Richard A. Mould. Basic Relativity. With 144 Figures. Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Size: px
Start display at page:

Download "Richard A. Mould. Basic Relativity. With 144 Figures. Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest"

Transcription

1 Richard A. Mould Basic Relativity With 144 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

2 Contents Preface vii PARTI 1. Principles of Relativity Galileo's Principle A Century of Electricity and Magnetism Maxwell's Equations Stellar Aberration The Michelson-Morley Experiment The Trouton-Noble Experiment 13 Problems The Physical Arguments Physical Ideas Some Applications Velocity Addition The Twin Paradox The Pole in the Barn Paradox Coordinate Frames of Reference 42 Problems The Algebraic and Graphic Arguments The Lorentz Transformation Other Applications Velocity Addition The Invariant Interval The Minkowski Diagram Use of the Minkowski Diagram Four-Vectors Velocity and Acceleration Four-Vectors The Propagation Four-Vector Doppler Effect Experimental Evidence Kinematics 81 Problems Mathematical Tools Matrices The Lorentz Transformation Vector Operators Tensors 103 XI

3 xii Contents 4.5 The Metric Inequality 107 Summary 109 Problems Dynamics The Physical Assumptions The Euler-Lagrange Formalism The Momentum Four-Vector The Four-Force Torque Collisions Experimental Evidence Dynamics 142 Problems Electromagnetic Theory ^^^_ Electric and Magnetic Fields Lorentz Force Moving Magnet Problem Trouton-Noble Experiment Maxwell's Equations Electromagnetic Potentials Energy-Momentum Tensor 168 Problems 170 PART II 7. Differential Geometry I The Scalar Invariant The Metric Tensor Vectors The Rectilinear Case The Polar Case Contravariant Metric Tensor Tensors 191 Summary of Tensor Algebra Parallel Displacement The Geodesic Path Parallel Displacement of Covariant Vectors Covariant Derivatives Space-Time Differential Geometry 213 Summary of Four-Vectors 217 Problems Uniform Acceleration Nonrigid Bodies Accelerating a Point Mass 224

4 Contents xiii 8.3 A Uniformly Accelerated Frame Uniformly Accelerated Coordinates The Matter of Metric 232 Summary of Metric Relationships Kinematic Characteristics of the System Falling Bodies Geodesic Paths Falling Clocks A Supported Object Local Coordinates 253 Summary of Kinematic Relationships Dynamics Gravitational Force and Constants of Motion 261 Problems Rotation and the Electromagnetic Field The Rotation Transformation Physical Interpretation The Geodesic Equation Dynamics General Electromagnetic Fields Nongeodesic Paths Generally Covariant Field Equations 283 Problems The Material Medium The Energy-Momentum Tensor Dust Particles Ideal Gas Internal Forces The Total Tensor 295 Problems Differential Geometry II: Curved Surfaces A Spherical Surface A Curvature Criterion Curvature Tensor on a Sphere Ricci Tensor and the Scalar Curvature 307 Problems General Relativity The Principle of Equivalence Einstein's Field Equation Evaluation of the Constant The Schwarzschild Solution Kinematic Characteristics of the Field 324

5 xiv Contents 12.6 Falling Bodies Four-Velocity Dynamics Theory as Construct Three Tests of General Relativity New Tests and Challenges 344 Problems Astrophysics Compact Objects Black Holes Rotating Black Holes Evidence for Compact Objects Gravity Waves 377 Problems Cosmology The Cosmological Principle The Cosmological Constant Three-Dimensional Hypersurface General Solution of the Field Equation Einstein and de Sitter Solutions The Matter-Dominated Universe Critical Mass Measuring a Flat, Matter-Dominated Universe The Inflationary Universe 416 Problems 423 Appendixes A. The Lorentz Transformation 425 B. Calculus of Variations 427 С The Geodesic Equations 430 D. The Geodesic Equation in Coordinate Form 431 E. Uniformly Accelerated Transformation Equations 432 F. The Riemann-Christoffel Curvature Tensor 434 G. Transformation to the Tangent Plane 436 H. General Lorentz Transformation and the Stress Tensor Answers to Selected Problems 439 References 443 Index 445

Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas

Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS Contents Introduction l 1 From absolute space

More information

Tensor Calculus, Relativity, and Cosmology

Tensor Calculus, Relativity, and Cosmology Tensor Calculus, Relativity, and Cosmology A First Course by M. Dalarsson Ericsson Research and Development Stockholm, Sweden and N. Dalarsson Royal Institute of Technology Stockholm, Sweden ELSEVIER ACADEMIC

More information

Giinter Ludyk. Einstein in Matrix. Form. Exact Derivation of the Theory of Special. without Tensors. and General Relativity.

