523

Index

aberration – radio galaxies, 174 – of starlight, 224 – Seyfert galaxies, 174 – relativistic, 140 – unified model of, 175 abundance adiabatic – 4He, 53, 60, 64 – expansion, 7, 10, 40 – 7Li, 66 – perturbations, 89, 103, 104, 127, – cosmic, 36, 43 499 – deuterium, 52, 59, 64 affine – hydrogen, 66 – connection, 303, 342, 361 – neutron, 54, 56 – geometry, 334 – of lightest elements, 62 – manifold, 342 – primordial, 49, 61, 63, 195 – parameter, 305, 373, 505 accretion, 171, 177 – space, 334 – characteristic temperature, 179 Aharonov–Bohm effect, 330 – disk, 180 Alfven´ wave, 172 – magnetohydrodynamic instabil- annihilation ity, 180 – electron–positron, 44 – viscous torque, 183 – gluon–gluon, 193 acoustic – particle–antiparticle, 10, 20, 42, – damping, 131 221 – oscillations, 12, 118, 129, 135, antenna 487, 503 – beam solid angle, 151 – peak at 150 Mpc, 105, 118 – collecting area, 151 – peaks, 130, 145 – effective aperture, 151 – waves, 88, 104, 144 – feedhorn, 153 action integral, 267, 273 – main beam, 150 active galactic nuclei, 79, 171 – power pattern, 150 – blazars, 173 – side lobes, 150 – – BL Lacertae, 174 – temperature, 152 – quasars, 172 antimatter

Relativity, Astrophysics and Cosmology Radoje Beluseviˇ c´ Copyright c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  ISBN: 978-3-527-40764-4 524 Index

– presence in the universe, 21, 43, – – ergosphere, 172 198 – – frame dragging, 172 astronomical unit (AU), 421 – supermassive, 170–172, 182, 187, atomic 190 – clock, 214, 250, 252 black-hole era, 198 – energy levels bolometer, 14, 153 – – conjectured by Boltzmann, 2 Boltzmann – – in cyanogen, 16 – constant, 8 – equation, 73, 504 baryon, 29, 37 – – for photons, 508, 509 – bulk velocity, 508 – factor, 50 – density, 22, 128 BOOMERANG telescope, 13, 148 – density parameter, 38 Bose–Einstein – loading, 512 – distribution, 508 baryon-number conservation, 198 boson, 21, 42 baryon-photon fluid, 512 – star, 192 baryon-to-photon density ratio, brightness 510 – distribution of the sky, 157 baryon-to-photon ratio, 66 – function, 508 basis – of blackbody radiation, 133 – anholonomic, 375 – temperature, 152 – coordinate, 336, 407 – dyad, 338 Cartan – holonomic, 380 – calculus, 342, 361 – one-form, 335 – formula, 364 – orthonormal, 380 – formulation of Newtonian grav- – tensors, 338 ity, 31 – vectors, 334 – structural equations, 377, 407 Bianchi identity, 300, 306, 313, 352 Cassini spacecraft, 183 – contracted, 315, 317 CBI telescope array, 15, 148, 153, – first, 374, 378 159 – second, 374, 379 CCD array, 416 Big Bang, 4, 28, 192 celestial binding energy – coordinates, 141 – 4He, 52 – equator, 160, 424 – deuterium, 51, 53 – sphere, 148, 165, 424, 475 – gravitational, 247 Cepheids, see stars – of lightest nuclei, 60 chemical Birkhoff’s theorem, 24, 257 – equilibrium, 38, 44 black hole, 163, 171, 198 – potential, 21, 42 – accretion, 182, 187 Cherenkov – rotating (Kerr) – radiation, 228 Index 525

