Relativity Demystified

Relativity Demystified
DAVID McMAHON
CONTENTS
Preface xi
CHAPTER 1 A Quick Review of Special Relativity 1
Frame of Reference 5
Clock Synchronization 5
Inertial Frames 6
Galilean Transformations 7
Events 7
The Interval 8
Postulates of Special Relativity 9
Three Basic Physical Implications 13
Light Cones and Spacetime Diagrams 17
Four Vectors 19
Relativistic Mass and Energy 20
Quiz 21
CHAPTER 2 Vectors, One Forms, and the Metric 23
Vectors 23
New Notation 25
Four Vectors 27
The Einstein Summation Convention 28
Tangent Vectors, One Forms, and the
Coordinate Basis 29
Coordinate Transformations 31
For more information about this title, click herevi CONTENTS
The Metric 32
The Signature of a Metric 36
The Flat Space Metric 37
The Metric as a Tensor 37
Index Raising and Lowering 38
Index Gymnastics 41
The Dot Product 42
Passing Arguments to the Metric 43
Null Vectors 45
The Metric Determinant 45
Quiz 45
CHAPTER 3 More on Tensors 47
Manifolds 47
Parameterized Curves 49
Tangent Vectors and One Forms, Again 50
Tensors as Functions 53
Tensor Operations 54
The Levi-Cevita Tensor 59
Quiz 59
CHAPTER 4 Tensor Calculus 60
Testing Tensor Character 60
The Importance of Tensor Equations 61
The Covariant Derivative 62
The Torsion Tensor 72
The Metric and Christoffel Symbols 72
The Exterior Derivative 79
The Lie Derivative 81
The Absolute Derivative and Geodesics 82
The Riemann Tensor 85
The Ricci Tensor and Ricci Scalar 88
The Weyl Tensor and Conformal Metrics 90
Quiz 91CONTENTS vii
CHAPTER 5 Cartan’s Structure Equations 93
Introduction 93
Holonomic (Coordinate) Bases 94
Nonholonomic Bases 95
Commutation Coefficients 96
Commutation Coefficients and Basis
One Forms 98
Transforming between Bases 100
A Note on Notation 103
Cartan’s First Structure Equation and the
Ricci Rotation Coefficients 104
Computing Curvature 112
Quiz 120
CHAPTER 6 The Einstein Field Equations 122
Equivalence of Mass in Newtonian Theory 123
Test Particles 126
The Einstein Lift Experiments 126
The Weak Equivalence Principle 130
The Strong Equivalence Principle 130
The Principle of General Covariance 131
Geodesic Deviation 131
The Einstein Equations 136
The Einstein Equations with Cosmological
Constant 138
An Example Solving Einstein’s Equations
in 2 + 1 Dimensions 139
Energy Conditions 152
Quiz 152
CHAPTER 7 The Energy-Momentum Tensor 155
Energy Density 156
Momentum Density and Energy Flux 156
Stress 156
Conservation Equations 157viii CONTENTS
Dust 158
Perfect Fluids 160
Relativistic Effects on Number Density 163
More Complicated Fluids 164
Quiz 165
CHAPTER 8 Killing Vectors 167
Introduction 167
Derivatives of Killing Vectors 177
Constructing a Conserved Current
with Killing Vectors 178
Quiz 178
CHAPTER 9 Null Tetrads and the Petrov
Classification 180
Null Vectors 182
A Null Tetrad 184
Extending the Formalism 190
Physical Interpretation and the Petrov
Classification 193
Quiz 201
CHAPTER 10 The Schwarzschild Solution 203
The Vacuum Equations 204
A Static, Spherically Symmetric Spacetime 204
The Curvature One Forms 206
Solving for the Curvature Tensor 209
The Vacuum Equations 211
The Meaning of the Integration Constant 214
The Schwarzschild Metric 215
