<<

. 23
( 24)



>>

metric tensor alone is

S= ’g d4 x

Show that the corresponding field equations are given by ’gg = 0 and clearly
do not constitute a viable theory of gravity.
19.22 Under a general infinitesimal coordinate transformation of the form x = x +
x , show that

≡g x ’g x =’ +
g

19.23 Consider a general action for gravity in vacuo of the form

S= d4 x
g g g

By considering a general infinitesimal coordinate transformation of the form x =
x+ x , where the x vanish on the boundary , show that the metric and
its derivatives must satisfy the differential constraints

=0
g
where / g is the variational derivative of the Lagrangian density with respect
to the metric. Hence show that for the Einstein“Hilbert action these differential
constraints lead to the contracted Bianchi identities G = 0.
19.24 Show explicitly that the quadratic action

¯
S= ’ ’g d4 x
g

is not a scalar with respect to general coordinate transformations. Show further
that varying this action with respect to the metric and its first derivative leads to
the Einstein field equations in vacuo, provided that the variation in the metric and
its first derivative vanish on the boundary .
19.25 Obtain an expression for the dynamical energy“momentum tensor of the complex
scalar field considered in Exercise 19.15 and that of the massive vector field
considered in Exercise 19.19.
Bibliography




Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Physics, Dover, 1972.
Chandrasekhar, S., An Introduction to the Study of Stellar Structure, Dover, 1958.
Chandrasekhar, S., The Mathematical Theory of Black Holes, Oxford University Press,
1983.
Clarke, C., On the global isometric embedding of pseudo-Riemannian manifolds,
Proceedings of the Royal Society A314, 417“28, 1970.
d™Inverno, R., An Introduction to Einstein™s Relativity, Oxford University Press, 1992.

Princeton University Press, 1996.
Feynman, R. P., Morinigo, F. B. & Wagner, W. G., Feynman Lectures on Gravitation,
Addison-Wesley, 1995.
Foster, J. & Nightingale, J. D., A Short Course in General Relativity, Springer-Verlag,
1995.
Islam, J. N., An Introduction to Mathematical Cosmology, Cambridge University Press,
1992.
Liddle, A. & Lyth, D., Cosmological Inflation and Large-Scale Structure, Cambridge
University Press, 2000.
Misner, C. W., Thorne, K. S. and Wheeler, J. A., Gravitation, Freeman, 1973.
Mukhanov, V. F., Feldman, H. A. & Brandenburger, R. H., Theory of cosmological
perturbations, Physics Reports 215, 203“333, 1992.
Nash, J., The imbedding problem for Riemannian manifolds, Annals of Mathematics 63,
20“63, 1956.
Padmanabhan, T., Structure Formation in the Universe, Cambridge University Press,
1993.
Padmanabhan, T., From Gravitons to Gravity: Myths and Reality, abs/grqc/0409089.
Peacock, J., Cosmological Physics, Cambridge University Press, 1999.
Rindler, W., Relativity: Special, General and Cosmological, Oxford University Press,
2001.
Ryder, R. H., Quantum Field Theory, Cambridge University Press, 1985.
Schutz, B. F., Geometrical Methods of Mathematical Physics, Cambridge University
Press, 1980.
Schutz, B. F., A First Course in General Relativity, Cambridge University Press, 1985.
Tanaka, Y. et al., Nature 375, 659, 1995.
Wald, R. M., General Relativity, University of Chicago Press, 1984.
Weinberg, S., Gravitation and Cosmology, Wiley, 1972.
Will, C., Theory and Experiment in Gravitational Physics, Cambridge University Press,
1981.

555
Index




Page links added automatically. Attention: for a range of pages the link to the upper
limit of the range is incorrect when not all digits of the page number are shown.
An italic page number indicates that there is a figure related to the topic on this page.


Arcminute Cosmology Bolometer Array Receiver
absolute luminosity, 372
(ACBAR), 460
accelerating observers, 125“8
area, manifolds, 38“42
acceleration
atlas, 27
and instantaneous rest frames, 126“7
in special relativity, 19“20
radial, 335 basis tensors, 100
three-acceleration, 123 basis vectors, 56“9, 69
uniform, 21 and coordinate transformations, 60“1
universe, 187 Cartesian, 113“14
see also four-acceleration derivatives, 62“4, 84
accretion discs dual, 56“7, 84
around compact objects, 240“4, 277 orthonormal, 59
radiation efficiency, 215“16, 240, 338 polar coordinates, 83
accretion power, of black holes, 216 timelike, 152
action, 525, 526“7 see also coordinate basis vectors
and classical field theory, 527“9 Bianchi identity, 161, 162
equivalent, 533“4 big-bang origin, 399“400, 404, 409, 419
and general relativity in vacuo, 542“3 big-bang theory, 394, 398“400
stationary, 529“30 big-crunch theory, 395, 398“400
see also Einstein“Hilbert action binary system, 508“9
advanced time parameter, 255 compact, 277“9
affine connection, 62“4 spin-up, 516“17
and metric functions, 65“7 Birkhoff™s theorem, 202
definition, 63 black hole, 240, 270
symmetry, 65 accretion power, 216
transformation properties, 64 angular momenta, 260
affine parameters, 75“6, 117, 120, 221“2, 340 charged, Reissner“Nordström geometry,
geodesics, 76“7 300“2
amplitude tensor, 499“500 definition, 257
Andromeda galaxy, 355 detection, 277“9
angular diameter distance, 371, 373“4, 411“13 dynamical mass limits, 279
angular momentum barrier, 213, 214 existence of, 258, 260
angular speed formation of and gravitational collapse,
coordinate, 245 259“64, 277
proper, 245 Hawking effect, 274“7
antiparticles, 275 in binary systems, 277, 278
aphelion, 230 singularities, 258, 270


