DOCSLIB.ORG

Geodesic Dynamics in Reissner-Nordström Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geodesic Dynamics in Reissner-Nordström Geometry Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 Original Research Article Geodesic Dynamics in Reissner-Nordström Geometry Daniel Abramson* 9/72 Soi Kasemsan 3, Rama 1 Road, Wangmai, Pathumwan, Bangkok 10330 *Corresponding author E-mail: [email protected] Received: 28th May 2021, Revised: 22nd July 2021, Accepted: 24th July 2021 Abstract We analyze geodesics (trajectories of free-falling bodies) in the geometry created by a Reissner-Nordström black hole. Bound and unbound radial orbits are obtained first in the superextremal case, then in the extremal and subextremal cases where geodesics penetrate a horizon, inside of which an outside observer cannot see. We determine the location of geodesics in conformal (Penrose) spacetime diagrams. We also deduce that the “river model”, a useful idea to explain the behavior of Schwarzschild black holes, fails in the case where electric charge is present. Keywords: Reissner-Nordström, Black hole, Geodesic, Penrose diagram Introduction The solution of Einstein’s field equations for a spherically symmetric mass distribution was found by Karl Schwarzschild in 1916 [1], leading to the theoretical possibility of black holes. The spacetime geometry created by a static, non-rotating black hole with mass 푚 and electric charge 푞 was determined independently by Hans Reissner in 1916 [2] and Gunnar Nordström in 1918 [3].1,2 The geometry created by such a charged black hole is determined by the metric 2푚 푞2 푑푟2 푑푠2 = − (1 − + ) 푑푡2 + + 푟2푑휃2 + (푟 sin 휃)2푑휙2 . 푟 푟2 1 − 2푚⁄푟 + 푞2⁄푟2 _____________________ 1 Geometrized units where 퐺 = 푐 = 1 are used in this paper, so that mass and charge have units of length. These quantities can be converted to units of length by multiplying by the conversion factors 퐺⁄푐2 = 7.43 × 10−28 m⁄kg 4 −18 √퐺⁄4휋휖0푐 = 8.62 × 10 m⁄C 2 We investigate only the case of static (eternal), spherically symmetric, non-rotating black holes. Black holes created at a finite prior time (by stellar collapse or some other means) are not considered. 69 Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 This is not a vacuum solution since the geometry is determined by both the mass and the electric field with energy density 퐄2⁄8휋 = 푞2⁄8휋푟4. Reissner-Nordström black holes receive cursory treatments in the text literature [4,5] or are introduced as exercises [6,7]. This can be attributed to circumstances and hypotheses which suggest that the phenomena associated with charged black holes are likely to be unphysical.3 Notwithstanding this, the Reissner-Nordström solution has intrinsic interest and shares features of the Kerr solution for rotating black holes. Useful Formulas and Identities We establish several elementary results which will be used repeatedly. A. Notation 푑휏푔 denotes proper time along a geodesic. When there is no possibility of confusion, we will write 푑휏. 푑휏푟 denotes proper time for a stationary shell observer 풮푟 located at radius 푟. ̇ A dot ⬚ always denotes differentiation with respect to proper time along a geodesic: 푟̇ ≡ 푑푟⁄푑휏푔. B. Shell observer speed We consider metrics of the form 푑푟2 푑푠2 = −푓(푟) 푑푡2 + + 푟2푑휃2 + (푟 sin 휃)2푑휙2 . (1) 푓(푟) We assume 푓(∞) = 1, so the spacetime geometry is asymptotically flat. In the case of radial motion (푑휃 = 푑휙 = 0) the metric simplifies to 푑푟2 푑푠2 = −푓(푟) 푑푡2 + . 