Christoffel symbol, 275, 278, 293, conformal 295, 303, 376, 495 – mapping, 459 CMB, 4, 82 – time, 458 – angular power spectrum, 15, congruence of geodesics, 372 130, 137, 146 – connecting vector, 372 – dipole anisotropy, 121, 141 connection – discovery of, 10, 195 – coefficients, 294, 367, 368, 371 – energy density, 18, 36 – Koszul, 365 – foregrounds, 146, 147 – one-form, 376, 407 – lensing, 445, 478 continuity equation, 25, 85, 87, – map, 15, 149 122, 305, 324, 353 – multipole expansion, 135 contravariant – observational techniques, 146 – basis vectors, 270 – photons, 90 – components, 277, 296 – polarimeter, 485 – transformation rule, 277, 288 – polarization, 14, 81, 485, 489 convolution theorem, 158 – – (E,B)-type patterns, 487 coordinate transformations – – anisotropy, 478, 487 – Bardeen variables, 494 – – power spectrum, 480, 488 – gauge freedom, 491 – – WMAP measurement, 487 coordinates – primary anisotropies, 127 – Cartesian, 32, 89, 265, 278, 280, – secondary anisotropies, 131, 146 299, 347 – spectrum, 13 – celestial, 141 – temperature, 8, 12 – curvilinear, 240, 280 – – anisotropy, 142, 149, 157, 487 – galactic, 141 – – power spectrum, 15, 130, 137, – generalized, 265 143, 488 – spherical, 240 CMBFAST code, 105, 114 – system of, 334 co-moving Coriolis force, 282 – coordinates, 387, 458 correlation function, 107, 137 – distance, 412 cosmic – frame, 86, 387 – baryons, 67 – observer, 387, 390, 505 – dark age, 481 – wavevector, 88, 502 – energy density, 29 COBE telescope, 13, 148 – fluid, 130, 387, 395, 398, 489 collision rate, 37 – lens, 461 color index, 417 – photosphere, 11, 106, 195 Compton – rays, 213 – scattering, 12 – shear, 468, 471 – – inverse, 132, 436 – time, 6, 9, 30, 387, 439 configuration space, 265 – variance, 138, 139, 488 526 Index

cosmic expansion covariant – accelerated, 27, 196, 398, 402, 443 – basis vectors, 269 – decelerated, 395, 443 – components, 270, 271, 277, 296 – deceleration parameter, 391 – derivative, 296, 308, 366 – scale parameter, 7 – formulation, 232 cosmic inflation, 514 – tensor, 290 – chaotic, 517 – transformation rule, 277, 288 – density perturbations, 517 Crab Nebula, 207 – quantum fluctuations, 514 critical density, 29, 396 – reheating, 516 curvature – scalar field φ, 514 – extrinsic, 311 – slow-roll approximation, 516 – forms, 361 cosmic microwave background, – intrinsic, 302, 307, 311, 330 see CMB – of manifold, 331 cosmological – of spacetime, 35, 111, 253, 319, – constant, 27, 30, 104, 142, 401, 396 402, 440, 515 – parameter, 27, 30 – fluctuations – perturbations, 489, 494, 518 – – acoustic dynamics, 509, 511 – radius of, 397 – – adiabatic, 499 – scalar, 496 – – dust-dominated universe, 501 – tensor, 369 – – Einstein equations, 495 cyanogen excitation, 17 – – gauge transformation, 491 cyclotron frequency, 234 – – Newtonian theory, 82 – – radiation-dominated universe, d’Alembertian, 299 503 dark energy, 30, 196, 402 – – relativistic theory, 489 dark halo, 77, 101 – – scalar, 491 dark matter, 29, 77, 92, 96, 103, – – tensor, 491 104, 162 – – tight-coupling limit, 510 dark-sky paradox, 17 ––vector,491 DASI telescope array, 14, 148, 159, – hypothesis, 4, 82, 86 485 – parameters, 142 de Broglie Coulomb – wavelength, 9 – barrier, 52 Debye – force, 21 – length, 23 – potential, 23 – shielding, 21 – scattering, 235 degenerate covariance – era, 198 – matrix, 158 density – of tensors, 288 – contrast, 87, 106 – energy, 39 Index 527