556
557
Index
tidal forces near, 264“5 comoving coordinates, 443, 467n
see also Kerr black hole; Schwarzschild black and fundamental observers, 356“9
hole; supermassive black hole; white hole; comoving Hubble distance, 421
wormhole compact-source approximation, 481“3, 507“8
blackbody complex functions, analytic continuation, 254
energy spectrum, 276 components
radiation, 388“9 metric, 83“4
temperature, 277 mixed, 94, 97
bounce model, 398 tensor, 93“4, 100, 102, 103“4
Boyer“Lindquist coordinates, 318, 319, 320, 322, vector field, 73
344, 347 see also contravariant components; covariant
Boyer“Lindquist form, 318 components
Brans“Dicke theory, 191“2, 235, 236 Compton effect, 124
Buchdahl™s theorem, 296 Compton scattering, and relativistic collisions,
bug, two-dimensional, confined to two-dimensional 123“5
surface, 33“4, 54 configuration space, 79, 525
congruence of timelike worldlines, 356a
connection coefficients, 85, 202, 244, 317
calculus of variations, 87“8
of general static isotropic metric, 200“1
Cartesian basis vectors, 113“14
conservation of energy for perfect fluid, 179“81
Cartesian coordinates, 1, 27, 33“5, 320, 346“7, 525
conservation of momentum for perfect fluid,
advantages, 128
179“81
local, 42“4, 46“7, 48, 67“8, 160
continuity, equation of, 180
Lorentz transformation, 112“13
contraction, 66
Minkowski spacetime, 111“12
Lorentz, 177
rotations, 26
tensor, 99“100
Cartesian inertial frames, 47, 112, 122, 141,
contravariant components, 56, 57, 59“60, 68
142, 149
tensors, 93“4, 100
global, 151
coordinate angular speed, 245a
local, 150“1, 153, 177“8, 179
coordinate basis vectors, 57“9, 113“14, 115“16
centre of mass, worldline of, 170
spacelike, 275
centre-of-momentum coordinates, 482, 507
timelike, 275
centrifugal force, repulsive, 217
coordinate distance, 371
Cepheid variable, 370
coordinate patches, 27
Chandrasekhar, Subrahmanyan (1910“95), 259
coordinate singularity, 37, 250, 341
Chandrasekhar limit, 259, 277
coordinate transformations, 5, 8
charge and electromagnetic force, 135“6
and basis vectors, 60“1
charge density, 136“7
infinitesimal general, 468, 469“70
proper, 508
manifolds, 28“30
charge density distribution, 508
tensor, 101“2
charged particle, equations of motion, 144“5
coordinates, 26“52
Christoffel symbol, 106
arbitrary, 128“31, 142“4, 145, 151, 155“6, 179
of the first kind, 66
Boyer“Lindquist, 318, 319, 320, 322, 344, 347
of the second kind, 63
characterisation, 248
see also affine connection
concept of, 27
circular motion, 209, 335
Kruskal, 258, 266“71, 273
massive particle, 212“13
Minkowski, 297“8
photon, 219“20
momentum, 26
see also equatorial circular motion
non-degenerate, 27
circular orbits, 213, 243“4
Novikov, 254n
stable, 214, 215, 243
null, 248, 258
unstable, 214
quasi-Minkowski, 467“8, 485
circularly polarised mode, 507
spacelike, 248, 249“50, 266, 273
classical field theory, 524
tensor, 102“3
and action, 527“9
timelike, 248, 249“50, 256, 258, 266, 273
clocks, ideal, 11
cold dark matter (CDM), 387 unique values, 27
558 Index
closed timelike, 301, 327
coordinates (cont.)
non-null, 75“6
see also Cartesian coordinates; comoving
null, 75“6
coordinates; Eddington“Finkelstein
parametric, 28
coordinates; inertial coordinates;
tangent vector, 55“6
polar coordinates; Schwarzschild
coordinates curved spacetime, 181, 244
corona, solar, 239 electromagnetism in, 155“6
cosmic censorship hypothesis, 301, 324 geodesic motion, 188
cosmic microwave background (CMB) observers in, 152“3
characterisation, 388“9 tidal forces in, 167“70
distribution, 389 see also spacetime curvature
cyclic identity, 160
power spectrum, 459“62
cylinder, 34
cosmic time, 419
parallel transport around, 165
cosmological constant, 185“8, 376, 386,
407“8, 432
effective, 430 d™Alembertian operator, 71, 140, 144, 148,
size of, 188 471, 475
cosmological field equations, 386, 392“3, 407, covariant, 432, 535
433, 434 dark matter, 387
derivation, 376“9 de Sitter model, 398
cosmological fluids properties, 407
components, 386“9 deceleration parameter, 368“71
equations of motion, 379“80 delta function, four-dimensional, 189
multiple-component, 381 density parameters, 390“2, 410
scalar fields as, 431“2 curvature, 391
cosmological models, 386“421 evolution, 415“17
analytical, 400“8 present-day, 415
cosmological parameters, 390“2 total, 392
cosmological principle, 355“6 derivatives
cosmological redshift, 367“8 absolute, 72
covariant components, 57, 59“60, 81 of basis vectors, 62“4, 84
of tensors, 93“4, 100, 112 see also covariant derivative; directional
covariant derivative, 68“70, 85“6 derivative; intrinsic derivative
of tensor, 104“7 development angle, 402
critical density, 392 differential manifold, 26
curl, vector field, 71 directional derivative, 56
curvature, 33 vectors as, 81“2
and geodesic deviation, 165“7 distance“redshift relation, 411“13
and parallel transport, 163“5 divergence, of vector field, 70
Gaussian, 161, 171 divergence theorem, 532
Doppler effect, 16“18, 240
of manifold, 157“8
and relativistic aberration, 120“1
see also spacetime curvature; spatial curvature
formula for, 18
curvature density parameter, 391
dual basis vectors, 56“7, 84
curvature perturbations
dust, 178, 182
and gauge invariance, 446“9
spherically symmetric collapse of, 260“4
evolution, 449“52
use of term, 176
initial conditions, 452“5
worldline, 190
normalisation, 452“5
power spectrum, 456“7
curvature scalar, see Ricci scalar eclipses
curvature spectrum, definition, 456 lunar, 235
curvature tensor, 75, 158“9, 182, 250, 267 solar, 235
properties, 159“61 Eddington, Sir Arthur Stanley (1882“1944),
in Schwarzschild coordinates, 264 235, 542
spherical surfaces, 161, 170 Eddington“Finkelstein coordinates, 254“9, 303
curve in manifold, 27“8 advanced, 258, 261, 262, 270, 303, 346
559
Index
definition, 254“9 antisymmetric, 136
components, 142, 176
limitations, 266
definition, 136
and Kerr geometry, 344“6
electromagnetic forces, 148
retarded, 270
