Schwarzschild and Kerr Solutions of Einstein's Field Equation
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A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
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. -
Schwarzschild Solution to Einstein's General Relativity
Schwarzschild Solution to Einstein's General Relativity Carson Blinn May 17, 2017 Contents 1 Introduction 1 1.1 Tensor Notations . .1 1.2 Manifolds . .2 2 Lorentz Transformations 4 2.1 Characteristic equations . .4 2.2 Minkowski Space . .6 3 Derivation of the Einstein Field Equations 7 3.1 Calculus of Variations . .7 3.2 Einstein-Hilbert Action . .9 3.3 Calculation of the Variation of the Ricci Tenosr and Scalar . 10 3.4 The Einstein Equations . 11 4 Derivation of Schwarzschild Metric 12 4.1 Assumptions . 12 4.2 Deriving the Christoffel Symbols . 12 4.3 The Ricci Tensor . 15 4.4 The Ricci Scalar . 19 4.5 Substituting into the Einstein Equations . 19 4.6 Solving and substituting into the metric . 20 5 Conclusion 21 References 22 Abstract This paper is intended as a very brief review of General Relativity for those who do not want to skimp on the details of the mathemat- ics behind how the theory works. This paper mainly uses [2], [3], [4], and [6] as a basis, and in addition contains short references to more in-depth references such as [1], [5], [7], and [8] when more depth was needed. With an introduction to manifolds and notation, special rel- ativity can be constructed which are the relativistic equations of flat space-time. After flat space-time, the Lagrangian and calculus of vari- ations will be introduced to construct the Einstein-Hilbert action to derive the Einstein field equations. With the field equations at hand the Schwarzschild equation will fall out with a few assumptions. -
Black Holes. the Universe. Today’S Lecture
Physics 311 General Relativity Lecture 18: Black holes. The Universe. Today’s lecture: • Schwarzschild metric: discontinuity and singularity • Discontinuity: the event horizon • Singularity: where all matter falls • Spinning black holes •The Universe – its origin, history and fate Schwarzschild metric – a vacuum solution • Recall that we got Schwarzschild metric as a solution of Einstein field equation in vacuum – outside a spherically-symmetric, non-rotating massive body. This metric does not apply inside the mass. • Take the case of the Sun: radius = 695980 km. Thus, Schwarzschild metric will describe spacetime from r = 695980 km outwards. The whole region inside the Sun is unreachable. • Matter can take more compact forms: - white dwarf of the same mass as Sun would have r = 5000 km - neutron star of the same mass as Sun would be only r = 10km • We can explore more spacetime with such compact objects! White dwarf Black hole – the limit of Schwarzschild metric • As the massive object keeps getting more and more compact, it collapses into a black hole. It is not just a denser star, it is something completely different! • In a black hole, Schwarzschild metric applies all the way to r = 0, the black hole is vacuum all the way through! • The entire mass of a black hole is concentrated in the center, in the place called the singularity. Event horizon • Let’s look at the functional form of Schwarzschild metric again: ds2 = [1-(2m/r)]dt2 – [1-(2m/r)]-1dr2 - r2dθ2 -r2sin2θdφ2 • We want to study the radial dependence only, and at fixed time, i.e. we set dφ = dθ = dt = 0. -
Review Study on “The Black Hole”
IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 10 | March 2016 ISSN (online): 2349-6010 Review Study on “The Black Hole” Syed G. Ibrahim Department of Engineering Physics (Nanostructured Thin Film Materials Laboratory) Prof. Ram Meghe College of Engineering and Management, Badnera 444701, Maharashtra, India Abstract As a star grows old, swells, then collapses on itself, often you will hear the word “black hole” thrown around. The black hole is a gravitationally collapsed mass, from which no light, matter, or signal of any kind can escape. These exotic objects have captured our imagination ever since they were predicted by Einstein's Theory of General Relativity in 1915. So what exactly is a black hole? A black hole is what remains when a massive star dies. Not every star will become a black hole, only a select few with extremely large masses. In order