– fluctuations, 106 – tensor, 316, 382, 406, 496 – number, 39 Einstein, Albert, 316 – parameter, 29 Einstein–de Sitter model, 89, 123 – phase-space, 55 electromagnetic detailed balance, 54, 74 – four-potential, 272 differential forms, 342 electromagnetic field – coderivative operator, 354 – tensor, 273 – exterior derivative, 343, 349 electroweak – Hodge star operator, 347 – symmetry, 192 – inner product, 358 ellipse, 465 – interior product, 363 ensemble – one-form, 333, 342 – average, 108, 137, 138, 487 – two-form, 343, 377, 408 – average power, 109 – wedge product, 344 entropy differential geometry, 331 – conservation of, 7, 42 Dirac – density, 22, 40, 41 – delta function, 109 equinox, 423 – monopole, 330 equivalence principle, 247 directional derivative, 332 – strong, 247 distance modulus, 415, 424, 427, – weak, 247 440 ergodic hypothesis, 108, 138 distribution function, 38, 45, 508 ether, 204, 215, 228, 229 Doppler effect, 97, 129, 134, 140, Euclidean 220, 221, 226, 249, 262 – geometry, 394 dual field tensor, 306, 353 – space, 25, 27, 31, 239, 299, 333 Euler’s equation, 25, 85, 87, 324 ecliptic Euler–Lagrange equations, 265 – coordinates, 456 extragalactic distance – definition of, 424 – angular-diameter distance, 413, – latitude, 223 449 – plane of, 224, 423 – co-moving distance, 412 Eddington – distance ladder, 420 – limit, 178 – expanding photosphere method, Einstein 434 – field equations, 320, 496 – fundamental plane relation, 432 – – conservation laws, 319 – fundamental rung, 427 – – coordinate transformations, – luminosity distance, 412, 419, 317 440 – – for empty space, 316 – primary indicator, 435 – – from variational principle, 324 – proper distance, 389 – – weak-field limit, 320 – summation convention, 239, 271 528 Index

– secondary indicator, 426, 427, – winds, 188 432, 438 galaxy – to Large Magellanic Cloud, 426 – Andromeda, 28, 165 – Tully–Fisher relation, 123, 430 – biasing, 119 – classification, 168 Fermat’s principle, 253, 269 – correlation function, 105, 107 fermion, 21, 42 – dwarf, 98, 159 Feynman – elliptical, 98, 159, 169, 188 – path integral approach, 325 – irregular, 170 first stars, 77, 195 – lenticular, 169 Fizeau’s experiment, 229 – Local Group, 141, 165 flux – Magellanic Clouds, 165 – definition of, 413 – Milky Way, 121, 160, 189 – of angular momentum, 184 – morphological evolution, 187 flux density, 399, 413 – nonlinear evolution, 119 – incident on antenna, 150 – peculiar motion, 121 – of starlight, 414 – redshift distortions, 121 Foucault pendulum, 302 – – finger-of-God effect, 126 Fourier – rotation curve, 97 – decomposition, 135 – Sombrero, 170 – modes, 108, 110 – spiral, 97, 168, 188 – series, 88 – spiral arms, 160, 168 – space, 108 – starburst, 188 – transform, 109, 157 – Whirlpool, 170 Fresnel drag coefficient, 229 galaxy cluster, 98, 195 Friedmann–Lemaˆıtre – Abell 2029, 99 – equations, 27, 387, 395, 406, 410, – Centaurus, 141 500 – Coma, 165 – – derivation of, 402 – Virgo, 124, 165 – model, 4, 82, 459, 475 galaxy redshift survey – universe, 27, 395, 407, 440, 472 – 2MASS, 167 fusion – 2dFGRS, 83, 115 – Gamow – CfA, 116 – – condition, 37 – DEEP2, 116 – SDSS, 115 galactic Galilean relativity, 203 – bulge, 160 Galilei, Galileo, 162 – center, 160 gamma-ray – disk, 160 – bursters, 208 – maps, 147 – emission from – mergers, 188 – – blazars, 174 – plane, 160 Index 529 gauge – critical surface density, 452 – longitudinal, 494 – deflection angle, 448, 474 – symmetries, 325 – discovery by Walsh et al., 444 gauge invariance – effect on SNe Ia, 442 – in general relativity, 491 – effective convergence, 474 Gauss’s theorem, 305, 327, 357 – effective potential, 460, 463 Gaussian – Einstein radius, 450, 452, 454 – initial conditions, 110 – Einstein ring, 445 – perturbations, 115, 137 – galactic, 452 – random field, 136 – geometrical time delay, 459 – random variables, 136 – in cosmology, 458 generalized momentum, 269, 273 – lens equation, 449 geodesic, 31, 254 – MACHO, 101, 453 –deviation,31 – magnification factor, 451 – – equation of, 284, 373, 472 – power spectrum, 476 – equation, 254, 275, 301, 372 – shear, 465 – – for photons, 507 – shear power spectrum, 476 – – in RW spacetime, 388 – shear variance, 476 – frame, 282, 297 – strong limit, 467 – hypothesis, 307, 320 – weak limit, 466 – null, 254, 304 gravitational microlens, 101, 444, – space-like, 303 467 – time-like, 303 – light curve, 453 geometric phase, 330 – magnification pattern, 467 gravitational – optical depth, 457 – collapse, 91, 321, 434 – parallax, 454 – constant, 24 graviton, 518 – energy gravity – – released by accretion, 177 – Newtonian, 31 – energy–momentum tensor, 319, – – Cartan’s formulation of, 31 323, 328 Gravity Probe A, 262 – stress–energy tensor, 323 Great Attractor, 124, 141 gravitational field Great Wall, 116, 160 – static, 260, 262 group – stationary, 260 – defintiion of, 243 gravitational lensing, 132, 443 – Lorentz, 211, 244, 245 – arcs, 467 – non-Abelian, 246 – caustics, 466 – Poincare,´ 246 – convergence, 464, 465 – rotation, 245 – cosmic shear, 468 Gunn–Peterson – critical curves, 466 –effect,79 530 Index