and charge, 135“6
definition, 257“9
electromagnetic radiation, generation of, 508
limitations, 266
electromagnetic waves and gravitational waves
effective potential, 335
compared, 501
general relativistic, 214“15
electromagnetism, 135“46, 508
Newtonian, 213“14
and Lorentz gauge conditions, 139“41
Einstein, Albert (1879“1955)
and special relativity, 135
and cosmological constant, 185“8, 407“8
consistent theory of, 135
and special relativity, 22“3
from variational principles, 536“9
elevator experiment, 148“50
in arbitrary coordinates, 142“4
general relativity theory, 150, 233
in curved spacetime, 155“6
˜On the Electrodynamics of Moving Bodies™
electron degeneracy pressure, 259
(1905), 22
elevator experiment, 148“50
Einstein“Cartan theories, 193
ellipticity of planetary orbits, 230, 231
Einstein“de-Sitter (EdS) model, 398, 402, 410, 413
emitters
radiation-dominated, 420
four-velocity, 241
Einstein equations, 176, 181“3
gravitational redshift, 202“5, 315
and cosmological term, 185“8
empty-space field equations, 288
and geodesic motion, 188“90
solutions, 198“202, 248
exact, 487
energy, 118“19
in empty space, 183, 202
conservation of, 179“81
limitations, 317
potential, 535
non-linearity, 473
energy density
perturbed, 443“5
of universe, 390, 433
solving, 196, 198“202, 248, 270
of vacuum, 187, 390
for spherically symmetric geometries, 288“305
energy equation for particle motion, 213“14
weak-field limit, 184“5
energy“momentum invariant, 119
see also gravitational field equations
energy“momentum tensor, 176“8, 182, 188,
Einstein“Hilbert action
192, 484
and general relativity in vacuo, 539“42
and spacetime curvature, 176
Lagrangian density, 544
canonical, 549
stationarity, 545
dynamical, 546“9
variation, 540
for electromagnetic field, 548“9
Einstein“Hilbert Lagrangian density, 543, 544
for gravitational field, 486“90, 511“13
Einstein“Maxwell coupled equations, 297“8, 300
for matter, 546
Einstein“Maxwell formulation of linearised
for multiple-component fluids, 381
gravity, 490“2
for perfect fluid, 178“9, 187, 377“9, 432
Einstein“Rosen bridge
for scalar field, 432, 444“5
and wormholes, 271“4
non-zero, 288, 296“7, 475
structure, 273
of vacuum, 187“8
Einstein tensor, 162, 183, 442“3, 444, 546
symmetry of, 179
linearised, 487
epoch, 418, 19
Einstein™s static universe, 407“8
inflationary, 433, 437
electric fields in inertial frames, 141“2
of recombination, 420
electrodynamics, 22, 189“90
equation of continuity, 180
electromagnetic field, energy“momentum tensor
for, 548“9 equation of state, 292
electromagnetic field equations, 138“9, 176 polytropic, 293
derivation, 136“7 equation-of-state parameter, 380
in arbitrary coordinates, 142“4, 155“6 equations of motion, 148
simplification, 140“1 and Newtonian gravity, 209
see also Maxwell™s equations Euler™s, 181
electromagnetic field tensor, 138, 139, 140, 156 for charged particles, 144“5
560 Index
non-linearity of, 196
equations of motion (cont.)
perturbed, 443“5
for cosmological fluid, 379“80
for perfect fluids, 179“81 vacuum, 249
for photons, 119 see also cosmological field equations;
for scalar field, 433 electromagnetic field equations;
geodesics, 188“90 empty-space field equations; gravitational
Newtonian, 154“5, 230, 486 field equations; linearised field
radial, 304“5 equations
relativistic, 180 field Lagrangian, 528“9
equatorial circular motion field theories
massive particle, 335“6 Minkowski spacetime, 487
photon, 341 of real scalar fields, 534“6
equatorial orbits see also classical field theory
massive particle, stability, 337“8 field-strength tensor, 537, 538
photon, stability, 342“4 fixed spatial coordinates, 223, 315
equatorial planes, geodesics in, 330“2 flatness, conformal, 267, 282“3
equatorial trajectories flatness problem, 418, 428
energy equation, 331 solving, 429“30, 436, 442
massive particle, 332“3 fluid
photon, 338“9 four-velocity of, 177“8
equivalence principle, 148“50, 191 in instantaneous rest frame, 176“7, 178“9
equivalent mass densities, 390, 392 Lorentz contraction of, 177
ergoregion, 324, 325“7 multiple-component, 381
Euclidean geometry, 4 see also cosmological fluid; perfect fluid
Euclidean metric tensor, 505
fluorescence, 240
Euclidean space, 27
force
four-dimensional, 37“8
gravitational, 147, 148
pseudo-, 47, 54, 114
pure, 122
three-dimensional, 26, 33, 36“7, 40“2, 70, 271“2
repulsive centrifugal, 217
Euler angles, 26
see also electromagnetic force; four-force;
Euler“Lagrange (EL) equations, 78, 88, 209, 349,
three-force; tidal forces
432, 525, 529“30
four-acceleration, 123, 125“7, 152, 153
alternative forms, 531“2, 538
orthogonal, 125
alternatives to, 331
four-current density, 136“7, 156, 176
substitution of ˜Lagrangian™ into, 79“80, 199,
components, 137
205“6
four-dimensional rotations, Lorentz transformations
Euler™s equation of motion, 181
as, 5“6
event horizons, 257, 269, 274“5, 301, 315“16, 420
four-force, 122, 135“6, 156
formation, 260
pure, 123
in Kerr metric, 323
four-gradient, 138
smooth closed convex, 317
four-momentum, 123, 126, 144, 207, 274“5, 331
in special relativity, 21“2
and Compton scattering, 123“5
events, unique specification, 1
conservation along geodesic, 312“13
expansion problem, 428
of massive particle, 118“19
experimental tests
of photon, 119“20, 222“3, 224, 242, 244
and Schwarzschild geometry, 230
four-potential, 297
of general relativity, 230“46
four-tensors, 136, 152, 250
exponential expansion, 439“40
four-vector potential, 142, 143
extrinsic geometry, 33“6
four-vectors, 120, 136, 138“9, 349
and gyroscopes, 244“6
Fermi“Dirac statistics, 259
and lightcones, 115“16
Fermi energy, 259
and Lorentz transformations, 116
Fermi“Walker transportation, 127, 152, 153
as geometrical entities in spacetime, 115
field equations, 524
four-velocity, 168“9, 190, 276,
dynamical, 536
325“6, 349
homogeneous, 476
561
Index
Fourier space, 458 Milky Way, 186, 280, 355
perturbation equations in, 445“6 proper time, 357“8