푓(푟) A shell observer 풮푟, who is stationary on the sphere of radius 푟, measures the speed of an object in (timelike) motion in his frame of reference: 푑푠| proper length 푑푡=0 푣푟 = = . proper time √−푑푠2| 푑퐱=0 _____________________ 3 Such as the cosmic censorship conjecture, mass inflation instability, and the sheer amount of excess charge needed for the inner event horizon radius to reach the Compton wavelength of an elementary particle or even the Planck length. 70 Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 Therefore 푑푟2⁄푓(푟) 푑푟2⁄푑푡2 1 푑푟 푣2 = = ⟹ 푣 = . (2) 푟 푓(푟) 푑푡2 푓(푟)2 푟 푓(푟) 푑푡 4 We choose the positive square root in (2) so that 푣푟 has the same sign as 푑푟⁄푑푡 when 푓(푟) > 0. The proper time element 푑휏푟 for a shell observer 풮푟 is 2 푑휏푟 = √−푑푠 | = √푓(푟) 푑푡 . (3) 푑퐱=0 Therefore, by (2) and (3): 2 2 1 푑푟 1 푑푟 푑휏푔 푑휏푟 푟̇√1 − 푣푟 √푓(푟) 푟̇√1 − 푣푟 푣푟 = = = = . 푓(푟) 푑푡 푓(푟) 푑휏푔 푑휏푟 푑푡 푓(푟) √푓(푟) 2 Solving for 푣푟 : 푟̇2 푣2 = . (4) 푟 푓(푟) + 푟̇2 It follows that |푣푟| < 1 if and only if 푓(푟) > 0. C. Geodesic motion Timelike geodesic trajectories in the equatorial plane 휃 = 휋⁄2 satisfy the equations of motion:5 ℓ 휙̇ = 푟2 푒 푡̇ = (5) 푓(푟) ℓ2 푟̇2 = 푒2 − (1 + ) 푓(푟) 푟2 _____________________ 4 푓(푟) < 0 corresponds to a black or white hole; see (37) and Figure 13. Along a timelike path in such a region, 푑푟 never changes sign — 푑푟 < 0 in a black hole — but 푑푡 can be positive, negative or zero. Thus “ingoing” objects (푑푟⁄푑푡 < 0) and “outgoing” objects (푑푟⁄푑푡 > 0) both move in the direction of decreasing 푟. 5 A straightforward generalization of the derivation given by Dray [8] (pp. 21-23) for Schwarzschild geometry. 71 Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 where 푒 and ℓ are constants of the motion.6 For a geodesic that starts from rest at infinity, the third equation in (5) shows that 푒2 = 푓(∞) = 1. For a radial geodesic, 휙̇ = 0 implies ℓ = 0, whence 푟̇2 = 푒2 − 푓(푟). Inserting this into (4) yields two useful relations (taking 푒 > 0): 푟̇ 푣 = (6) 푟 푒 푓(푟) 푣 = 1 − (7) 푟 푒2 From (7) we may deduce that 푒 is the ratio of total energy 퐸 to mass 푀 of a free-falling object measured by a distant, stationary observer.7 We verify this as follows. In a potential energy field 푈 the total energy is 퐸 = 훾푀 + 푈, where 훾 = (1 − 푣2)−1⁄2 and 푈 are both functions of position. Hence: 푀 2 1 2 푓(푟) 푓(푟) ( ) = ( ) = 1 − 푣2 = 1 − (1 − ) = . 퐸 − 푈 훾 푒2 푒2 Thus (퐸 − 푈)⁄푀 = 푒⁄√푓(푟). Letting 푟 → ∞ we obtain 퐸⁄푀 = 푒. Henceforth all geodesics will henceforth be assumed to be radial. Preliminaries The Reissner-Nordström metric is given by (1) with 2푚 푞2 2푚(푟 − 훽) 푓(푟) = 1 − + = 1 − (8) 푟 푟2 푟2 where 훽 ≡ 푞2⁄2푚. The radial equation of motion along a geodesic is given by (5): 2푚(푟 − 훽) 푟̇2 = 푒2 − (1 − ) . (9) 푟2 For an object that starts from rest at infinity, 푒 = 1, so its equation of motion is 2푚(푟 − 훽) 푟̇2 = . (10) 푟2 _____________________ 6 ℓ is the angular momentum per unit mass. 7 All test “objects” are assumed to be uncharged, and therefore do not interact with the electric field. 72 Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 Since the trajectory is ingoing, we conclude: √2푚(푟 − 훽) 푟̇ = − . (11) 푟 We see that the object comes to a stop at 푟 = 훽 and moves no closer to the singularity. A shell observer 풮푟 measures the object’s speed from (6): 푟̇ √2푚(푟 − 훽) 푣 = = − (12) 푟 푒 푟 Since a shell observer cannot observe an object with nonzero mass traveling with speed |푣푟| ≥ 1, it must be that 2푚(푟 − 훽) 2푚(푟 − 훽) < 1 ⟺ 0 < 1 − = 푓(푟) ⟺ 푃(푟) = 푟2 − 2푚푟 + 푞2 > 0 . 푟2 푟2 2 2 푃(푟) has discriminant 4(푚 − 푞 ), so there are three distinct scenarios: (i) If |푞| > 푚, then 푃(푟) has no real zeros and |푣푟| < 1 for all 푟. This is a superextremal singularity. 