The Time Coordinate 215
The Schwarzschild Radius 215
Geodesics in the Schwarzschild Spacetime 216
Particle Orbits in the Schwarzschild
Spacetime 218CONTENTS ix
The Deflection of Light Rays 224
Time Delay 229
Quiz 230
CHAPTER 11 Black Holes 233
Redshift in a Gravitational Field 234
Coordinate Singularities 235
Eddington-Finkelstein Coordinates 236
The Path of a Radially Infalling Particle 238
Eddington-Finkelstein Coordinates 239
Kruskal Coordinates 242
The Kerr Black Hole 244
Frame Dragging 249
The Singularity 252
A Summary of the Orbital Equations
for the Kerr Metric 252
Further Reading 253
Quiz 254
CHAPTER 12 Cosmology 256
The Cosmological Principle 257
A Metric Incorporating Spatial
Homogeneity and Isotropy 257
Spaces of Positive, Negative, and
Zero Curvature 262
Useful Definitions 264
The Robertson-Walker Metric and the
Friedmann Equations 267
Different Models of the Universe 271
Quiz 276
CHAPTER 13 Gravitational Waves 279
The Linearized Metric 280
Traveling Wave Solutions 284
The Canonical Form and Plane Waves 287x CONTENTS
The Behavior of Particles as a
Gravitational Wave Passes 291
The Weyl Scalars 294
Review: Petrov Types and the
Optical Scalars 295
pp Gravity Waves 297
Plane Waves 301
The Aichelburg-Sexl Solution 303
Colliding Gravity Waves 304
The Effects of Collision 311
More General Collisions 312
Nonzero Cosmological Constant 318
Further Reading 321
Quiz 322
Final Exam 323
Quiz and Exam Solutions 329
References and Bibliography 333
Index 337
INDEX
absolute derivative
geodesics, 82
active gravitational mass, 123. See also
Newtonian theory
affine connection, 65
affine parameter, 82
anisotropic space, 256. See also isotropic space
astigmatic, 313
basis
coordinate basis, 29
one forms, 30, 94, 98–99
transformation, 32, 101–102
set
holonomic basis, 94
vectors, 24, 29, 31, 94
differentiation, 64, 65
Bianchi identities, 87
Big Bang, 276
black hole, 224. See also special relativity
classification, 230
coordinate singularities, 233–235
Eddington-Finkelstein coordinates,
236–239
Kruskal coordinates, 242–244
event horizon, 241
infinite redshift, 234
Kerr black hole, 233, 234, 244–245, 250
Boyer-Lindquist coordinates, 245
Cauchy horizon, 248, 249
cross terms, 245
ergosphere, 245
frame dragging, 249, 250
horizons, 245–246
lense-thirring effect, 250
metric, 244
orbital equations, 251, 252
Penrose process, 245
reduced circumference, 251
singularity, 253
solution, 233
Reissner-Nordstrøm, 233
Schwarzschild black hole
coordinate singularities, 234
Eddington-Finkelstein coordinates, 236,
239
infinite redshift, 234
Kruskal coordinates, 242, 243
radially infalling particle, 238, 239
solution, 233
tortoise coordinate, 237, 239
Bondi metric, 32. See also metric
Boyer-Lindquist coordinates, 240
Brinkmann metric, 195
Cartan’s structure equation, 93–121, 139.
See also curvature tensor
bases transformation, 100–103
inverse matrix, 101–102
tetrad, 100
basis one forms, 98–100
commutation coefficients, 96–98
curvature tensor, 139–146
first structure equation, 104, 207
holonomic bases, 94–95
basis vectors, 94
337
Copyright © 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use.338 Index
Cartan’s structure equation (contd.)