and gyroscopes, 244“6 spectra, 243
definition, 116“18 worldlines, 356
normalised, 126, 152 Galilean transformations, 3, 4
of charged particle, 135“6, 144, 156 Galilei, Galileo (1564“1642), 148
of emitter, 241 gauge
of fluid, 177“8 choice of, 442“3
of free particle, 504 longitudinal, 443
of massive particle, 207, 304 see also Lorentz gauge conditions;
transverse-traceless (TT) gauge
of perfect fluid, 290
gauge freedom, 140
spatial components of, 223
gauge invariance, 536, 548, 549
four-wavevector, 499“500, 501
and curvature perturbation, 446“9
and Doppler effect, 120“1
gauge transformation, 140, 472
concept of, 120
Gauss™ theorem, 482“3
Fowler, Ralph Howard (1889“1944), 259
Gaussian curvature, 161, 171
frames of reference, 1
general relativity
free particle, 123
and matter, 545“6
gravitational-wave effects, 504“7
experimental tests of, 230“46
freely falling frame (FFF), 152“3
frequency shift, 240 in vacuo
and Einstein“Hilbert action, 539“42
see also Doppler effect; redshift
equivalent action, 542“3
Friedmann equations, 379
Palatini approach, 543“5
Friedmann expansion, 440
linearised, 467“92
Friedmann“Lema®tre equations, 379
predictions, 235
Friedmann models, 400“3, 419
sign conventions, 193
dust-only, 401
theory of, 150, 183, 233
radiation-only, 403, 430, 436
variational approach, 524“49
early-time, 439
see also special relativity
spatially flat, 402, 403
geodesic convergence, 167
Friedmann“Robertson“Walker (FRW) geometry,
355“81, 467n geodesic coordinates, local, 68“9
spatial curvature geodesic deviation
negative, 364“5 and curvature, 165“7
positive, 363“4 equation of, 165, 167, 168
zero, 364 geodesic equations, 77, 78“9, 145, 154,
number densities, 374“6 189“90, 504
proper volume, 375 alternative forms, 81
volume element, 375 integration, 206
Friedmann“Robertson“Walker metric, 362, geodesic motion and Einstein equations, 188“90
386, 442 geodesic precession effect, 246
geodesics in, 365“7 geodesics, 76“7
geometric properties, 362“5 congruence, 356
Friedmann“Robertson“Walker universes, in equatorial plane, 330“2
properties, 393“4 in Friedmann“Robertson“Walker metric, 365“7
fundamental observers and comoving coordinates, in Minkowski spacetime, 128
357“8 in Schwarzschild geometry, 205“7
future-pointing vectors, 116 Lagrangian procedures, 78“80, 199, 205
non-null, 80, 123, 206, 332
G2000 + 25, 279 stationary property, 77“8
null, 80, 123, 203, 206, 256
Galactic centre, 282
principal, 339
galaxies, 186“7, 355, 420
polar coordinates, 86“7
and fundamental observers, 358
timelike, 168, 244
Andromeda, 355
geometry
comoving coordinates, 357
distribution of, 462 Euclidean, 4
562 Index
existence, 498
geometry (cont.)
generation, 507“11
extrinsic, 33“6
intrinsic, 33“6 polarisation, 510“11
Kerr, 230, 310“50 see also plane gravitational waves
Newtonian, 3 gravitational-wave luminosity, 514, 515
of manifolds, 31 gravitoelectric fields, 491
Riemannian, 32“3 gravitomagnetic fields, 491
spacetime, 3“5 gravity, 135
see also Friedmann“Robertson“Walker as spacetime curvature, 150“1
geometry; Minkowski geometry; strong-field regime, 240
non-Euclidean geometry; theories of, 524
Reissner“Nordström (RN) geometry; Brans“Dicke, 191“2, 235, 236
Schwarzschild geometry relativistic, 191“3
gradient scalar, 190
four-gradient, 138 scalar“tensor, 192
in scalar field, 70 self-consistent, 542
grand unified theories (GUTs), 431 see also linearised gravity; Newtonian gravity
phase transitions in, 436, 438“9 gravity“electromagnetism coupling, 191
gravitational binding energy, 337“8 gravity“matter coupling strength, 192
gravitational collapse Gravity Probe B (GP-B), 246
and black-hole formation, 259“60, 261“4, 277 Green™s functions, 475“8
and redshift, 263“4 Guth model, 438
concept of, 259 gyroscopes, geodesic precession, 244, 246
free-fall, 263 slow-rotation limit, 347“50
gravitational deflection formula, 234
gravitational effects included in field equations, 156
Hamilton™s principle, 524“7
gravitational field equations, 176“93, 376
Harrison“Zel™dovich spectrum, 458, 459
in empty space, 183
Hawking, Stephen (1942“), 274
non-linearity of, 196, 467
Hawking effect, 274“7
see also Einstein equations; linearised field
definition, 275
equations
Hawking temperature, 276“7
gravitational field tensor, 514
Heaviside functions, 478
gravitational fields
Heisenberg™s uncertainty principle, 274
energy-momentum tensor, 486“90, 511“13
Higgs field, 431, 438, 440
non-vanishing, 184
horizon problem, 419“20, 428
weak, 153“5, 467“70
solving, 437, 442
gravitational focussing, 413
hot dark matter (HDM), 387
gravitational forces, 147, 148
Hubble, Edwin (1889“1953), 186“7, 369
gravitational Lorentz force law, 491
Hubble distance, 420“1
gravitational mass, 147, 149
comoving, 421, 429, 450, 451
gravitational matter density, 147“8
Hubble parameter, 368“71, 390, 392, 407,
gravitational Maxwell equations, 491
435, 444
gravitational perturbations, 502
and redshift, 393
gravitational potential, 147, 155, 168, 185, 201
periods when constant, 434“5
Newtonian, 486
Hubble Space Telescope, 280
gravitational radiation, 508, 516“17
Hubble time, 397, 398, 400
gravitational redshift, 486
and age of universe, 408“10
for fixed emitter or receiver, 202“5, 315
Hubble™s law, 370
general approach, 221“4
Hulse, Russell Alan (1950“), 517
gravitational waves, 498“520
hydrogen, nuclear burning of, 216, 240
and electromagnetic waves compared, 501
hyperbolae, 21, 268
and linear strain, 518“19
invariant, 11“12
detection, 517“20
hypersurfaces, 28, 248, 271“2, 477, 483
effect on free particle, 504“7
non-intersecting spacelike, 356
emission, energy loss, 513“16
hypervolumes, four-dimensional, 476
energy flow, 511“13
563
Index
intrinsic derivative, 71“3
impact parameter, 220“1
tensor, 107“8
indices
intrinsic geometry, 33“6
dummy, 31, 94, 535, 544“6
invariant hyperbolae, 11“12
free, 30“1
inverse transformations, 29“30
lowering, 60
iron, spectral lines of, 240, 243
raising, 60
isotropic metric
tensor, 97
