The Point-Coincidence Argument and Einstein's Struggle with The
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Schwarzschild's Solution And
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server RECONSIDERING SCHWARZSCHILD'S ORIGINAL SOLUTION SALVATORE ANTOCI AND DIERCK EKKEHARD LIEBSCHER Abstract. We analyse the Schwarzschild solution in the context of the historical development of its present use, and explain the invariant definition of the Schwarzschild’s radius as a singular surface, that can be applied to the Kerr-Newman solution too. 1. Introduction: Schwarzschild's solution and the \Schwarzschild" solution Nowadays simply talking about Schwarzschild’s solution requires a pre- liminary reassessment of the historical record as conditio sine qua non for avoiding any misunderstanding. In fact, the present-day reader must be firstly made aware of this seemingly peculiar circumstance: Schwarzschild’s spherically symmetric, static solution [1] to the field equations of the ver- sion of the theory proposed by Einstein [2] at the beginning of November 1915 is different from the “Schwarzschild” solution that is quoted in all the textbooks and in all the research papers. The latter, that will be here al- ways mentioned with quotation marks, was found by Droste, Hilbert and Weyl, who worked instead [3], [4], [5] by starting from the last version [6] of Einstein’s theory.1 As far as the vacuum is concerned, the two versions have identical field equations; they differ only because of the supplementary condition (1.1) det (g )= 1 ik − that, in the theory of November 11th, limited the covariance to the group of unimodular coordinate transformations. Due to this fortuitous circum- stance, Schwarzschild could not simplify his calculations by the choice of the radial coordinate made e.g. -
General Relativity As Taught in 1979, 1983 By
General Relativity As taught in 1979, 1983 by Joel A. Shapiro November 19, 2015 2. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro c Joel A. Shapiro, 1979, 2012 Contents 0.1 Introduction............................ 4 0.2 SpecialRelativity ......................... 7 0.3 Electromagnetism. 13 0.4 Stress-EnergyTensor . 18 0.5 Equivalence Principle . 23 0.6 Manifolds ............................. 28 0.7 IntegrationofForms . 41 0.8 Vierbeins,Connections . 44 0.9 ParallelTransport. 50 0.10 ElectromagnetisminFlatSpace . 56 0.11 GeodesicDeviation . 61 0.12 EquationsDeterminingGeometry . 67 0.13 Deriving the Gravitational Field Equations . .. 68 0.14 HarmonicCoordinates . 71 0.14.1 ThelinearizedTheory . 71 0.15 TheBendingofLight. 77 0.16 PerfectFluids ........................... 84 0.17 Particle Orbits in Schwarzschild Metric . 88 0.18 AnIsotropicUniverse. 93 0.19 MoreontheSchwarzschild Geometry . 102 0.20 BlackHoleswithChargeandSpin. 110 0.21 Equivalence Principle, Fermions, and Fancy Formalism . .115 0.22 QuantizedFieldTheory . .121 3 4. Last Latexed: November 19, 2015 at 11:03 Joel A. Shapiro Note: This is being typed piecemeal in 2012 from handwritten notes in a red looseleaf marked 617 (1983) but may have originated in 1979 0.1 Introduction I am, myself, an elementary particle physicist, and my interest in general relativity has come from the growth of a field of quantum gravity. Because the gravitational inderactions of reasonably small objects are so weak, quantum gravity is a field almost entirely divorced from contact with reality in the form of direct confrontation with experiment. There are three areas of contact 1. In relativistic quantum mechanics, one usually formulates the physical quantities in terms of fields. A field is a physical degree of freedom, or variable, definded at each point of space and time. -
General Relativity