2 2 (ii) If |푞| < 푚, then 푃(푟) = 0 at 푟± = 푚 ± √푚 − 푞 . In the region 푟− < 푟 < 푟+ we find 푓(푟) < 0. In this region |푣푟| > 1 so no shell observers can exist. The radii 푟− and 푟+ are the inner and outer event horizons. This case is called subextremal. (iii) If |푞| = 푚, then 푃(푟) = 0 at 푟 = 푚. There is only a single horizon, which separates regions where 푃(푟) > 0. This case is called extremal. We consider these cases separately. Superextremal Singularities A. Geodesics originating from 풓 = ∞ In the superextremal case, 푓(푟) > 0 for all 푟, so by (4) we have |푣푟| < 1 for all 훽 ≤ 푟 < ∞. This is apparent in Figure 1, which shows 푣푟 given by (12) for various values of 푞. The curves for 푞 > 푚 = 1 correspond to superextremal singularities; the other curves will be referenced later. 73 Thai Journal of Physics Vol. 38 No. 2 (2021) 69-98 Figure 1 Shell-observer speed 푣푟 for a free-falling object starting from rest at infinity. 푞 > 0 is Reissner-Nordström geometry; 푞 = 0 is Schwarzschild geometry. 푞 > 푚 (푚 = 1) corresponds to a superextremal singularity.8 The trajectory is obtained by integrating 푑푟 √2푚(푟 − 훽) 푟̇ = = − (13) 푑휏 푟 to obtain 1 −푟푑푟 2 2 휏 = ∫ = − ( ) √푟 − 훽 (푟 + 2훽) + 퐶 . (14) √2푚(푟 − 훽) 9푚 Choosing 퐶 = 0 is equivalent to the condition 푟 = 훽 at 휏 = 0. Eqn. (14) can be inverted to obtain the trajectory 푟(휏). Free fall times. A freely falling object starting from rest at infinity falls from 푟 = 푅 to 푟 = 훽 in finite proper time: 1 2 2 ∆휏 = ( ) √푅 − 훽 (푅 + 2훽) . (15) 9푚 _____________________ 8 We choose 푚 = 1 in all figures in this paper. 74 Thai Journal of Physics Vol.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • The Point-Coincidence Argument and Einstein's Struggle with The
    Nothing but Coincidences: The Point-Coincidence Argument and Einstein’s Struggle with the Meaning of Coordinates in Physics Marco Giovanelli Forum Scientiarum — Universität Tübingen, Doblerstrasse 33 72074 Tübingen, Germany [email protected] In his 1916 review paper on general relativity, Einstein made the often-quoted oracular remark that all physical measurements amount to a determination of coincidences, like the coincidence of a pointer with a mark on a scale. This argument, which was meant to express the requirement of general covariance, immediately gained great resonance. Philosophers like Schlick found that it expressed the novelty of general relativity, but the mathematician Kretschmann deemed it as trivial and valid in all spacetime theories. With the relevant exception of the physicists of Leiden (Ehrenfest, Lorentz, de Sitter, and Nordström), who were in epistolary contact with Einstein, the motivations behind the point-coincidence remark were not fully understood. Only at the turn of the 1960s did Bergmann (Einstein’s former assistant in Princeton) start to use the term ‘coincidence’ in a way that was much closer to Einstein’s intentions. In the 1980s, Stachel, projecting Bergmann’s analysis onto his historical work on Einstein’s correspondence, was able to show that what he started to call ‘the point-coincidence argument’ was nothing but Einstein’s answer to the infamous ‘hole argument.’ The latter has enjoyed enormous popularity in the following decades, reshaping the philosophical debate on spacetime theories. The point-coincidence argument did not receive comparable attention. By reconstructing the history of the argument and its reception, this paper argues that this disparity of treatment is not justied.