to have the ability to become a black hole, a star will have to have about 20 times the mass of our Sun. No known process currently active in the universe can form black holes of less than stellar mass. This is because all present black hole formation is through gravitational collapse, and the smallest mass which can collapse to form a black hole produces a hole approximately 1.5-3.0 times the mass of the sun .Smaller masses collapse to form white dwarf stars or neutron stars. Keywords: Escape Velocity, Horizon, Schwarzschild Radius, Black Hole _______________________________________________________________________________________________________ I. INTRODUCTION Soon after Albert Einstein formulated theory of relativity, it was realized that his equations have solutions in closed form. -
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. -
Spacetimes with Singularities Ovidiu Cristinel Stoica
Spacetimes with Singularities Ovidiu Cristinel Stoica To cite this version: Ovidiu Cristinel Stoica. Spacetimes with Singularities. An. Stiint. Univ. Ovidius Constanta, Ser. Mat., 2012, pp.26. hal-00617027v2 HAL Id: hal-00617027 https://hal.archives-ouvertes.fr/hal-00617027v2 Submitted on 30 Nov 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spacetimes with Singularities ∗†‡ Ovidiu-Cristinel Stoica Abstract We report on some advances made in the problem of singularities in general relativity. First is introduced the singular semi-Riemannian geometry for met- rics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein’s equation, which works for degenerate metric too. Once we make the singularities manageable from mathematical view- point, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. -
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 . -
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 513–554 http://dx.doi.org/10.1090/bull/1544 Article electronically published on August 2, 2016 THE EMERGENCE OF GRAVITATIONAL WAVE SCIENCE: 100 YEARS OF DEVELOPMENT OF MATHEMATICAL THEORY, DETECTORS, NUMERICAL ALGORITHMS, AND DATA ANALYSIS TOOLS MICHAEL HOLST, OLIVIER SARBACH, MANUEL TIGLIO, AND MICHELE VALLISNERI In memory of Sergio Dain Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detec- tion of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole bina- ries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. -
Generalizations of the Kerr-Newman Solution
Generalizations of the Kerr-Newman solution Contents 1 Topics 1035 1.1 ICRANetParticipants. 1035 1.2 Ongoingcollaborations. 1035 1.3 Students ............................... 1035 2 Brief description 1037 3 Introduction 1039 4 Thegeneralstaticvacuumsolution 1041 4.1 Line element and field equations . 1041 4.2 Staticsolution ............................ 1043 5 Stationary generalization 1045 5.1 Ernst representation . 1045 5.2 Representation as a nonlinear sigma model . 1046 5.3 Representation as a generalized harmonic map . 1048 5.4 Dimensional extension . 1052 5.5 Thegeneralsolution . 1055 6 Tidal indicators in the spacetime of a rotating deformed mass 1059 6.1 Introduction ............................. 1059 6.2 The gravitational field of a rotating deformed mass . 1060 6.2.1 Limitingcases. 1062 6.3 Circularorbitsonthesymmetryplane . 1064 6.4 Tidalindicators ........................... 1065 6.4.1 Super-energy density and super-Poynting vector . 1067 6.4.2 Discussion.......................... 1068 6.4.3 Limit of slow rotation and small deformation . 1069 6.5 Multipole moments, tidal Love numbers and Post-Newtonian theory................................. 1076 6.6 Concludingremarks . 1077 7 Neutrino oscillations in the field of a rotating deformed mass 1081 7.1 Introduction ............................. 1081 7.2 Stationary axisymmetric spacetimes and neutrino oscillation . 1082 7.2.1 Geodesics .......................... 1083 1033 Contents 7.2.2 Neutrinooscillations . 1084 7.3 Neutrino oscillations in the Hartle-Thorne metric . 1085 7.4 Concludingremarks . 1088 8 Gravitational field of compact objects in general relativity 1091 8.1 Introduction ............................. 1091 8.2 The Hartle-Thorne metrics . 1094 8.2.1 The interior solution . 1094 8.2.2 The Exterior Solution . 1096 8.3 The Fock’s approach . 1097 8.3.1 The interior solution . -
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 .