GENERALRELATIVITY t h i m o p r e i s1 17th April 2020 1 [email protected] CONTENTS 1 differential geomtry 3 1.1 Differentiable manifolds 3 1.2 The tangent space 4 1.2.1 Tangent vectors 6 1.3 Dual vectors and tensor 6 1.3.1 Tensor densities 8 1.4 Metric 11 1.5 Connections and covariant derivatives 13 1.5.1 Note on exponential map/Riemannian normal coordinates - TO DO 18 1.6 Geodesics 20 1.6.1 Equivalent deriavtion of the Geodesic Equation - Weinberg 22 1.6.2 Character of geodesic motion is sustained and proper time is extremal on geodesics 24 1.6.3 Another remark on geodesic equation using the principle of general covariance 26 1.6.4 On the parametrization of the path 27 1.7 An equivalent consideration of parallel transport, geodesics 29 1.7.1 Formal solution to the parallel transport equa- tion 31 1.8 Curvature 33 1.8.1 Torsion and metric connection 34 1.8.2 How to get from the connection coefficients to the connection-the metric connection 34 1.8.3 Conceptional flow of how to add structure on our mathematical constructs 36 1.8.4 The curvature 37 1.8.5 Independent components of the Riemann tensor and intuition for curvature 39 1.8.6 The Ricci tensor 41 1.8.7 The Einstein tensor 43 1.9 The Lie derivative 43 1.9.1 Pull-back and Push-forward 43 1.9.2 Connection between coordinate transformations and diffeomorphism 47 1.9.3 The Lie derivative 48 1.10 Symmetric Spaces 50 1.10.1 Killing vectors 50 1.10.2 Maximally Symmetric Spaces and their Unique- ness 54 iii iv co n t e n t s 1.10.3 Maximally symmetric spaces and their construc- tion -
Aspectos De Relatividade Numérica Campos Escalares E Estrelas De Nêutrons
UNIVERSIDADE DE SÃO PAULO INSTITUTO DE FÍSICA Aspectos de Relatividade Numérica Campos escalares e estrelas de nêutrons Leonardo Rosa Werneck Orientador: Prof. Dr. Elcio Abdalla Uma tese apresentada ao Instituto de Física da Universidade de São Paulo como parte dos requisitos necessários para obter o título de doutor em Física. Banca examinadora: Prof. Dr. Elcio Abdalla (IF-USP) – Presidente da banca Prof. Dr. Arnaldo Gammal (IF-USP) Prof. Dr. Daniel A. Turolla Vanzela (IFSC-USP) Prof. Dr. Alberto V. Saa (IFGW-UNICAMP) Profa. Dra. Cecilia Bertoni Chirenti (UFABC/UMD/NASA) São Paulo 2020 FICHA CATALOGRÁFICA Preparada pelo Serviço de Biblioteca e Informação do Instituto de Física da Universidade de São Paulo Werneck, Leonardo Rosa Aspectos de relatividade numérica: campos escalares e estrelas de nêutrons / Aspects of Numerical Relativity: scalar fields and neutron stars. São Paulo, 2020. Tese (Doutorado) − Universidade de São Paulo. Instituto de Física. Depto. Física Geral. Orientador: Prof. Dr. Elcio Abdalla Área de Concentração: Relatividade e Gravitação Unitermos: 1. Relatividade numérica; 2. Campo escalar; 3. Fenômeno crítico; 4. Estrelas de nêutrons; 5. Equações diferenciais parciais. USP/IF/SBI-057/2020 UNIVERSITY OF SÃO PAULO INSTITUTE OF PHYSICS Aspects of Numerical Relativity Scalar fields and neutron stars Leonardo Rosa Werneck Advisor: Prof. Dr. Elcio Abdalla A thesis submitted to the Institute of Physics of the University of São Paulo in partial fulfillment of the requirements for the title of Doctor of Philosophy in Physics. Examination committee: Prof. Dr. Elcio Abdalla (IF-USP) – Committee president Prof. Dr. Arnaldo Gammal (IF-USP) Prof. Dr. Daniel A. Turolla Vanzela (IFSC-USP) Prof. -
Symbols 1-Form, 10 4-Acceleration, 184 4-Force, 115 4-Momentum, 116
Index Symbols B 1-form, 10 Betti number, 589 4-acceleration, 184 Bianchi type-I models, 448 4-force, 115 Bianchi’s differential identities, 60 4-momentum, 116 complex valued, 532–533 total, 122 consequences of in Newman-Penrose 4-velocity, 114 formalism, 549–551 first contracted, 63 second contracted, 63 A bicharacteristic curves, 620 acceleration big crunch, 441 4-acceleration, 184 big-bang cosmological model, 438, 449 Newtonian, 74 Birkhoff’s theorem, 271 action function or functional bivector space, 486 (see also Lagrangian), 594, 598 black hole, 364–433 ADM action, 606 Bondi-Metzner-Sachs group, 240 affine parameter, 77, 79 boost, 110 alternating operation, antisymmetriza- Born-Infeld (or tachyonic) scalar field, tion, 27 467–471 angle field, 44 Boyer-Lindquist coordinate chart, 334, anisotropic fluid, 218–220, 276 399 collapse, 424–431 Brinkman-Robinson-Trautman met- ric, 514 anti-de Sitter space-time, 195, 644, Buchdahl inequality, 262 663–664 bugle, 69 anti-self-dual, 671 antisymmetric oriented tensor, 49 C antisymmetric tensor, 28 canonical energy-momentum-stress ten- antisymmetrization, 27 sor, 119 arc length parameter, 81 canonical or normal forms, 510 arc separation function, 85 Cartesian chart, 44, 68 arc separation parameter, 77 Casimir effect, 638 Arnowitt-Deser-Misner action integral, Cauchy horizon, 403, 416 606 Cauchy problem, 207 atlas, 3 Cauchy-Kowalewski theorem, 207 complete, 3 causal cone, 108 maximal, 3 causal space-time, 663 698 Index 699 causality violation, 663 contravariant index, 51 characteristic hypersurface, 623 contravariant -