    [Show full text]
  • Events in Science, Mathematics, and Technology | Version 3.0
    EVENTS IN SCIENCE, MATHEMATICS, AND TECHNOLOGY | VERSION 3.0 William Nielsen Brandt | [email protected] Classical Mechanics -260 Archimedes mathematically works out the principle of the lever and discovers the principle of buoyancy 60 Hero of Alexandria writes Metrica, Mechanics, and Pneumatics 1490 Leonardo da Vinci describ es capillary action 1581 Galileo Galilei notices the timekeeping prop erty of the p endulum 1589 Galileo Galilei uses balls rolling on inclined planes to show that di erentweights fall with the same acceleration 1638 Galileo Galilei publishes Dialogues Concerning Two New Sciences 1658 Christian Huygens exp erimentally discovers that balls placed anywhere inside an inverted cycloid reach the lowest p oint of the cycloid in the same time and thereby exp erimentally shows that the cycloid is the iso chrone 1668 John Wallis suggests the law of conservation of momentum 1687 Isaac Newton publishes his Principia Mathematica 1690 James Bernoulli shows that the cycloid is the solution to the iso chrone problem 1691 Johann Bernoulli shows that a chain freely susp ended from two p oints will form a catenary 1691 James Bernoulli shows that the catenary curve has the lowest center of gravity that anychain hung from two xed p oints can have 1696 Johann Bernoulli shows that the cycloid is the solution to the brachisto chrone problem 1714 Bro ok Taylor derives the fundamental frequency of a stretched vibrating string in terms of its tension and mass p er unit length by solving an ordinary di erential equation 1733 Daniel Bernoulli
    [Show full text]
  • General Relativity
    PX436: General Relativity Gareth P. Alexander University of Warwick email: [email protected] office: D1.09, Zeeman building http://go.warwick.ac.uk/px436 Wednesday 6th December, 2017 Contents Preface iii Books and other reading iv 1 Gravity and Relativity2 1.1 Gravity........................................2 1.2 Special Relativity...................................6 1.2.1 Notation....................................8 1.3 Geometry of Minkowski Space-Time........................ 10 1.4 Particle Motion.................................... 12 1.5 The Stress-Energy-Momentum Tensor....................... 14 1.6 Electromagnetism................................... 15 1.6.1 The Transformation of Electric and Magnetic Fields........... 17 1.6.2 The Dynamical Maxwell Equations..................... 17 1.6.3 The Stress-Energy-Momentum Tensor................... 20 Problems.......................................... 23 2 Differential Geometry 27 2.1 Manifolds....................................... 27 2.2 The Metric...................................... 29 2.2.1 Lorentzian Metrics.............................. 31 2.3 Geodesics....................................... 32 2.4 Angles, Areas, Volumes, etc.............................. 35 2.5 Vectors, 1-Forms, Tensors.............................. 36 2.6 Curvature....................................... 39 2.7 Covariant Derivative................................. 46 2.7.1 Continuity, Conservation and Divergence................. 47 2.8 Summary....................................... 49 Problems.........................................