general static, 196“8
vector, 57, 59“60
connection coefficients, 199“200
inertial coordinates
stationary, 198
Cartesian, 140, 144
isotropy of universe, 355
local, 151“2
inertial frames, 117
Jacobian, 29“30, 48
and principle of relativity, 1“2
concept of, 1
Kepler™s laws, 277“8
dragging of, 312“14, 346, 347, 350
Kerr, Roy P. (1934“), 321
electric fields in, 141“2
Kerr black holes
four-current density in, 136“7
binding energy, 338
in standard configuration, 2, 113
extreme, 323
magnetic fields in, 141“2
rotational energy, 325, 327“9
transformations between, 6
structure, 322“7
see also Cartesian inertial frames; instantaneous
Kerr geometry, 230, 310“50
rest frames
Kerr metric, 243, 246, 317“19, 322
inertial mass, 148, 149
event horizon, 323
infinite redshift surfaces, 315, 324, 419
extension, 327
infinitesimal general coordinate transformations,
limits of, 319“20
468, 469“70
Kerr“Schild form, 321“2
inflation
Kerr solution, 345“6, 347
amount of, 435“7
frame-dragging effect, 346, 347
chaotic, 437“8, 440“1
kinetic energy, 535
definition, 428“9
Klein“Gordon equation, 536
ending, 435, 440
covariant, 432
new, 437, 438“40
Kronecker delta, 30
periods of, 429, 430
Kruskal, Martin David (1925“), 266
perturbations from, 442
Kruskal coordinates, 258, 266“71, 273
predictions, 456“7
Kruskal extension, 301
starting, 437“8 Kruskal spacetime diagrams, 269, 270, 273
stochastic, 438, 441“2
inflationary cosmology, 420, 428“62 Lagrangian, 209, 525, 535, 536“7, 539
models, 437 field, 528“9
theory vs. observation, 459“62 gravitational, 542
inflationary epoch, 433, 437 substitution of, 79“80, 199, 205“6
inflaton field, 431“2 Lagrangian density, 526, 528, 529“30, 531“2
instantaneous rest frame (IRF), 15, 20, 168“9 in Einstein“Hilbert action, 542“3
and acceleration, 126“7 modified, 533
definition, 125 variational derivative of, 530
fluid in, 176“7, 178“9 Lagrangian formalism, 524
integration constant as new coordinate, 255 Lagrangian procedures, 349
interferometers, 519 for geodesics, 78“80, 199, 205
interval Laplacian, 148
and lightcone, 6“8 four-dimensional, 140
infinitesimal, 13“14 scalar field, 70, 71
lightlike, 7, 14 spatial, 444
quadratic, 32 symbols for, 71
spacelike, 7“8, 14 laser Michelson interferometers, 519“20
timelike, 7“8, 14 Leibnitz™ rule, 526, 542
564 Index
Leibnitz™ theorem, 63, 530, 532, 544, 547 differentials, 18
global, 468, 488
Lemaitre models, 393“4
homogeneous, 6, 113
matter-only, 404“6
inertial frames, 313
spatially flat, 406“7, 410, 419
inhomogeneous, 6, 113
properties, 404
Lorentz-boost transformations, 4, 5, 8, 9, 11
see also de Sitter model
Lorenz gauge conditions, 501, 510
length
and electromagnetism, 139“41
coordinate, 39
definition, 140
in manifolds, 38“42
in arbitrary coordinates, 143“4
proper, 10, 39
linearised gravity in, 472, 473“4
length contraction, 10“11
satisfying, 478“9, 498“9, 502, 511“12
Lens“Thirring effect, 350
luminosity
Lie derivative, 447n
absolute, 372
light, bending of, 233“6, 486, see also speed
and gravitational collapse, 263
of light
gravitational-wave, 514, 515
lightcones
luminosity distance, 372“4, 411“12
and four-vectors, 115“16
Lynden-Bell, Donald (1935“), 280
and intervals, 6“8
and Schwarzschild solution, 251“2
at Schwarzschild radius, 257 Mach™s principle, 149
future-pointing, 477 magnetic fields in inertial frames, 141“2
past, 479 magnetohydrodynamic instabilities, 215
special-relativistic, 218 manifold, 26“52
line element of Minkowski spacetime, 12“14, see arbitrary, 528
also Schwarzschild line element area, 38“42
linear strain and gravitational waves, 518“19 concept of, 26“7
linear transformations, 2, 46 coordinate transformations in, 28“30
linearised field equations, 467, 470“1, coordinates for, 26
487, 490“1 curvature of, 157“8
compact-source approximation, 481“3 differential, 26
empty-space, 502 dimensions, 26
far-field solution, 481“3 flat, 157, 159
general properties, 473“4 geometry of, 31
general solution, 475“80 length of, 38“42
multipole expansion for, 480“1 local geometry of, 31
in vacuo solution, 474“5 one-dimensional, 527
static source, 485“6 pseudo-Euclidean, 111
stationary source, 483“5 scalar fields, 53
linearised general relativity, 467“97 Schwarzschild, 272
linearised gravity, 472 signatures of, 47
Einstein“Maxwell formulation of, 490“2 tangent spaces to, 44“5, 47, 54, 59
Local Group, 355 tensor calculus on, 92“110
local theories, 50 tensor fields on, 92“3
longitudinal gauge, 443 topology, 49“50
look-back time, 408“10 torsionless, 65, 76
Lorentz contraction of fluid element, 177 two-dimensional, 54“5
Lorentz force law, 491 vector calculus, 53“91
gravitational, 492 vector fields on, 54“5
Lorentz invariant, 137 volume, 38“42
Lorentz symmetry, loss of in general relativity, 487 see also pseudo-Riemannian manifolds;
Lorentz transformation matrices, 125 Riemannian manifolds; submanifolds
Lorentz transformations, 4, 8, 13, 22, 127, 151 Mars, Viking lander, 239
and four-vectors, 116 masers, 281
and length contraction, 10 mass function, 278“9
as four-dimensional rotations, 5“6 massive particle
Cartesian coordinates, 112“13 circular motion, 212“13
565
Index
equatorial, 335“6 field theories, 487
fixed, 473
equatorial trajectories, 332“3
four-dimensional, 47, 364
four-momentum, 118“19
in arbitrary coordinates, 128“31, 142“4
orbits, stability of, 213“17
in Cartesian coordinates, 111“12
radial motion, 209“11
line element, 12“14
equatorial initially, 333“5
pseudo-Euclidean, 114, 151
trajectories, 304“5
symmetries, 486“7
trajectories, 207“9
tensorial equations, 135
matter
weak distortions, 487
and general relativity, 545“6
Minkowski 2-space, 266“7
baryonic, 387, 391
momentum coordinates, 26
dark, 387
Moon, eclipses, 235
energy“momentum tensor, 546
motion, equations of, see equations of motion
non-baryonic, 387
M-theory, 271
matter-density, 176“7, 387“8, 389, 393
gravitational, 147“8
matter-density distribution, quadrupole N Oph 77, 279
moments, 508 naked singularities, 301, 324