nonholonomic bases, 95–96
second structure equation, 209–211
Christoffel symbols, 65, 72, 84, 111, 198, 280,
281
correction terms, 68
for polar coordinates, 69
geodesic, 84
Kahn-Penrose metric, 75
Ricci rotation coefficients, 104
Riemann tensor, 113
transformation law, 68
clock synchronization, 5–6. See also special
relativity
commutation coefficients, 72, 96–98. See also
Cartan’s structure equations
commutator, 96
nonholonomic basis, 98
computing curvature, 112–119
conformal metrics, 85. See also tensor
calculus
congruence, 130. See also geodesic deviation
connecting vector, 131. See also geodesic
deviation
conservation equations, 155. See also
energy-momentum tensor
coordinate basis. See holonomic bases
coordinate patches, 47. See also manifolds
coordinate singularities, 233–235. See also
black hole
Kruskal coordinates, 242–244
Schwarzschild black hole
Eddington-Finkelstein coordinates,
236–239
coordinate system, 5. See also frame of
reference
coordinate transformations, 33. See also
special relativity
correction terms
Christoffel symbols, 68
cosmology, 256–278
anisotropic space, 256
closed universe, 266
constant, 137–138
cosmological principle, 257
critical density, 267
deceleration parameter, 266
density parameter, 267
different models of universe, 271–276
dust-filled universe, 276
flat universe, 267
Friedmann equations, 267–271
Gaussian normal coordinates, 257
Ricci tensor, 259, 260
homogeneous concept, 253
Hubble parameter, 265, 266
megaparsecs, 266
Hubble time, 266
Hubble’s law, 266
isotropic space, 256
matter density, 264
dust, 265
matter-dominated universes, 262
negative curvature, 262
open universe, 266
positive curvature, 262
radiation density, 265
radiation-dominated universes, 265
Robertson-Walker metric, 264–267
scale factor, 264
Schur’s theorem, 257
spatial homogeneity and isotropy, 257
vacuum density, 265
zero curvature, 262
curvature constant, 261
curvature one forms, 104, 204–206. See also
Schwarzschild solution
Ricci rotation coefficients, 110
symmetry relations, 105
curvature singularities, 235
curvature tensor, 85. See also Riemann tensor
in noncoordinate basis, 143
curvature two forms, 113, 118
curvature, computing, 112–119
d’Alembertian operator, 287
de Sitter universe, 272, 273, 274
differentiable manifold, 47. See also manifold
dot product, 42, 43
dummy index, 28
dust, 155–159. See also energy-momentum
tensor
Eddington-Finkelstein coordinates, 233–234,
236–239. See also black hole
tortoise coordinate, 237, 239Index 339
eigenbivectors, 194. See also Weyl tensor
multiplicity, 194
Einstein field equations, 28–29,
122–152
and Newtonian gravity, 136–139
constraint, 137
Einstein lift experiments, 126–130
energy conditions, 152
equivalence of mass, 123–126
geodesic deviation, 131–136
in 2+1 dimensional space, 139–152
principle of general covariance, 131
strong equivalence principle, 130
tensor, 89, 113
Cartan’s methods, 140
test particles, 126
weak equivalence principle, 129
with cosmological constant, 138–139
energy density, 155. See also
energy-momentum tensor
energy flux, 155. See also energy-momentum
tensor
energy-momentum tensor, 155–166. See also
special relativity
and number density, 163–164
conservation equations, 157–158
dust, 158–159
energy density, 158
energy flux, 156
equivalence of mass, 123–126
principle of general covariance, 131
strong equivalence principle, 130
test particles, 126
weak equivalence principle, 129
with cosmological constant, 138–139
events, 7
four vectors, 19–20
frame of reference, 5
inertial frames, 6–7
Galilean transformations, 7
killing vector, 81, 167–179, 219, 220
and Ricci tensor, 177
constructing conserved current with,
178
contravariant components, 176
derivatives of, 177
for 2 sphere, 170
isometry, 168
light cones, 17–19
Lorentz transformations, 13–17
manifolds, 47, 48
coordinate patches, 48
differentiable manifold, 48
Maxwell’s equations, 2
metric, 23, 32–45
arguments passing, 43, 44, 45
as tensor, 37
Bondi metric, 35
cylindrical coordinates, 34, 35
determinant, 45
dot product, 42, 43
flat space metric, 37
index gymnastics, 41, 42
index raising and lowering, 38–41
inverse of, 37
Kronecker delta function, 37
null vectors, 45
ordinary cartesian coordinates, 34
second rank tensor, 34
signature of, 36
spherical coordinates, 34, 35
Michelson-Morley experiment, 4
Lorentz transformations, 4
luminiferous ether, 4
momentum density, 156
nonperfect fluid, 164–166
null tetrads, 180–202
null vectors, 182–184
one forms, 23
parameterized curves, 49, 50
one forms, 50–53
tangent vectors, 50–53
perfect fluids, 160–162
Petrov classification, 180–202
posits of, 9–13
relativistic mass and energy, 20–21
Schwarzschild solution, 203
spacetime diagrams, 8, 17–19
spacelike, 19
timelike, 19
stress, 155, 156
tensors, 47–59
algebraic operations, 54–58
as functions, 53–54
Levi-Cevita tensor, 59
with mixed indices, 54340 Index
energy-momentum tensor (contd.)
