A Most Incomprehensible Thing · Notes Towards a Very Gentle Introduction to the Mathematics of Relativity by Collier, Peter -- Read -- Imperial Library of Trantor

Index

Title Page Copyright Page Dedication PREFACE INTRODUCTION 1 FOUNDATION MATHEMATICS

1.1 Introduction 1.2 Definitions and symbols 1.3 Units and constants 1.4 Expressions, equations and functions 1.5 Getting a feel for functions

1.5.1 Graphs of simple functions

1.6 Simplifying and rearranging equations and expressions 1.7 Trigonometric functions

1.7.1 Basic trigonometric functions 1.7.2 Radians 1.7.3 Trigonometric identities

1.8 Powers and logarithms

1.8.1 Powers 1.8.2 Logarithms 1.8.3 The constant e and natural logarithms 1.8.4 The exponential function

1.9 Coordinate systems

1.9.1 Cartesian coordinates 1.9.2 Plane polar coordinates 1.9.3 Spherical coordinates

1.10 Calculus

1.10.1 Differential calculus 1.10.2 Integral calculus 1.10.3 Taylor series

1.11 Parametric equations 1.12 Matrices

1.12.1 Matrix multiplication 1.12.2 Identity matrix 1.12.3 Inverse matrix 1.12.4 Symmetric matrices 1.12.5 Diagonal matrices

1.13 Introducing vectors

1.13.1 Vector fields 1.13.2 Gradient of a scalar field 1.13.3 Divergence of a vector field

1.14 Euclidean geometry

1.14.1 The Euclidean metric

1.15 Index notation

2 NEWTONIAN MECHANICS

2.1 Introduction 2.2 Newtonian space and time 2.3 Newtonian inertial frames of reference 2.4 Newton's three laws of motion

2.4.1 Newton's first law 2.4.2 Newton's second law 2.4.3 Newton's third law

2.5 Newton's law of universal gravitation

2.5.1 Statement of the law 2.5.2 How Newton derived his law of universal gravitation 2.5.3 Gravitational acceleration 2.5.4 How the law predicts elliptical orbits 2.5.5 The gravitational field 2.5.6 Tidal forces 2.5.7 Gravitational potential energy

3 SPECIAL RELATIVITY

3.1 Introduction 3.2 Basic concepts

3.2.1 Time 3.2.2 Spacetime 3.2.3 Events 3.2.4 Frames of reference 3.2.5 Inertial frames 3.2.6 Coordinate transformations 3.2.7 The Galilean transformations 3.2.8 The two postulates of special relativity

3.3 Spacetime diagrams

3.3.1 Units and notation 3.3.2 Spacetime diagrams - basic concepts 3.3.3 Adding a second observer 3.3.4 Simultaneity and causality 3.3.5 Invariance of the interval 3.3.6 Invariant hyperbolae

3.4 The Lorentz transformations

3.4.1 Deriving the Lorentz transformations 3.4.2 Lorentz transformation matrix 3.4.3 A second observer revisited 3.4.4 Interval transformation rules 3.4.5 Proper time and time dilation 3.4.6 Length contraction 3.4.7 The end of simultaneity 3.4.8 Velocity transformations 3.4.9 Invariance of the interval - again 3.4.10 The interval and proper time

3.5 The Minkowski metric 3.6 Mechanics in special relativity

3.6.1 Conservation laws 3.6.2 Invariance 3.6.3 Four-velocity 3.6.4 Relativistic momentum 3.6.5 Relativistic kinetic energy 3.6.6 Total relativistic energy 3.6.7 Four-momentum 3.6.8 Four-force 3.6.9 Energy-momentum relation

4 INTRODUCING THE MANIFOLD

4.1 Introduction 4.2 Riemannian manifolds and the metric 4.3 Surface of a sphere

5 SCALARS, VECTORS, ONE-FORMS AND TENSORS

5.1 Introduction 5.2 Scalars 5.3 Contravariant vectors and one-forms

5.3.1 Contravariant vectors 5.3.2 One-forms 5.3.3 A little more detail 5.3.4 Examples of one-forms and contravariant vectors 5.3.5 Definition of a contravariant vector 5.3.6 Definition of a one-form 5.3.7 More on the coordinate transformation of contravariant vectors 5.3.8 And more on the coordinate transformation of one-forms