– trough, 79 – expansion, 4, 27, 44, 49, 85, 196, gyration frequency, 234 390, 396, 401, 507 – flow, 86, 428 hadron era, 193 – horizon, 504 hadrons, 193 – law, 4, 25, 389, 390, 419 Hafele–Keating experiment, 266 – parameter, 36, 389, 396 Hamilton’s principle, see varia- – radius, 95, 489 tional principle – Space Telescope, 437 Hamilton–Jacobi equation, 272 – time, 459 Hawking hydrostatic equilibrium, 98 – radiation, 196, 198 heat ideal-gas law, 98 – capacity, 10, 19 inertial – flow, 321 – confinement, 23 –latent,7,23 – frame, 203, 208, 240 – specific, 251 – mass, 247, 273 helicity – observer, 254 – states, 41 inflation, see cosmic inflation helioseismology, 142 infrared astronomy, 163 helium synthesis, 194 interferometer, 156 Hertzsprung–Russell diagram, – baseline, 155 424 – complex visibility, 157 High-Z Supernova Search, 442 – double-slit, 330 Hilbert – fringe function, 155 – action, 326 – Michelson–Morley, 204 Hipparchus, Greek astronomer, – radio, 163, 436 414, 423 – signal correlation, 155 Hipparcos satellite, 423 interstellar Hodge star, see differential forms – extinction, 162, 207, 418, 440 holonomic frame, see coordinate – medium, 161 basis – reddening, 418 horizon invariant mass, 219 – distance, 93 ionization – event, 172, 178, 185 – fraction, 70 – particle, 69, 93, 96, 112 isocurvature perturbations, 103 – puzzle, 95, 514 isomorphism HST Key Project, 437 – of tensor spaces, 341 Hubble – of vectors and one-forms, 341 – classification, 168 isothermal perturbations, 103 – constant, 29, 411, 419, 437 – diagram, 428, 439 Jacobi identity, 314, 362 Jacobian, 290, 331, 393 Index 531