tensor calculus, 60–92
commutation coefficients, 72
conformal metrics, 90
covariant derivative, 62, 63, 65
exterior derivative, 79–81
Lie derivative, 81–85
metric and Christoffel symbols,
72–79
Riemann tensor, 85–88
tensor equations, 61–62
testing tensor character, 60–61
torsion tensor, 72
Weyl tensor, 90
vectors, 23, 25
basis vectors, 24
components of, 26
coordinate basis, 29–31
Einstein summation convention,
28–29
four vectors, 27, 28
one forms, 29–31
tangent vectors, 29–31
equivalence principle
Einstein lift experiments, 126–129
ergosphere, 245
event horizon, 241
events, 8. See also special relativity
exterior derivative, 79–81
flat space metric, 37. See also metric
four vector, 19–20, 27. See also special
relativity
four acceleration, 19
four velocity, 19
in Schwarzschild metric, 215
frame dragging, 244–245, 249. See also Kerr
black hole
lense-thirring effect, 250
frame of reference, 5. See also special relativity
inertial frame, 6
free index, 28
Friedmann equations, 161–162
Galilean transformations, 7. See also special
relativity
gauge transformation, 286
geodesic, 84, 216–218. See also Schwarzschild
solution
absolute derivative, 82
Christoffel symbols, 84
congruence, 131, 132
connecting vector, 132, 133
Euler-Lagrange equations, 216
for cylindrical coordinates, 83
geodesic deviation, 131–132
gravitational field
Einstein tensor, 113
energy-momentum tensor, 155
vacuum equations, 138
gravitational potential, 137
gravitational waves, 279–318
Aichelburg-Sexl solution, 300
Brinkmann metric, 280, 300
canonical form, 290
colliding gravity waves, 304
Dirac delta, 304, 311, 312
effects of collision, 311
general collisions, 312
gravitational wave passes, 291
linearized metric, 280–284
Narai spacetime, 318
Newton-Penrose scalar, 320
nonzero cosmological constant, 318
optical scalars, 293, 295
Petrov types, 295
Petrov type D, 295, 321
Petrov type II, 295
Petrov type III, 295
Petrov type N, 295
pp gravity waves, 297
Brinkmann metric, 300, 303
covariantly constant, 297, 301
plane waves, 287, 288, 298, 301
shock wave, 304
traveling wave solutions, 284
trace reverse, 284
Weyl scalars, 294
Newman-Penrose identities, 294
guess method, 109
hatted index, 105
heaviside step function, 304
holonomic bases, 29, 72, 93–94, 105. See also
Cartan’s structure equationsIndex 341
basis set, 94
basis vectors, 93
Einstein tensor
components of, 147–150
for spherical polar coordinates, 96
Weyl tensor, 191
inertial frame, 6–7, 9
inertial mass, 123. See also Newtonian theory
infinite redshift, 234, 236
inverse matrix
spherical polar coordinates, 103
isometry, 167. See also killing vector
isotropic space, 256. See also anisotropic space
Kahn-Penrose metric
Christoffel symbols, 75
Kerr black hole, 233, 244–245, 250. See also
black holes
Boyer-Lindquist coordinates, 245
Cauchy horizon, 248, 249
cross terms, 245
ergosphere, 245
frame dragging, 249, 250
horizons, 245–246
Kerr metric, 244
lense-thirring effect, 250
orbital equations, 251, 252
Penrose process, 245
reduced circumference, 251
singularity, 253
killing vector, 81, 167–179, 219, 220
and Ricci tensor, 177
constructing conserved current with, 178
contravariant components, 176
derivatives of, 177
for 2 sphere, 170
in Schwarzschild metric, 219
isometry, 168
Kronecker delta function, 30, 37
tensor, 56
trace, 285
Kruskal coordinates, 242–244
last line, 71
lense-thirring effect, 249, 250. See also frame
dragging
Levi-Cevita tensor, 59. See also tensors
Lie derivative. See also tensor calculus