The Reissner-Nordström Metric
The Reissner-Nordström metric Jonatan Nordebo March 16, 2016 Abstract A brief review of special and general relativity including some classi- cal electrodynamics is given. We then present a detailed derivation of the Reissner-Nordström metric. The derivation is done by solving the Einstein-Maxwell equations for a spherically symmetric electrically charged body. The physics of this spacetime is then studied. This includes gravitational time dilation and redshift, equations of motion for both massive and massless non-charged particles derived from the geodesic equation and equations of motion for a massive charged par- ticle derived with lagrangian formalism. Finally, a quick discussion of the properties of a Reissner-Nordström black hole is given. 1 Contents 1 Introduction 3 2 Review of Special Relativity 3 2.1 4-vectors . 6 2.2 Electrodynamics in Special Relativity . 8 3 Tensor Fields and Manifolds 11 3.1 Covariant Differentiation and Christoffel Symbols . 13 3.2 Riemann Tensor . 15 3.3 Parallel Transport and Geodesics . 18 4 Basics of General Relativity 19 4.1 The Equivalence Principle . 19 4.2 The Principle of General Covariance . 20 4.3 Electrodynamics in General Relativity . 21 4.4 Newtonian Limit of the Geodesic Equation . 22 4.5 Einstein’s Field Equations . 24 5 The Reissner-Nordström Metric 25 5.1 Gravitational Time Dilation and Redshift . 32 5.2 The Geodesic Equation . 34 5.2.1 Comparison to Newtonian Mechanics . 37 5.2.2 Circular Orbits of Photons . 38 5.3 Motion of a Charged Particle . 38 5.4 Event Horizons and Black Holes . 40 6 Summary and Conclusion 44 2 1 Introduction In 1915 Einstein completed his general theory of relativity. -
Dawn of Fluid Dynamics : a Discipline Between Science And
Titelei Eckert 11.04.2007 14:04 Uhr Seite 3 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology WILEY-VCH Verlag GmbH & Co. KGaA Titelei Eckert 11.04.2007 14:04 Uhr Seite 1 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology Titelei Eckert 11.04.2007 14:04 Uhr Seite 2 Related Titles R. Ansorge Mathematical Models of Fluiddynamics Modelling, Theory, Basic Numerical Facts - An Introduction 187 pages with 30 figures 2003 Hardcover ISBN 3-527-40397-3 J. Renn (ed.) Albert Einstein - Chief Engineer of the Universe 100 Authors for Einstein. Essays approx. 480 pages 2005 Hardcover ISBN 3-527-40574-7 D. Brian Einstein - A Life 526 pages 1996 Softcover ISBN 0-471-19362-3 Titelei Eckert 11.04.2007 14:04 Uhr Seite 3 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology WILEY-VCH Verlag GmbH & Co. KGaA Titelei Eckert 11.04.2007 14:04 Uhr Seite 4 The author of this book All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and Dr. Michael Eckert publisher do not warrant the information Deutsches Museum München contained in these books, including this book, to email: [email protected] be free of errors. Readers are advised to keep in mind that statements, data, illustrations, proce- Cover illustration dural details or other items may inadvertently be “Wake downstream of a thin plate soaked in a inaccurate. water flow” by Henri Werlé, with kind permission from ONERA, http://www.onera.fr Library of Congress Card No.: applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library. -
Conformal Field Theory and Black Hole Physics