matter density perturbations National Aeronautics and Space Administration
growing mode, 459 (NASA) (US), missions, 246
power spectrum, 458“9 neutrinos, 259, 388
matter power spectrum, 458 neutron, discovery of, 259
maximal analytic extension, 301 neutron star, 259“60, 288
maximally symmetric 3-space, 359“61 gravitational forces, 260
Maxwell™s equations, 22“3, 139, 141, 176, 189“90, in binary system, 277, 278
297“8 Newtonian dynamics, 213, 280
gravitational, 491 Newtonian gauge see longitudinal gauge
homogeneous, 538 Newtonian geometry of space and time, 3
inhomogeneous, 538 Newtonian gravity, 147“8, 153, 183, 185, 458
predictions, 498 and equations of motion, 208“9
see also electromagnetic field equations and planetary motion, 230
MCG-6-30-15, spectra, 243 and tidal forces, 167“8, 264
mechanics field equation, 181“2, 186
Newtonian, 524“7 relativistic generalisation, 191
quantum, 259, 274 Newtonian limit, 153“5, 180, 182, 185, 393“4, 509
relativistic, 122“3 and binary systems, 516
Mercury and static sources, 485“6
perihelion shift, 233, 235 Newtonian mechanics, Hamilton™s principle in,
precession, 233 524“7
retardation, 191 Newtonian potential, 443
metric components, 83“4 Newtonian theory, 147, 154“5, 180, 181, 183, 230
metric connection, use of term, 66 and special relativity compared, 2
metric function, 196 of stellar structure, 288
and affine connection, 65“7 Newton™s laws of motion, 1
metric tensor, 32, 93, 96, 112 NGC 4258, 281
Michelson“Morley experiment, 22, 23 Nobel Prize, 517
Milky Way Galaxy, 187, 280“1, 355 Noether™s theorem, 549
Minkowski, Hermann (1864“1909), 4 non-Euclidean geometry, 4
Minkowski coordinates, 297“8 examples, 36“8
Minkowski geometry, 5, 26, 317, 319“20 non-Euclidean space
spacetime, 153, 156 infinite, 42
use of term, 4 three-dimensional, 38, 41“2
Minkowski regions, 269“70 non-inertial frames, 129
Minkowski spacetime, 13, 123, 181, 251, 457, 468 normalised scale parameter, 389
as background, 469, 471, 476n, 484“5, 487, 498 Novikov coordinates, 254n
coordinate transformations, 5 null curve, 75“6
566 Index
null-cone, 116 equations of motion, 180“1
number densities four-velocity, 290
in Friedmann“Robertson“Walker geometry, perihelion, 230
374“6 shift, 233, 235, 486
proper, 375“6 perturbation equations, in Fourier space, 445“6
perturbations
from inflation, 442
observers
gravitational, 502
accelerating, 125“8
Newtonian potential, 443
in curved spacetime, 152“3
scalar-field, evolution, 442“6
fundamental, 356“7
see also curvature perturbations; matter density
Oppenheimer“Volkoff equation, 293, 294
perturbations
Oppenheimer“Volkoff limit, 260
phase transitions, 430“1
orbit
photon, 388
Newtonian, 208
circular motion, 219
non-circular, 216“17, 230
equatorial, 341
of massive particle, 212“17
equation of motion, 119
shape of, 208
four-momentum, 119“20, 222, 223, 241, 243
spiral, 215, 217
four-wavevector, 120“1
see also circular orbits; equatorial orbits; photon
radial motion, 218“19, 302“4
orbits; planetary orbits
radially outgoing, 257“8
orthogonal connecting vectors, 170
redshift, 204“5, 240, 243, 408
orthogonal coordinates, 39
trajectories, 217, 233“4
orthonormal basis vectors, 59
equatorial, 338“9
photon orbits
Palatini approach for general relativity in vacuo,
circular, 233
543“5
energy equation, 219“20, 236“7
Palatini equation, 541, 544
equatorial, stability of, 342“4
parallel transport, 222
general, 220
and curvature, 163“5
stability of, 220“1
and gyroscopes, 244
photon path deflection, 237“9
of tensor, 108
photon propagation, 342“3, 344
of vector, 73“5
photon worldlines, 14, 218, 242, 256
on spherical surface, 165
radial, in Schwarzschild coordinates, 251“2
path dependence, 74“5
Planck era, 430“1, 436
particle
Planck scales, 270“1
charged, 144“5
plane gravitational waves
four-momentum, 312“13
and polarisation states, 498“500
infalling, 210“11, 219, 252“9
effects on free particles, 504“7
non-interacting, 176
propagation, 505
tunnelling, 275
planetary motion and Newtonian gravity, 231
see also free particle; massive particle
planetary orbits
particle“antiparticle pairs, 274
ellipticity, 230, 231
particle horizon, 418“20
precession, 230“3
particle worldlines, 14“16, 116“17, 154, 156
Poincar© transformations, 6, 7, 113
radial, in Schwarzschild coordinates, 252“4
Poisson™s equation, 147, 183, 185, 458
past-pointing vectors, 116
polar coordinates
Pauli exclusion principle, 259
cylindrical, 272
Penrose, Sir Roger (1931“), 260, 301,
in a plane, 82“7
324, 325
polarisation states and plane gravitational waves,
Penrose process, 327“9, 344
498“500
perfect fluid, 289, 386“9
polarisation tensors, linear, 500, 511
and weak-field limit, 184“5
polytropic index, 293
conservation of energy“momentum, 179“81
position coordinates, 26
definition, 178“9
potential energy, 535
energy“momentum tensors, 178“9, 187,
376“9, 432 potential functions, 443
567
Index
power spectrum quark“hadron phase transition, 431
cosmic microwave background, 459“62 quasars
curvature perturbations, 456“7 discovery, 279“80
definition, 456 radio-wave deflection measurements, 236
matter density perturbations, 458“9 quasi-Minkowski coordinates, 467“8, 485
scale invariant, 456 quasi-stellar objects (QSOs), 279n
precession quotient theorem, 103“4
geodesic, 244“6
gyroscopes, 244“6 radar echoes, 236“9
slow-rotation limit, 347“50 radial coordinates, 209
planetary orbits, 230“3 radial distance, 209
primordial spectral index, 457“2, 458 radial motion
principal photon geodesics, equatorial, 339“41 equatorial initially, massive particle, 333“5
principle of relativity and inertial frames, 1“2 massive particle, 209“11
principal stresses, 170 photon, 218“19
projectiles and elevator experiment, 148“9 radiation density, 388“9, 393
proper angular speed, 245 radiation efficiency in accretion discs, 215“16,
proper charge density, 508 240, 338
proper density, 509 radio quasars, 235
proper distance, 371 radio sources, 235
