. The PASCOS in the dark-matter fraction with mass and scale. She also suggested symposium is the largest interdisciplinary gathering on the interface that gravitational lensing maps combined with large redshift sur­ of the three disciplines of cosmology, particle physics and string veys promise to measure the dark-matter distribution in the uni­ theory, which have become increasingly entwined in recent years. verse. The microwave background can also provide clues to inflation Topics at PASCOS 2004 included the large-scale structure of the in the early universe.
    [Show full text]