Giinter Ludyk. Einstein in Matrix. Form. Exact Derivation of the Theory of Special. without Tensors. and General Relativity. Giinter Ludyk Einstein in Matrix Form Exact Derivation of the Theory of Special and General Relativity without Tensors ^ Springer Contents 1 Special Relativity 1 1.1 Galilei Transformation 1 1.1.1 Relativity

More information

Class Notes Introduction to Relativity Physics 375R Under Construction

Class Notes Introduction to Relativity Physics 375R Under Construction Class Notes Introduction to Relativity Physics 375R Under Construction Austin M. Gleeson 1 Department of Physics University of Texas at Austin Austin, TX 78712 March 20, 2007 1 gleeson@physics.utexas.edu

More information

8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline

8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 1 Main Headings I Introduction and relativity pre Einstein II Einstein s principle of relativity and a new concept of spacetime III

More information

Relativity Discussion

Relativity Discussion Relativity Discussion 4/19/2007 Jim Emery Einstein and his assistants, Peter Bergmann, and Valentin Bargmann, on there daily walk to the Institute for advanced Study at Princeton. Special Relativity The

More information

Geometry for Physicists

Geometry for Physicists Hung Nguyen-Schafer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 4 i Springer Contents 1 General Basis and Bra-Ket Notation 1 1.1 Introduction to

More information

RELG - General Relativity

RELG - General Relativity Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2017 230 - ETSETB - Barcelona School of Telecommunications Engineering 749 - MAT - Department of Mathematics 748 - FIS - Department

More information

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific

More information

Relativity, Gravitation, and Cosmology

Relativity, Gravitation, and Cosmology Relativity, Gravitation, and Cosmology A basic introduction TA-PEI CHENG University of Missouri St. Louis OXFORD UNIVERSITY PRESS Contents Parti RELATIVITY Metric Description of Spacetime 1 Introduction

More information

2.5.1 Static tides Tidal dissipation Dynamical tides Bibliographical notes Exercises 118

2.5.1 Static tides Tidal dissipation Dynamical tides Bibliographical notes Exercises 118 ii Contents Preface xiii 1 Foundations of Newtonian gravity 1 1.1 Newtonian gravity 2 1.2 Equations of Newtonian gravity 3 1.3 Newtonian field equation 7 1.4 Equations of hydrodynamics 9 1.4.1 Motion of

More information

INTRODUCTION TO GENERAL RELATIVITY

INTRODUCTION TO GENERAL RELATIVITY INTRODUCTION TO GENERAL RELATIVITY RONALD ADLER Instito de Fisica Universidade Federal de Pemambuco Recife, Brazil MAURICE BAZIN Department of Physics Rutgers University MENAHEM SCHIFFER Department of

More information

Classical Field Theory

Classical Field Theory April 13, 2010 Field Theory : Introduction A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word classical is used in

More information

A GENERAL RELATIVITY WORKBOOK. Thomas A. Moore. Pomona College. University Science Books. California. Mill Valley,

A GENERAL RELATIVITY WORKBOOK. Thomas A. Moore. Pomona College. University Science Books. California. Mill Valley, A GENERAL RELATIVITY WORKBOOK Thomas A. Moore Pomona College University Science Books Mill Valley, California CONTENTS Preface xv 1. INTRODUCTION 1 Concept Summary 2 Homework Problems 9 General Relativity

More information

Preface Exposition and Paradoxes Organization of this Book... 5

Preface Exposition and Paradoxes Organization of this Book... 5 Contents I. A FIRST PASS 1 Preface 2 0.1 Exposition and Paradoxes................ 2 0.2 Organization of this Book................ 5 1 Introduction to the Paradoxes 11 1.1 Aristotle vs. Galileo...................