5.4 Tensors

5.4.1 Tensor transformation 5.4.2 The rules of tensor algebra

5.5 Trying to visualise all this

6 MORE ON CURVATURE

6.1 Introduction 6.2 Connection coefficients 6.3 Covariant differentiation 6.4 Parallel transport of vectors 6.5 Geodesics 6.6 The Riemann curvature tensor 6.7 The Ricci tensor and Ricci scalar

6.7.1 The Ricci tensor 6.7.2 The Ricci scalar

7 GENERAL RELATIVITY

7.1 Introduction 7.2 The three key principles

7.2.1 The principle of equivalence 7.2.2 The principle of general covariance 7.2.3 The principle of consistency

7.3 Spacetime 7.4 Geodesics in spacetime

7.4.1 Geodesics on Earth 7.4.2 Geodesic deviation

7.5 The energy-momentum tensor

7.5.1 Dust 7.5.2 Perfect fluid 7.5.3 Covariant divergence of the energy-momentum tensor

7.6 The Einstein field equations 7.7 The cosmological constant

8 THE NEWTONIAN LIMIT

8.1 Introduction 8.2 Newton's three laws of motion

8.2.1 Newton's first law of motion 8.2.2 Newton's second law of motion 8.2.3 Newton's third law of motion

8.3 Newton's field equation for how matter responds to gravity 8.4 Newton's law of universal gravitation 8.5 Poisson's equation

9 THE SCHWARZSCHILD METRIC

9.1 Introduction 9.2 Forms of the Schwarzschild metric 9.3 Derivation of the Schwarzschild metric

9.3.1 Vacuum field equations 9.3.2 Using our proposed metric 9.3.3 Finding the connection coefficients 9.3.4 The Ricci tensor components 9.3.5 Assembling the bits 9.3.6 Solving for b/r

9.4 More on Schwarzschild spacetime

9.4.1 The Schwarzschild radius 9.4.2 Gravitational time dilation 9.4.3 Gravitational redshift - the first test of general relativity 9.4.4 Using the equivalence principle to predict gravitational time dilation and gravitational redshift 9.4.5 Proper distance 9.4.6 Geodesics in Schwarzschild spacetime 9.4.7 Perihelion advance - the second test of general relativity 9.4.8 Gravitational deflection of light - the third test of general relativity 9.4.9 Gravitational time delay of signals passing the Sun - the fourth test of general relativity

10 SCHWARZSCHILD BLACK HOLES

10.1 Introduction 10.2 Falling into a black hole 10.3 Falling - seen from up close 10.4 Falling - seen from far, far away

10.4.1 Journey time of a light signal 10.4.2 Relationship between coordinate time and coordinate distance 10.4.3 Gravitational redshift and luminosity 10.4.4 Distant observer - summary

10.5 Coordinates and lightcones

11 COSMOLOGY

11.1 Introduction 11.2 Key observed properties of the Universe

11.2.1 The dark night sky 11.2.2 The cosmological principle 11.2.3 Cosmic microwave background radiation 11.2.4 Hubble's law

11.3 Robertson-Walker spacetime

11.3.1 The curvature parameter 11.3.2 Proper distance

11.4 Introducing the Friedmann equations

11.4.1 The cosmic energy-momentum tensor 11.4.2 More on density and pressure 11.4.3 The Friedmann equations 11.4.4 The empty universe model 11.4.5 The static Einstein model 11.4.6 The de Sitter model 11.4.7 The radiation only model 11.4.8 The Einstein-de Sitter model 11.4.9 Density parameters

11.5 More FRW models, and our Universe

BIBLIOGRAPHY APPENDIX ACKNOWLEDGEMENTS

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