CONFORMAL FIELD THEORY AND BLACK HOLE PHYSICS Steve Sidhu Bachelor of Science, University of Northern British Columbia, 2009 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfilment of the Requirements for the Degree MASTER OF SCIENCE Department of Physics and Astronomy University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Steve Sidhu, 2012 Dedication To my parents, my sister, and Paige R. Ryan. iii Abstract This thesis reviews the use of 2-dimensional conformal field theory applied to gravity, specifically calculating Bekenstein-Hawking entropy of black holes in (2+1) dimen- sions. A brief review of general relativity, Conformal Field Theory, energy extraction from black holes, and black hole thermodynamics will be given. The Cardy formula, which calculates the entropy of a black hole from the AdS/CFT duality, will be shown to calculate the correct Bekenstein-Hawking entropy of the static and rotating BTZ black holes. The first law of black hole thermodynamics of the static, rotating, and charged-rotating BTZ black holes will be verified. iv Acknowledgements I would like to thank my supervisors Mark Walton and Saurya Das. I would also like to thank Ali Nassar and Ahmed Farag Ali for the many discussions, the Theo- retical Physics Group, and the entire Department of Physics and Astronomy at the University of Lethbridge. v Table of Contents Approval/Signature Page ii Dedication iii Abstract iv Acknowledgements v Table of Contents vi 1 Introduction 1 2 Einstein’s field equations and black hole solutions 7 2.1 Conventions and notations . 7 2.2 Einstein’s field equations . -
The Formative Years of Relativity: the History and Meaning of Einstein's
© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 1 INTRODUCTION The Meaning of Relativity, also known as Four Lectures on Relativity, is Einstein’s definitive exposition of his special and general theories of relativity. It was written in the early 1920s, a few years after he had elaborated his general theory of rel- ativity. Neither before nor afterward did he offer a similarly comprehensive exposition that included not only the theory’s technical apparatus but also detailed explanations making his achievement accessible to readers with a certain mathematical knowledge but no prior familiarity with relativity theory. In 1916, he published a review paper that provided the first condensed overview of the theory but still reflected many features of the tortured pathway by which he had arrived at his new theory of gravitation in late 1915. An edition of the manuscript of this paper with introductions and detailed commentar- ies on the discussion of its historical contexts can be found in The Road to Relativity.1 Immediately afterward, Einstein wrote a nontechnical popular account, Relativity— The Special and General Theory.2 Beginning with its first German edition, in 1917, it became a global bestseller and marked the first triumph of relativity theory as a broad cultural phenomenon. We have recently republished this book with extensive commentaries and historical contexts that document its global success. These early accounts, however, were able to present the theory only in its infancy. Immediately after its publication on 25 November 1915, Einstein’s theory of general relativity was taken up, elaborated, and controversially discussed by his colleagues, who included physicists, mathematicians, astronomers, and philosophers. -
Black Hole Math Is Designed to Be Used As a Supplement for Teaching Mathematical Topics
National Aeronautics and Space Administration andSpace Aeronautics National ole M a th B lack H i This collection of activities, updated in February, 2019, is based on a weekly series of space science problems distributed to thousands of teachers during the 2004-2013 school years. They were intended as supplementary problems for students looking for additional challenges in the math and physical science curriculum in grades 10 through 12. The problems are designed to be ‘one-pagers’ consisting of a Student Page, and Teacher’s Answer Key. This compact form was deemed very popular by participating teachers. The topic for this collection is Black Holes, which is a very popular, and mysterious subject among students hearing about astronomy. Students have endless questions about these exciting and exotic objects as many of you may realize! Amazingly enough, many aspects of black holes can be understood by using simple algebra and pre-algebra mathematical skills. This booklet fills the gap by presenting black hole concepts