More information

Lecture Notes on General Relativity

Lecture Notes on General Relativity Lecture Notes on General Relativity Matthias Blau Albert Einstein Center for Fundamental Physics Institut für Theoretische Physik Universität Bern CH-3012 Bern, Switzerland The latest version of these

More information

Quantum Mechanics: Foundations and Applications

Quantum Mechanics: Foundations and Applications Arno Böhm Quantum Mechanics: Foundations and Applications Third Edition, Revised and Enlarged Prepared with Mark Loewe With 96 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

More information

Class Notes Introduction to Modern Physics Physics 321 Plan II Under Construction

Class Notes Introduction to Modern Physics Physics 321 Plan II Under Construction Class Notes Introduction to Modern Physics Physics 321 Plan II Under Construction Austin M. Gleeson 1 Department of Physics University of Texas at Austin Austin, TX 78712 January 15, 2010 1 gleeson@physics.utexas.edu

More information

The Theory of Relativity

The Theory of Relativity The Theory of Relativity Lee Chul Hoon chulhoon@hanyang.ac.kr Copyright 2001 by Lee Chul Hoon Table of Contents 1. Introduction 2. The Special Theory of Relativity The Galilean Transformation and the Newtonian

More information

General Relativity and Cosmology Mock exam

General Relativity and Cosmology Mock exam Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers

More information

Modern Geometric Structures and Fields

Modern Geometric Structures and Fields Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

More information

Fisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica

Fisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica Fisica Matematica Stefano Ansoldi Dipartimento di Matematica e Informatica Università degli Studi di Udine Corso di Laurea in Matematica Anno Accademico 2003/2004 c 2004 Copyright by Stefano Ansoldi and

More information

ENTER RELATIVITY THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION 8/19/2016

ENTER RELATIVITY THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION 8/19/2016 ENTER RELATIVITY RVBAUTISTA THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION The laws of mechanics must be the same in all inertial

More information

Title. Author(s)Greve, Ralf. Issue Date Doc URL. Type. Note. File Information. A material called spacetime

Title. Author(s)Greve, Ralf. Issue Date Doc URL. Type. Note. File Information. A material called spacetime Title A material called spacetime Author(s)Greve, Ralf Issue Date 2017-08-21 Doc URL http://hdl.handle.net/2115/67121 Type lecture Note Colloquium of Mechanics, Study Center Mechanics, Dar File Information

More information

Advanced Theoretical Physics A Historical Perspective. Nick Lucid

Advanced Theoretical Physics A Historical Perspective. Nick Lucid Advanced Theoretical Physics A Historical Perspective Nick Lucid June 2015 ii Contents Preface ix 1 Coordinate Systems 1 1.1 Cartesian............................. 2 1.2 Polar and Cylindrical.......................

More information

The 77th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world

The 77th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world The 77th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world Lecture 6: Space-time geometry General relativity and some topology Efi Efrati Simons postdoctoral

More information

Pedagogical Strategy

Pedagogical Strategy Integre Technical Publishing Co., Inc. Hartle November 18, 2002 1:42 p.m. hartlemain19-end page 557 Pedagogical Strategy APPENDIX D...as simple as possible, but not simpler. attributed to A. Einstein The

More information

Nonlinear Problems of Elasticity

Nonlinear Problems of Elasticity Stuart S. Antman Nonlinear Problems of Elasticity With 105 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Contents Preface vn Chapter I. Background

More information

Gravity and action at a distance

Gravity and action at a distance Gravitational waves Gravity and action at a distance Newtonian gravity: instantaneous action at a distance Maxwell's theory of electromagnetism: E and B fields at distance D from charge/current distribution:

More information

Einstein Toolkit Workshop. Joshua Faber Apr

Einstein Toolkit Workshop. Joshua Faber Apr Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms

More information

An introduction to gravitational waves. Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France)

An introduction to gravitational waves. Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France) An introduction to gravitational waves Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France) Outline of lectures (1/2) The world's shortest introduction to General Relativity The linearized

More information

A. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens , Greece

A. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens , Greece SPECIAL RELATIVITY A. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens 157 71, Greece Abstract We give an introduction to Einstein s Special Theory of Relativity.