killing vectors, 81
light cone, 180, 181
Schwarzschild coordinates, 238
lightlike vector, 183
line element, 33, 34, 35, 37, 103
Lorentz transformation, 12–15
length contraction, 14
time dilation, 13
velocity composition law, 14–17
luminiferous ether, 4
manifolds, 47, 48. See also special relativity
mass, active gravitational, 123. See also
Newtonian theory
Maxwell’s equations, 2
metric, 23, 32–45, 72
arguments passing, 43–45
as tensor, 37
basis one forms, 102
Bondi metric, 35
Brinkmann metric, 195
conformal metrics, 85
cylindrical coordinates, 34, 35
determinant, 45
dot product, 42, 43
flat space metric, 37
for Rindler space, 84
index, 38–42
inverse of, 37
Kahn-Penrose metric
Christoffel symbols, 75
Kronecker delta function, 37
Lie derivative, 168
Minkowski metric
in spherical coordinates, 204
null vectors, 45
ordinary cartesian coordinates, 34
second rank tensor, 34
signature of, 36
spherical coordinates, 34, 35
Michelson-Morley experiment, 4
Lorentz transformations, 4
luminiferous ether, 4
Minkowski metric
flat space metric, 37
in spherical coordinates, 204342 Index
momentum density, 155. See also
energy-momentum tensor
momentum four vector, 156
Newman-Penrose equations, 192, 280, 312.
See also null vector
Ricci scalar, 193
Weyl scalar, 191, 193
Newtonian theory, 122–125, 136–138.
See also Einstein field equations
equivalence of mass
active gravitational mass, 123
gravitational field, 123–126
inertial mass, 123
passive gravitational mass, 123
noncoordinate basis vectors, 94–95. See also
nonholonomic bases
basis vectors, 95, 97
commutator, 96
definition of, 100
spherical polar coordinates, 96
nonperfect fluid, 164. See also
energy-momentum tensor
null tetrad, 180, 182. See also special relativity
augmenting, 183
for Minkowski metric, 184
frame metric augmentation, 190
null vector, 45, 182–184. See also metric
Newman-Penrose formalism, 190
Petrov classification, 190
spin coefficients, 191
number density, 163. See also
energy-momentum tensor
one form transformation, 29, 32, 50–53.
See also parameterized curves
one forms curvature, 104, 204–206. See also
Schwarzschild solution
Ricci rotation coefficients, 110
symmetry relations, 105
orthogonal vectors, 95
orthonormal basis, 95, 104
hatted index, 105
orthonormal tetrad, 95, 103, 185, 186
curvature one forms, 206
parameterized curves, 49, 50. See also special
relativity
one forms, 50–53
tangent vectors, 50–53
passive gravitational mass, 123. See also
Newtonian theory
perfect fluid, 157, 160–162. See also special
relativity
Petrov classification, 190, 193–202. See also
null vector; Weyl tensor
principal null directions, 193
plain index, 105
pp-wave spacetimes (plane fronted waves)
Brinkmann metric, 300, 303
principal null directions, 193
principle of equivalence, 122
principle of general covariance, 136. See also
Einstein field equations
Pythagorean theorem, 32
rapidity, 11
relativistic velocity composition law. See
velocity composition law
rest mass, 20
Ricci rotation coefficients, 104, 106, 140, 141
Christoffel symbols, 104
inorthonormal basis, 105
Ricci scalar, 88–90, 116, 148. See also tensor
calculus
Ricci tensor, 61, 88–90, 192, 201, 259, 260,
261. See also tensor calculus
vacuum equations, 211
Riemann tensor, 85–88, 211, 257, 280, 282,
287. See also tensor calculus
Bianchi identities, 87
Christoffel symbols, 113
Ricci scalar, 88–90
Ricci tensor, 88–90
symmetries, 86
Robertson-Walker metric, 116, 264, 267
scale factor, 258,261,264
Schwarzschild black hole, 231, 234–241.