proper length, 10, 39 rapidities, addition of, 19
proper mass density, 508 rapidity parameter, 5, 18
proper motion of stars, 280“1 receiver and emitter fixed, gravitational redshift,
proper number densities, 375“6 202“5, 315
proper time, 14“16, 204, 206“7, 219, 252“3 recombination, 420
finite, 211, 254 red giants, 288
infinite, 211, 254 redshift, 187, 191, 395, 411
proper volume, 10 and gravitational collapse, 262“3
protons, 259 and Hubble parameter, 393
Schwarzschild radius, 249 cosmological, 355“6
pseudo-Euclidean geometry, see Minkowski infinite, 315, 324, 419
geometry photon, 204“5, 240, 242, 408
pseudo-Euclidean manifolds, four-dimensional, 111 quasar, 279“80
pseudo-Euclidean space, 47, 54, 114 see also gravitational redshift
pseudo-Riemannian manifold, 39, 45“7, 53, 54, reheating, 451
59, 157 use of term, 435
curved spacetime, 150“1 Reissner“Nordström (RN) black hole, extreme, 301
local Cartesian coordinates, 46“7, 67“8 Reissner“Nordström geometry
non-null curve, 75 charged black hole, 300“2
null curve, 75 radial massive particle trajectories, 304“5
use of term, 32 radial photon trajectories, 302“4
vectors, 62, 74 spacetime diagram, 303
pseudotensors, use of term, 468 relative three-vector, 117
PSR B1913+16 (binary pulsar), 517“18 relativistic aberration
pulsars, binary, 516“17 and Doppler effect, 120“1
pulsation, radial, 202 formula, 121
relativistic collisions and Compton scattering,
quadratic intervals, 32 123“5
quadrupole formula, 483, 507“8 relativistic gravitational equations
quadrupole-moment tensor, 483, 508, 509“10, static spherically symmetric charged body, 288,
514, 516 296“300
reduced, 514, 516 stellar interior, 288“92
quadrupole moments, transverse-traceless, 514“15 stellar structure, 292“4
quantum chromodynamics and inflation, 431 relativistic mechanics, 122“3
quantum gravity, theory, 188 relativistic theories of gravity, 191“3
quantum mechanics and white dwarfs, 259, 274 r-equation replacement, 206, 217
568 Index
resonant detectors, 518 for stellar interior, 294“5
retarded time parameter, 258 Schwarzschild coordinates, 248, 250, 261, 262,
264, 268
Ricci scalar, 250, 539“40
radial particle worldlines in, 252“4
definition, 161“2
radial photon worldlines in, 251“2
linearised, 470“1
timelike, 266
Ricci tensor, 182, 199, 202, 359“60, 540
Schwarzschild geometry, 196“224, 233“4, 248,
components, 200, 289“90, 298“9, 317, 378
288, 301
definition, 161“2
and experimental tests, 230
linearised, 470“1
geodesics in, 205“7
terms, 488“9
in Kruskal coordinates, 266“71
Riemann tensors, see curvature tensors
spacetime diagram, 256“7
Riemannian geometry, 32“3
static, 272
Riemannian manifolds, 26, 39, 61
tidal forces in, 264
definition, 32
Schwarzschild line element, 204, 205, 255, 258
local Cartesian coordinates, 42“4
derivation, 198“201
two-dimensional, 33, 44“5, 53, 311
Schwarzschild manifold, 272
conformal flatness, 267, 282“3
Schwarzschild metric, 211, 240, 242“3, 266,
vectors, 74
277, 443
see also pseudo-Riemannian manifolds
connection coefficients, 244
rotating bodies
singularities in, 249“50
characterisation, 310
spherical symmetry, 206, 289, 292, 296
slow, 347“50
validity, 201“2
spacetime geometry, 310“50
Schwarzschild radius, 202, 249, 251, 252
lightcone structure at, 257
scalar density, 529
Schwarzschild solution, 260, 486
scalar field, 430“1, 527“9
lightcone structure of, 251“2
as cosmological fluid, 431“2
maximal extension, 270
energy“momentum tensor, 432, 444“5
Schwarzschild spacetime, 202“3
equations of motion, 433
Shapiro effect, 486
field theories, 534“6
short X-ray transients, 279
gradient, 70
sign conventions, in general relativity, 193
Higgs-like, 438
signatures, of manifolds, 47
Laplacian, 70, 71
simultaneity, concept of, 9
on manifolds, 53
singularities, 38, 269“71
quantum irregularities, 442
black-hole, 259, 271
reheating, 435, 451
coordinate, 37, 250, 341
scalar multiplication, tensors, 98
intrinsic, 250, 288
scalar parameters, 431
naked, 301, 324
scalar product
real, 252
positive definite, 61
ring, 327
vectors, 58
in Schwarzschild metric, 249“50
scalar“tensor theory of gravity, 191“2
spacelike, 252, 345“6
scalar theory of gravity, 191
timelike, 345“6
scalar, covariant derivatives, 69“70, see also Ricci
white-hole, 258, 270
scalar
singularity theorems, 260
scale factor, 376“80
evolution of, 397“400 Sirius, 259
scale fluctuations, super-horizon, 451 Sirius B, 259
scale invariance, 456 slow-roll approximation, 434“5, 440
scale-invariant spectrum, 459 source term in field equations, 136, 176
scaling factors, conformal, 266“7 space
Schmidt, Maarten (1929“), 279“80 empty, 183
Schwarzschild, Karl (1873“1916), 196 pseudo-Euclidean, 47, 54, 114
Schwarzschild black holes, 202, 240, 248“83, 288 with Newtonian geometry, 3
formation of, 260“3, 296 see also Euclidean space; Fourier space;
non-Euclidean space
Schwarzschild constant-density solution, 296
569
Index
light deflection, 235, 236
spacetime, 26
Schwarzschild metric, 201
empty, 184
four-dimensional, 158 spherical surfaces, 36“7, 171“2
geometry of, 240 curvature tensor, 161, 171
Minkowski geometry, 153, 156 four-dimensional, 37“8
of special relativity, 1“25 geodesic convergence, 167
paths in, 13“14 parallel transport, 165
rotations in, 127 three-dimensional, 35, 40“2
Schwarzschild, 202“3 two-dimensional, 35
static, 196, 273 vector field, 54
stationary, 196, 275, 315 spherical symmetry, 202, 206, 288“305, 310
symmetries, 196 spherically symmetric collapse, 260“4
see also curved spacetime; Minkowski spacetime spin, quantum mechanical, 192“3
spacetime curvature, 181, 190, 250 spontaneous symmetry breaking, 431
and energy“momentum tensors, 176 stars
gravity as, 150“1 age of, 410
see also curved spacetime gravitational collapse, 259“60, 261“3
spacetime diagrams, 7, 8“9, 258, 261, 267 maximum mass, 260
Kerr solution, 346 proper motion of, 280“1
Reissner“Nordström geometry, 303 radial pulsation, 202
Schwarzschild geometry, 256“7 velocity dispersion, 280