More information

Practical Quantum Mechanics

Practical Quantum Mechanics Siegfried Flügge Practical Quantum Mechanics With 78 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Volume I I. General Concepts 1. Law of probability

More information

Talking about general relativity Important concepts of Einstein s general theory of relativity. Øyvind Grøn Berlin July 21, 2016

Talking about general relativity Important concepts of Einstein s general theory of relativity. Øyvind Grøn Berlin July 21, 2016 Talking about general relativity Important concepts of Einstein s general theory of relativity Øyvind Grøn Berlin July 21, 2016 A consequence of the special theory of relativity is that the rate of a clock

More information

First structure equation

First structure equation First structure equation Spin connection Let us consider the differential of the vielbvein it is not a Lorentz vector. Introduce the spin connection connection one form The quantity transforms as a vector

More information

Lorentz transformation

Lorentz transformation 13 Mar 2012 Equivalence Principle. Einstein s path to his field equation 15 Mar 2012 Tests of the equivalence principle 20 Mar 2012 General covariance. Math. Covariant derivative è Homework 6 is due on

More information

Basic Algebraic Geometry 1

Basic Algebraic Geometry 1 Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Table of Contents Volume 1

More information

Bibliography. Introduction to General Relativity and Cosmology Downloaded from

Bibliography. Introduction to General Relativity and Cosmology Downloaded from Bibliography Abbott, B. P. et al. (2016). Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102. Ade, P. A. R. et al. (2015). Planck 2015 results. XIII. Cosmological

More information

The Schwarzschild Metric

The Schwarzschild Metric The Schwarzschild Metric The Schwarzschild metric describes the distortion of spacetime in a vacuum around a spherically symmetric massive body with both zero angular momentum and electric charge. It is

More information

CLASSICAL ELECTRICITY

CLASSICAL ELECTRICITY CLASSICAL ELECTRICITY AND MAGNETISM by WOLFGANG K. H. PANOFSKY Stanford University and MELBA PHILLIPS Washington University SECOND EDITION ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo

More information

PH5011 General Relativity

PH5011 General Relativity PH5011 General Relativity Martinmas 2012/2013 Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk 0 General issues 0.1 Summation convention dimension of coordinate space pairwise

More information

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor. Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση

More information

RELATIVITY Basis of a Theory of Relativity

RELATIVITY Basis of a Theory of Relativity 30 000 by BlackLight Power, Inc. All rights reserved. RELATIVITY Basis of a Theory of Relativity 1 To describe any phenomenon such as the motion of a body or the propagation of light, a definite frame

More information

Geometry of the Universe: Cosmological Principle

Geometry of the Universe: Cosmological Principle Geometry of the Universe: Cosmological Principle God is an infinite sphere whose centre is everywhere and its circumference nowhere Empedocles, 5 th cent BC Homogeneous Cosmological Principle: Describes

More information

GRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are.

GRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are. GRAVITATION F0 S. G. RAJEEV Lecture. Maxwell s Equations in Curved Space-Time.. Recall that Maxwell equations in Lorentz covariant form are. µ F µν = j ν, F µν = µ A ν ν A µ... They follow from the variational

More information

Sliding Modes in Control and Optimization

Sliding Modes in Control and Optimization Vadim I. Utkin Sliding Modes in Control and Optimization With 24 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Parti. Mathematical Tools 1

More information

ON VARIATION OF THE METRIC TENSOR IN THE ACTION OF A PHYSICAL FIELD

ON VARIATION OF THE METRIC TENSOR IN THE ACTION OF A PHYSICAL FIELD ON VARIATION OF THE METRIC TENSOR IN THE ACTION OF A PHYSICAL FIELD L.D. Raigorodski Abstract The application of the variations of the metric tensor in action integrals of physical fields is examined.