See also black holes
Schwarzschild solution, 203–230. See also
special relativity
curvature one forms, 206–208
curvature tensor
Cartan’s second structure equation,
209–211Index 343
integration constant, 214–215
Kruskal coordinates, 242, 243
radially infalling particle, 238, 239
Schwarzschild metric, 215, 231, 234, 235
deflection of light rays, 224–229
four velocity, 219
geodesics, 216
killing vectors, 219, 220
particle orbits in, 218–224
Schwarzschild radius, 213–216
time coordinate, 215
time delay, 229–230
spherically symmetric spacetime, 204–206
vacuum equations, 203, 210–214
second rank tensor, 34
shear tensor, 165
signature, 36. See also metric
singularities, 236. See also black hole
coordinate, 233–236
Eddington-Finkelstein, 236–239
Kruskal, 242–244
curvature, 235
space metric, flat, 37. See also metric
spacelike vector, 185
spacetime diagram, 180
special relativity, 1–22
black holes, 233–254
Cartan’s structure equations, 93–119
clock synchronization, 5–6
coordinate transformations, 31–32
cartesian coordinates to polar coordinates,
31, 32
cosmology, 257–278
Einstein field equations, 122–152
and Newtonian gravity, 136–139
Einstein lift experiments, 126–130
energy conditions, 152
geodesic deviation, 131–136
in 2+1 dimensional space, 139–152
gravitational waves, 279–321
spin coefficients, 199
null tetrad, 190
Ricci rotation coefficients, 191
static limit, 247, 249
stress, 155. See also energy-momentum
tensor
stress-energy tensor, 137, 155. See
energy-momentum tensor
symmetry
killing vector, 167
synchronization, clock, 5–6. See also special
relativity
tangent vectors, 50–53. See also parameterized
curves
tensor calculus, 60–92
Christoffel symbols, 65, 67–70
commutation coefficients, 72
conformal metrics, 90
covariant derivative, 62, 63, 65
curvature tensor, 85
in noncoordinate basis, 143
exterior derivative, 79–81
of one form, 81
wedge product, 80
Levi-Cevita tensor, 59
Lie derivative, 81–85
absolute derivative, 82
metric, 72–79
Riemann tensor, 85–88
Bianchi identities, 87
Ricci scalar, 88–90
Ricci tensor, 88–90
tensor equations, 61–62
testing tensor character, 60–61
torsion tensor, 72
Weyl tensor, 90
tensors, 47–59. See also special relativity
algebraic operations, 54–58
addition, 54
antisymmetric, 55, 56, 57
contraction, 55
Kronecker delta, 56
scalar multiplication, 55
subtraction, 55
symmetric, 55, 56, 58
as functions, 53–54
Levi-Cevita tensor, 59
metric as, 37
with mixed indices, 54
tetrad, 100, 104
Riemann tensor, 115
timelike vector, 185
torsion tensor, 72
tortoise coordinate, 237, 239. See also
Eddington-Finkelstein coordinates344 Index
transformation matrix, 31, 112
two forms curvature, 113, 118
vacuum energy density, 271
vacuum equations, 138, 203, 211–214.
See also Schwarzschild solution
vector, 23, 25
basis vectors, 24
components of, 26
connecting, 131
contravariant, 32
coordinate basis, 29–31
covariant derivative, 133
Einstein summation convention,
28–29
four vectors, 19–20, 27
four acceleration, 19
four velocity, 19
in Schwarzschild metric, 215
killing, 81, 167–179
in Schwarzschild metric, 219
one forms, 29–31
tangent vectors, 29–31
velocity composition law
derivation of, 14–17
wave vector, 288
weak equivalence principle, 129
wedge product, 80. See also exterior derivative
antisymmetry, 99
Weyl scalar, 90, 190, 192, 197, 199, 294, 295,
305. See also Newman-Penrose
formalism; tensor calculus
eigenbivectors, 193
in coordinate basis, 191
Petrov classification, 193–201
Weyl scalars, 190
worldlie, 18
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