see also Kruskal spacetime diagrams see also binary system; neutron star; white dwarf
spacetime geometry static metric, 196“7
dynamics, 376 static source
of special relativity, 3“5 and Newtonian limit, 485“6
rotating bodies, 310“50 non-relativistic, 484
spacetime indices, 528 static spherically symmetric charged body,
relativistic gravitational equations, 288,
spacetime torsion, 193
296“300
spatial amplitude tensor, 503
stationary axisymmetric metric, general, 310“12
spatial curvature
stationary limit surface, 314“15, 324
evolution, 417“18
stationary source, 483“5
negative, 364“5, 391
Stefan“Boltzmann constant, 277, 388
positive, 363“4, 391
stellar interior
zero, 364
relativistic gravitational equations, 288“92
spatial momentum, 275
Schwarzschild constant-density solution, 294“5
spatial projection tensors, 503
stellar structure
spatial velocity fields, 484
Newtonian theory of, 288
special relativity, 111“13
relativistic gravitational equations, 292“4
and electromagnetism, 135
stress“energy tensor, see energy“momentum tensor
and elevator experiment, 149“50
submanifold, 28
and Newtonian theory compared, 2
integration over, 47“9
acceleration in, 19“20
subtraction, tensor, 98
Einstein™s route to, 22“3
summation convention, 30“1
event horizon in, 21“2
Sun
spacetime geometry of, 3“5
corona, 239
spacetime of, 1“25
eclipses, 235
velocity addition, 18“19
gravitational collapse, 259
spectrum, 240, 243
gravitational redshift, 486
curvature, 456
and light bending, 233“6
Harrison-Zel™dovich, 458, 459
photon path deflection, 237“9
matter power, 458
Schwarzschild radius, 249
see also power spectrum
super-horizon scale fluctuations, 451
speed of light, 14, 23
supermassive black holes, 265, 279“82
constant, 3“4
existence, 280
spherical mass, 486
and gravitational redshift, 202“5 potential, 282
570 Index
supernovae, 188 scalar multiplication, 98
superstring theory (M-theory), 271 spatial, 503
surfaces in manifolds, 27“8 subtraction, 98
infinite redshift surfaces, 315, 325, 419 symmetries, 94“6
parametric, 28 tidal stress, 168
stationary limit, 314“15, 324 torsion, 65
three-surfaces, 315“16 zero-rank, 93
two-surfaces, 272 see also curvature tensor; electromagnetic field
see also hypersurfaces; spherical surfaces tensor; energy“momentum tensor; metric
tensor; quadrupole-moment tensor
tetrads, 125, 126, 127, 152
tangent space to manifold, 44“5, 47, 54, 59
threading for spacetime, 356“7
tangent vectors, 57, 76, 80, 123, 248
three-acceleration, 123
covariant components, 81
three-force, 122
as directional derivative, 81“2
electromagnetic, 147
length, 75
three-momentum, 119
to curve, 55“6
three-space, 272
Taylor, Joseph Hooton, Jr (1941“), 517
see also maximally symmetric 3-space
tensor calculus on manifolds, 92“110
three-space vectors, 127
tensor equations, 102“3
three-spheres, 37“8
tensor fields, 487, 528
three-surfaces, null, 315“16
on manifolds, 92“3
three-vector potential, 297“8
symmetric, 498
three-vectors, 130, 135, 138, 141
tensor product, 98“9
relative, 117
tensorial equation, 135
unit, 120
tensorial operations
three-velocity, 118
definition, 98
spatial, 290
elementary, 98“100
three-wavevectors, 499
tensors
tidal forces, 149
addition, 98
and Newtonian gravity, 167“8, 264
amplitude, 499“500
gravitational, 167
and coordinate transformations, 101“2
in binary systems, 277
arbitrary, 104
in curved spacetime, 167“70
as geometrical objects, 100“1
in Schwarzschild geometry, 264
basis, 100
components, 93“4, 100, 102, 103“4, 112 near black holes, 264“5
concept of, 92 tidal stress tensor, 168
contraction, 99“100 time
coordinates, 102“3 cosmic, 419
covariant derivatives, 104“7 look-back, 408“10
definition, 93 Newtonian geometry, 3
field-strength, 537, 538 retarded, 478, 479
four-tensors, 136, 152, 250 see also Hubble time; proper time;
spacetime
gravitational field, 514
time dilation, 10, 11
indices, 97
in weak gravitational field, 155
inner product, 99“100
intrinsic derivatives, 107“8 timelike curves, closed, 301, 327
linear polarisation, 500, 511 topology of manifolds, 49“50
mapping, 97“8 torsion tensor, 65
metric, 96 torsion theories, 192“3
outer product, 98“9 tortoise coordinate, 266
parallel transport, 108 total density parameter, 392
rank of, 93 trajectories
rank-1, 93 of infalling particle of, 210, 218, 219,
252“9
rank-2, 94, 95, 100, 105, 177, 182
of massive particle, 207“9
definition, 98“9
of photon, 217, 233“4
rank-3, 99
571
Index
variations, calculus of, 87“8
radial
of massive particles, 304“5 vector calculus, on manifolds, 53“91
of photons, 302“4 vector fields, 92“3, 191, 527“8
see also equatorial trajectories components, 73
transformation matrices, 29 contravariant components, 56, 57
transformations covariant components, 57
Galilean, 3, 4 curl, 71
gauge, 140, 472 divergence, 70
inverse, 29“30 on manifold, 54“5
Jacobian of, 29“30 parallel, 73
linear, 2, 46 vector operator, component form,
Poincar©, 6, 7, 113 70“1
see also coordinate transformations; Lorentz vectors
transformation angle between, 62
transverse-traceless (TT) gauge, 500, 504, 505 as directional derivatives, 81“2
transformation into, 502“3, 510“11 as linear function, 92
tunnelling of particles, 275 concept of, 53
turbulent viscosity, 215 derivatives
two-spheres, 34“5, 41“2, 250 covariant, 68“70
two-surfaces, 272 intrinsic, 71“3
future-pointing, 116
ultra-stiff equations, 294
indices, 57, 59“60
uncertainty principle, 274, 276
length, 62
unified electroweak theory, 431
local, 54, 56
unit vectors, 62
null, 62, 115, 116, 475
timelike, 126
orthogonal, 62, 170
universe
parallel transport, 73“5
and Friedmann“Robertson“Walker geometry,
past-pointing, 116
355“81
properties, coordinate-independent, 61“2
acceleration, 188
reciprocal systems, 57
age of, 398, 400, 408“10
scalar product, 58
collapse, 398
spacelike, 115, 116, 152
dynamics, 401
tangent, 55“6
energy density of, 390, 433
three-space, 127
expansion, 367, 420

<<

. 23
( 24)



>>