More information

EXCISION TECHNIQUE IN CONSTRAINED FORMULATIONS OF EINSTEIN EQUATIONS

EXCISION TECHNIQUE IN CONSTRAINED FORMULATIONS OF EINSTEIN EQUATIONS EXCISION TECHNIQUE IN CONSTRAINED FORMULATIONS OF EINSTEIN EQUATIONS Journée Gravitation et Physique Fondamentale Meudon, 27 May 2014 Isabel Cordero-Carrión Laboratoire Univers et Théories (LUTh), Observatory

More information

An Introduction to Riemann-Finsler Geometry

An Introduction to Riemann-Finsler Geometry D. Bao S.-S. Chern Z. Shen An Introduction to Riemann-Finsler Geometry With 20 Illustrations Springer Contents Preface Acknowledgments vn xiii PART ONE Finsler Manifolds and Their Curvature CHAPTER 1 Finsler

More information

Michael Tsamparlis. Special Relativity. An Introduction with 200 Problems and Solutions

Michael Tsamparlis. Special Relativity. An Introduction with 200 Problems and Solutions Special Relativity Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions 123 Dr. Michael Tsamparlis Department of Astrophysics, Astronomy and Mechanics University of Athens

More information

Index. Cambridge University Press A First Course in General Relativity: Second Edition Bernard F. Schutz. Index.

Index. Cambridge University Press A First Course in General Relativity: Second Edition Bernard F. Schutz. Index. accelerated particle, 41 acceleration, 46 48 absolute, 2 of the universe, 351 353 accretion disk, 317 active gravitational mass, 197, 202, 355 adiabatic, 103 affine parameter, 161, 166, 175 angular diameter

More information

Keywords: Golden metric tensor, Geodesic equation, coefficient of affine connection, Newton s planetary equation, Einstein s planetary equation.

Keywords: Golden metric tensor, Geodesic equation, coefficient of affine connection, Newton s planetary equation, Einstein s planetary equation. A first approximation of the generalized planetary equations based upon Riemannian geometry and the golden metric tensor N. Yakubu 1, S.X.K Howusu 2, L.W. Lumbi 3, A. Babangida 4, I. Nuhu 5 1) Department

More information

Uniqueness theorems, Separation of variables for Poisson's equation

Uniqueness theorems, Separation of variables for Poisson's equation NPTEL Syllabus Electrodynamics - Web course COURSE OUTLINE The course is a one semester advanced course on Electrodynamics at the M.Sc. Level. It will start by revising the behaviour of electric and magnetic

More information

Two postulates Relativity of simultaneity Time dilation; length contraction Lorentz transformations Doppler effect Relativistic kinematics

Two postulates Relativity of simultaneity Time dilation; length contraction Lorentz transformations Doppler effect Relativistic kinematics Two postulates Relativity of simultaneity Time dilation; length contraction Lorentz transformations Doppler effect Relativistic kinematics Phys 2435: Chap. 37, Pg 1 Two postulates New Topic Phys 2435:

More information

ELECTROMAGNETIC FIELDS AND RELATIVISTIC PARTICLES

ELECTROMAGNETIC FIELDS AND RELATIVISTIC PARTICLES ELECTROMAGNETIC FIELDS AND RELATIVISTIC PARTICLES Emil J. Konopinski Professor of Physics Indiana University McGraw-Hill Book Company New York St. Louis San Francisco Auckland Bogota Hamburg Johannesburg

More information

The Mathematics of Minkowski Space-Time

The Mathematics of Minkowski Space-Time Frontiers in Mathematics The Mathematics of Minkowski Space-Time With an Introduction to Commutative Hypercomplex Numbers Bearbeitet von Francesco Catoni, Dino Boccaletti, Roberto Cannata, Vincenzo Catoni,

More information

Fundamental Theories of Physics in Flat and Curved Space-Time

Fundamental Theories of Physics in Flat and Curved Space-Time Fundamental Theories of Physics in Flat and Curved Space-Time Zdzislaw Musielak and John Fry Department of Physics The University of Texas at Arlington OUTLINE General Relativity Our Main Goals Basic Principles

More information

A Dyad Theory of Hydrodynamics and Electrodynamics

A Dyad Theory of Hydrodynamics and Electrodynamics arxiv:physics/0502072v5 [physics.class-ph] 19 Jan 2007 A Dyad Theory of Hydrodynamics and Electrodynamics Preston Jones Department of Mathematics and Physics University of Louisiana at Monroe Monroe, LA

More information

INVESTIGATING THE KERR BLACK HOLE USING MAPLE IDAN REGEV. Department of Mathematics, University of Toronto. March 22, 2002.