    [Show full text]
  • Arxiv:1703.09118V2
    Foundation of Physics manuscript No. (will be inserted by the editor) The Milky Way’s Supermassive Black Hole: How good a case is it? A Challenge for Astrophysics & Philosophy of Science Andreas Eckarta,b Andreas H¨uttemannc, Claus Kieferd, Silke Britzenb, Michal Zajaˇceka,b, Claus L¨ammerzahle, Manfred St¨ocklerf , Monika Valencia-S.a, Vladimir Karasg, Macarena Garc´ıa-Mar´ınh Received: date / Accepted: date 6 Abstract The compact and, with 4.3±0.3×10 M⊙, very massive object located at the center of the Milky Way is currently the very best candidate for a super- massive black hole (SMBH) in our immediate vicinity. The strongest evidence for this is provided by measurements of stellar orbits, variable X-ray emission, and strongly variable polarized near-infrared emission from the location of the radio source Sagittarius A* (SgrA*) in the middle of the central stellar cluster. Simul- taneous near-infrared and X-ray observations of SgrA* have revealed insights into the emission mechanisms responsible for the powerful near-infrared and X-ray flares from within a few tens to one hundred Schwarzschild radii of such a putative SMBH at the center of the Milky Way. If SgrA* is indeed a SMBH it will, in projection onto the sky, have the largest event horizon and will certainly be the first and most important target of the Event Horizon Telescope (EHT) Very Long Baseline In- terferometry (VLBI) observations currently being prepared. These observations in combination with the infrared interferometry experiment GRAVITY at the Very Large Telescope Interferometer (VLTI) and other experiments across the electro- magnetic spectrum might yield proof for the presence of a black hole at the center of the Milky Way.
    [Show full text]
  • Pos(Multif2019)028 ∗ [email protected] Speaker
    Astrophysical Black Holes: A Review PoS(Multif2019)028 Cosimo Bambi∗ Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China E-mail: [email protected] In this review, I have tried to focus on the development of the field, from the first speculations to the current lines of research. According to Einstein’s theory of general relativity, black holes are relatively simple objects and completely characterized by their mass, spin angular momentum, and electric charge, but the latter can be ignored in the case of astrophysical macroscopic objects. Search for black holes in the sky started in the early 1970s with the dynamical measurement of the mass of the compact object in Cygnus X-1. In the past 10-15 years, astronomers have developed some techniques for measuring the black hole spins. Recently, we have started using astrophysical black holes for testing fundamental physics. Multifrequency Behaviour of High Energy Cosmic Sources - XIII - MULTIF2019 3-8 June 2019 Palermo, Italy ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Astrophysical Black Holes Cosimo Bambi 1. Introduction 1.1 Historical overview Roughly speaking, a black hole is a region of the spacetime in which gravity is so strong that nothing can escape or communicate with the exterior region. Speculations on the existence of similar systems can be dated back to the end of the XVIII century. John Michell in 1783 and, independently, Pierre-Simon Laplace in 1796 discussed the possibility of the existence of extremely compact objects, so compact that the escape velocity on their surface could exceed the speed of light.
    [Show full text]
  • Geodesic Structure in Schwarzschild Geometry with Extensions in Higher Dimensional Spacetimes
    Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2018 GEODESIC STRUCTURE IN SCHWARZSCHILD GEOMETRY WITH EXTENSIONS IN HIGHER DIMENSIONAL SPACETIMES Ian M. Newsome Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Elementary Particles and Fields and String Theory Commons, and the Other Physics Commons © Ian M Newsome Downloaded from https://scholarscompass.vcu.edu/etd/5414 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. c Ian Marshall Newsome, May 2018 All Rights Reserved. GEODESIC STRUCTURE IN SCHWARZSCHILD GEOMETRY WITH EXTENSIONS IN HIGHER DIMENSIONAL SPACETIMES A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by IAN MARSHALL NEWSOME August 2016 to May 2018 Director: Robert H. Gowdy, Associate Professor, Associate Chair, Department of Physics Virginia Commonwewalth University Richmond, Virginia May, 2018 i Acknowledgements I would like to thank Dr. Robert Gowdy for his guidance during the creation of this thesis. His patience and assistance have been paramount to my understanding of the fields of differential geometry and the theory of relativity. I would also like to thank my mother and father for their continuing support during my higher education, without which I may have faltered long ago. ii TABLE OF CONTENTS Chapter Page Acknowledgements :::::::::::::::::::::::::::::::: ii Table of Contents :::::::::::::::::::::::::::::::: iii List of Tables ::::::::::::::::::::::::::::::::::: iv List of Figures :::::::::::::::::::::::::::::::::: v Abstract ::::::::::::::::::::::::::::::::::::: ix 1 Introduction ::::::::::::::::::::::::::::::::: 1 1.1 Background .
    [Show full text]