in their simplest mathematical form. General Approach: The activities are organized according to progressive difficulty in mathematics. Students need to be familiar with scientific notation, and it is assumed that they can perform simple algebraic computations involving exponentiation, square-roots, and have some facility with calculators. The assumed level is that of Grade 10-12 Algebra II, although some problems can be worked by Algebra I students. Some of the issues of energy, force, space and time may be appropriate for students taking high school Physics. For more weekly classroom activities about astronomy and space visit the NASA website, http://spacemath.gsfc.nasa.gov Add your email address to our mailing list by contacting Dr. -
Arch and Scaffold: How Einstein Found His Field Equations Michel Janssen, and Jürgen Renn
Arch and scaffold: How Einstein found his field equations Michel Janssen, and Jürgen Renn Citation: Physics Today 68, 11, 30 (2015); doi: 10.1063/PT.3.2979 View online: https://doi.org/10.1063/PT.3.2979 View Table of Contents: http://physicstoday.scitation.org/toc/pto/68/11 Published by the American Institute of Physics Articles you may be interested in The laws of life Physics Today 70, 42 (2017); 10.1063/PT.3.3493 Hidden worlds of fundamental particles Physics Today 70, 46 (2017); 10.1063/PT.3.3594 The image of scientists in The Big Bang Theory Physics Today 70, 40 (2017); 10.1063/PT.3.3427 The secret of the Soviet hydrogen bomb Physics Today 70, 40 (2017); 10.1063/PT.3.3524 Physics in 100 years Physics Today 69, 32 (2016); 10.1063/PT.3.3137 The top quark, 20 years after its discovery Physics Today 68, 46 (2015); 10.1063/PT.3.2749 Arch and scaffold: How Einstein found his field equations Michel Janssen and Jürgen Renn In his later years, Einstein often claimed that he had obtained the field equations of general relativity by choosing the mathematically most natural candidate. His writings during the period in which he developed general relativity tell a different story. his month marks the centenary of the pothesis he adopted about the nature of matter al- Einstein field equations, the capstone on lowed him to change those equations to equations the general theory of relativity and the that are generally covariant—that is, retain their highlight of Albert Einstein’s scientific form under arbitrary coordinate transformations. -
Untying the Knot: How Einstein Found His Way Back to Field Equations Discarded in the Zurich Notebook
MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science 2004 PREPRINT 264 Michel Janssen and Jürgen Renn Untying the Knot: How Einstein Found His Way Back to Field Equations Discarded in the Zurich Notebook MICHEL JANSSEN AND JÜRGEN RENN UNTYING THE KNOT: HOW EINSTEIN FOUND HIS WAY BACK TO FIELD EQUATIONS DISCARDED IN THE ZURICH NOTEBOOK “She bent down to tie the laces of my shoes. Tangled up in blue.” —Bob Dylan 1. INTRODUCTION: NEW ANSWERS TO OLD QUESTIONS Sometimes the most obvious questions are the most fruitful ones. The Zurich Note- book is a case in point. The notebook shows that Einstein already considered the field equations of general relativity about three years before he published them in Novem- ber 1915. In the spring of 1913 he settled on different equations, known as the “Entwurf” field equations after the title of the paper in which they were first pub- lished (Einstein and Grossmann 1913). By Einstein’s own lights, this move compro- mised one of the fundamental principles of his theory, the extension of the principle of relativity from uniform to arbitrary motion. Einstein had sought to implement this principle by constructing field equations out of generally-covariant expressions.1 The “Entwurf” field equations are not generally covariant. When Einstein published the equations, only their covariance under general linear transformations was assured. This raises two obvious questions. Why did Einstein reject equations of much broader covariance in 1912-1913? And why did he return to them in November 1915? A new answer to the first question has emerged from the analysis of the Zurich Notebook presented in this volume.