INVESTIGATING THE KERR BLACK HOLE USING MAPLE IDAN REGEV. Department of Mathematics, University of Toronto. March 22, 2002. INVESTIGATING THE KERR BLACK HOLE USING MAPLE 1 Introduction IDAN REGEV Department of Mathematics, University of Toronto March 22, 2002. 1.1 Why Study the Kerr Black Hole 1.1.1 Overview of Black Holes

More information

Curved Spacetime III Einstein's field equations

Curved Spacetime III Einstein's field equations Curved Spacetime III Einstein's field equations Dr. Naylor Note that in this lecture we will work in SI units: namely c 1 Last Week s class: Curved spacetime II Riemann curvature tensor: This is a tensor

More information

THE SPECIAL THEORY OF RELATIVITY

THE SPECIAL THEORY OF RELATIVITY THE SPECIAL THEORY OF RELATIVITY A TitIe in the Nature-Macmillan Physics Series THE SPECIAL THEORY OF RELATIVITY H. Muirhead University of Liverpool Macmillan Education H. Muirhead 1973 Softcover reprint

More information

A Brief Introduction to Mathematical Relativity

A Brief Introduction to Mathematical Relativity A Brief Introduction to Mathematical Relativity Arick Shao Imperial College London Arick Shao (Imperial College London) Mathematical Relativity 1 / 31 Special Relativity Postulates and Definitions Einstein

More information

Integrated Phd in Physical Sciences PH 516: General Relativity and Cosmology

Integrated Phd in Physical Sciences PH 516: General Relativity and Cosmology Indian Association for the Cultivation of Science Department of Theoretical Physics Integrated Phd in Physical Sciences PH 516: General Relativity and Cosmology Instructor: Soumitra SenGupta and Sumanta

More information

Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity

Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity Rob Salgado UW-La Crosse Dept. of Physics outline motivations the Cayley-Klein Geometries (various starting points)

More information

Jose Luis Blázquez Salcedo

Jose Luis Blázquez Salcedo Physical Review Letters 112 (2014) 011101 Jose Luis Blázquez Salcedo In collaboration with Jutta Kunz, Eugen Radu and Francisco Navarro Lérida 1. Introduction: Ansatz and general properties 2. Near-horizon

More information

ANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS

ANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS ANALYTICAL MECHANICS LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS Preface xi 1 LAGRANGIAN MECHANICS l 1.1 Example and Review of Newton's Mechanics: A Block Sliding on an Inclined Plane 1

More information

Problem 1, Lorentz transformations of electric and magnetic

Problem 1, Lorentz transformations of electric and magnetic Problem 1, Lorentz transformations of electric and magnetic fields We have that where, F µν = F µ ν = L µ µ Lν ν F µν, 0 B 3 B 2 ie 1 B 3 0 B 1 ie 2 B 2 B 1 0 ie 3 ie 2 ie 2 ie 3 0. Note that we use the

More information

General Relativity and Differential

General Relativity and Differential Lecture Series on... General Relativity and Differential Geometry CHAD A. MIDDLETON Department of Physics Rhodes College November 1, 2005 OUTLINE Distance in 3D Euclidean Space Distance in 4D Minkowski

More information

INDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226

INDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226 INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence

More information

Analytical Mechanics for Relativity and Quantum Mechanics

Analytical Mechanics for Relativity and Quantum Mechanics Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION:

More information

Cosmological Nonlinear Density and Velocity Power Spectra. J. Hwang UFES Vitória November 11, 2015

Cosmological Nonlinear Density and Velocity Power Spectra. J. Hwang UFES Vitória November 11, 2015 Cosmological Nonlinear Density and Velocity Power Spectra J. Hwang UFES Vitória November 11, 2015 Perturbation method: Perturbation expansion All perturbation variables are small Weakly nonlinear Strong

More information

From An Apple To Black Holes Gravity in General Relativity

From An Apple To Black Holes Gravity in General Relativity From An Apple To Black Holes Gravity in General Relativity Gravity as Geometry Central Idea of General Relativity Gravitational field vs magnetic field Uniqueness of trajectory in space and time Uniqueness

More information

Lecture 1 General relativity and cosmology. Kerson Huang MIT & IAS, NTU

Lecture 1 General relativity and cosmology. Kerson Huang MIT & IAS, NTU A Superfluid Universe Lecture 1 General relativity and cosmology Kerson Huang MIT & IAS, NTU Lecture 1. General relativity and cosmology Mathematics and physics Big bang Dark energy Dark matter Robertson-Walker

More information

carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general

carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general http://pancake.uchicago.edu/ carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been

More information

4/13/2015. Outlines CHAPTER 12 ELECTRODYNAMICS & RELATIVITY. 1. The special theory of relativity. 2. Relativistic Mechanics

4/13/2015. Outlines CHAPTER 12 ELECTRODYNAMICS & RELATIVITY. 1. The special theory of relativity. 2. Relativistic Mechanics CHAPTER 12 ELECTRODYNAMICS & RELATIVITY Lee Chow Department of Physics University of Central Florida Orlando, FL 32816 Outlines 1. The special theory of relativity 2. Relativistic Mechanics 3. Relativistic

More information

A873: Cosmology Course Notes. II. General Relativity

A873: Cosmology Course Notes. II. General Relativity II. General Relativity Suggested Readings on this Section (All Optional) For a quick mathematical introduction to GR, try Chapter 1 of Peacock. For a brilliant historical treatment of relativity (special

More information

has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity.

has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. http://preposterousuniverse.com/grnotes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been framed

More information

Notes on General Relativity Linearized Gravity and Gravitational waves

Notes on General Relativity Linearized Gravity and Gravitational waves Notes on General Relativity Linearized Gravity and Gravitational waves August Geelmuyden Universitetet i Oslo I. Perturbation theory Solving the Einstein equation for the spacetime metric is tremendously

More information

Theoretical Aspects of Black Hole Physics

Theoretical Aspects of Black Hole Physics Les Chercheurs Luxembourgeois à l Etranger, Luxembourg-Ville, October 24, 2011 Hawking & Ellis Theoretical Aspects of Black Hole Physics Glenn Barnich Physique théorique et mathématique Université Libre

More information

Chapter 1. Introduction

Chapter 1. Introduction Chapter 1 Introduction The book Introduction to Modern Physics: Theoretical Foundations starts with the following two paragraphs [Walecka (2008)]: At the end of the 19th century, one could take pride in

More information

Null Cones to Infinity, Curvature Flux, and Bondi Mass

Null Cones to Infinity, Curvature Flux, and Bondi Mass Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,

More information

Week 9: Einstein s field equations

Week 9: Einstein s field equations Week 9: Einstein s field equations Riemann tensor and curvature We are looking for an invariant characterisation of an manifold curved by gravity. As the discussion of normal coordinates showed, the first

More information

Superluminal motion in the quasar 3C273

Superluminal motion in the quasar 3C273 1 Superluminal motion in the quasar 3C273 The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is

More information

Black Hole Physics. Basic Concepts and New Developments KLUWER ACADEMIC PUBLISHERS. Valeri P. Frolov. Igor D. Nbvikov. and

Black Hole Physics. Basic Concepts and New Developments KLUWER ACADEMIC PUBLISHERS. Valeri P. Frolov. Igor D. Nbvikov. and Black Hole Physics Basic Concepts and New Developments by Valeri P. Frolov Department of Physics, University of Alberta, Edmonton, Alberta, Canada and Igor D. Nbvikov Theoretical Astrophysics Center, University

More information

arxiv: v1 [gr-qc] 17 May 2008

arxiv: v1 [gr-qc] 17 May 2008 Gravitation equations, and space-time relativity arxiv:0805.2688v1 [gr-qc] 17 May 2008 L. V. VEROZUB Kharkov National University Kharkov, 61103 Ukraine Abstract In contrast to electrodynamics, Einstein

More information

Tutorial I General Relativity

Tutorial I General Relativity Tutorial I General Relativity 1 Exercise I: The Metric Tensor To describe distances in a given space for a particular coordinate system, we need a distance recepy. The metric tensor is the translation

More information

Group Theory and Its Applications in Physics

Group Theory and Its Applications in Physics T. Inui Y Tanabe Y. Onodera Group Theory and Its Applications in Physics With 72 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Contents Sections marked with

More information

An introduction to General Relativity and the positive mass theorem

An introduction to General Relativity and the positive mass theorem An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of

More information

Lecture IX: Field equations, cosmological constant, and tides

Lecture IX: Field equations, cosmological constant, and tides Lecture IX: Field equations, cosmological constant, and tides Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: October 28, 2011) I. OVERVIEW We are now ready to construct Einstein

More information

An Introduction to General Relativity and Cosmology

An Introduction to General Relativity and Cosmology An Introduction to General Relativity and Cosmology Jerzy Plebariski Centro de Investigacion y de Estudios Avanzados Instituto Politecnico Nacional Apartado Postal 14-740, 07000 Mexico D.F., Mexico Andrzej

More information

Jose Luis Blázquez Salcedo

Jose Luis Blázquez Salcedo Jose Luis Blázquez Salcedo In collaboration with Jutta Kunz, Francisco Navarro Lérida, and Eugen Radu GR Spring School, March 2015, Brandenburg an der Havel 1. Introduction 2. General properties of EMCS-AdS

More information

Frank Y. Wang. Physics with MAPLE. The Computer Algebra Resource for Mathematical Methods in Physics. WILEY- VCH WILEY-VCH Verlag GmbH & Co.

Frank Y. Wang. Physics with MAPLE. The Computer Algebra Resource for Mathematical Methods in Physics. WILEY- VCH WILEY-VCH Verlag GmbH & Co. Frank Y. Wang Physics with MAPLE The Computer Algebra Resource for Mathematical Methods in Physics WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA k Preface Guide for Users Bibliography XI XVII XIX 1 Introduction

More information

ELECTRICITY AND MAGNETISM

ELECTRICITY AND MAGNETISM THIRD EDITION ELECTRICITY AND MAGNETISM EDWARD M. PURCELL DAVID J. MORIN Harvard University, Massachusetts Щ CAMBRIDGE Ell UNIVERSITY PRESS Preface to the third edition of Volume 2 XIII CONTENTS Preface

More information

Fundamental Cosmology: 4.General Relativistic Cosmology

Fundamental Cosmology: 4.General Relativistic Cosmology Fundamental Cosmology: 4.General Relativistic Cosmology Matter tells space how to curve. Space tells matter how to move. John Archibald Wheeler 1 4.1: Introduction to General Relativity Introduce the Tools

More information

Marcelo Alonso. Edward J. Finn. Georgetown University. Prentice Hall

Marcelo Alonso. Edward J. Finn. Georgetown University. Prentice Hall PHYSICS Marcelo Alonso Florida Institute of Technology Edward J. Finn Georgetown University PEARSON Prentice Hall Harlow, England " London " New York " Boston " San Francisco -Toronto Sydney " Tokyo "

More information

The Boltzmann Equation and Its Applications

The Boltzmann Equation and Its Applications Carlo Cercignani The Boltzmann Equation and Its Applications With 42 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo CONTENTS PREFACE vii I. BASIC PRINCIPLES OF THE KINETIC

More information

16. Einstein and General Relativistic Spacetimes

16. Einstein and General Relativistic Spacetimes 16. Einstein and General Relativistic Spacetimes Problem: Special relativity does not account for the gravitational force. To include gravity... Geometricize it! Make it a feature of spacetime geometry.

More information

1.4 LECTURE 4. Tensors and Vector Identities

1.4 LECTURE 4. Tensors and Vector Identities 16 CHAPTER 1. VECTOR ALGEBRA 1.3.2 Triple Product The triple product of three vectors A, B and C is defined by In tensor notation it is A ( B C ) = [ A, B, C ] = A ( B C ) i, j,k=1 ε i jk A i B j C k =

More information

Chapter 11. Special Relativity

Chapter 11. Special Relativity Chapter 11 Special Relativity Note: Please also consult the fifth) problem list associated with this chapter In this chapter, Latin indices are used for space coordinates only eg, i = 1,2,3, etc), while

More information

Chapter 7 Curved Spacetime and General Covariance

Chapter 7 Curved Spacetime and General Covariance Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME

More information