– matrix, 464 linear algebra, 332 Jeans linear perturbation theory, 88, 489 – instability, 92 Local Supercluster, 165 – length, 89 locally inertial frame, 281 – mass, 89, 91 lookback time, 410 Jupiter’s moons, 224 Lorentz Jupiter–Earth distance, 224 – boost, 211, 243 – force, 232, 272 K-correction, 417 – frame, 253 Kaiser effect, 122 – gamma factor, 209 Kepler’s – gauge, 354 – third law, 97, 259 – group kinetic – – homogeneous, 211 – energy, 217 – – orthochronous, 211 – – of molecular rotation, 8 – transformation, 208 – – released in fission, 221 Lorentz-invariant formulation, – equilibrium, 38, 43 216 – Sunyaev–Zeldovich effect, 134 luminosity, 399 – theory of gases, 2 – bolometric, 171 – viscosity, 184 – definition of, 413 Kronecker delta, 240, 270, 297 – distance, 399, 419, 440 – – definition of, 412 Lagrange point, 149 – Eddington limit, 178 Lagrangian, 265 lunar laser ranging, 248 – density, 307, 325 Lyman-α Laplace’s equation, 317 – emission, 65 Laplacian, 88 – forest, 79 – generalized, 354 – photons, 71 Legendre polynomial, 510 – reaction, 73 Leibniz rule, 297, 333, 361, 366 – resonance, 73 – generalized, 350 Levi–Civita Mobius¨ band, 331 – (pseudo)tensor, 292, 354 Mossbauer¨ effect, 250 – connection coefficients, 376 MACHO, see gravitational lensing – symbol, 347, 359 magnetic – tensor density, 291 – dipole moment, 235 Lie – moment – algebra, 362 – – anomalous, 236 – bracket, 362 magnitude, 414 – derivative, 361, 364 – absolute, 415 light cone, 212 – apparent, 414, 428, 439 – under conformal mapping, 459 – bolometric apparent, 415 532 Index

– Pogson’s scale, 414 – experiment, 204 main-sequence fitting, 424 – interferometer, 204 Malmquist bias, 441 Minkowski manifold, 331 – coordinates, 239 – differentiable, 331 – four-vector, 216 – orientable, 331 – interval, 208 – pseudo-Riemannian, 329 – metric tensor, 239 maser – spacetime, 208, 232, 239 – hydrogen, 263 morphological – infrared, 205 – classification, 168 – interstellar, 435 – K-correction, 168 – water-vapor, 436 – sequence, 187 mass multicolor photometry, 418 – fraction muon ––4He, 53, 61 – g 2 experiment, 236 − – – hydrogen, 53 – decay, 231, 237 mass fluctuation σ8, 114 muonium spectroscopy, 238 mass–energy conversion, 171 mass–luminosity ratio, 97 nebula, see galaxy matter-dominated era, 38, 70, 89, neutrino 397 – decoupling, 42 MAXIMA telescope, 13, 148 – number of species, 68, 103 Maxwell’s equations neutron – and gauge invaiance, 351 – β-decay, 50, 54 – from variational principle, 357 – interferometer, 230 – inhomogeneous, 352 neutron star, 198 metric neutron-to-proton ratio, 53, 194 – as a mapping, 338 Newton, Isaac, 32 – coefficients, 270, 299, 342 Newtonian – connection, 342, 369 – acceleration, 259 – pseudo-Riemannian, 342 – cosmology, 34 – Robertson–Walker, 26, 386, 407, – fluid, 324 458, 489, 505 – gravity, 31 – Schwarzschild, 256, 379 – – Cartan’s formulation of, 31 – signature of, 241 – limit, 276, 317, 324, 447 – space, 239, 342 – mechanics, 35, 204 – tensor, 278, 288, 338, 342 – perturbations, 82, 89 – – inverse, 338 – potential, 448, 459 – theory of gravity, 247 – spacetime, 32, 35, 286 – with spherical symmetry, 255 – universe, 26 Michelson–Morley Noether – theorem, 325 Index 533

Nordtvedt effect, 248 – constant, 2 nuclear – era, 517 – “clock”, 230 – formula, 8 – fission – function, 151 – – of uranium nuclei, 221 – length, 517 – magnetic resonance, 237 – mass, 42, 199, 517 nucleosynthesis – time, 517 – epoch of, 63 plasma, 23 – nonequilibrium, 48 – jet, 172 – primordial, 9, 30, 37, 43, 47, 54, Poincare’s´ lemma, 351 59, 63, 67, 144, 194 Poisson – distribution, 107 observer – random variable, 106 – co-moving, 387, 390, 505 Poisson’s equation, 24, 85, 87, 112, – fundamental, 387 474, 502 – inertial, 208, 227, 254 – modified, 395 optical depth, 79, 133 positronium, 199 orthomode transducer, 150, 485 Pound–Rebka experiment, 250 power spectrum, 104, 109, 112 Palatini’s equation, 327 – Harrison–Zeldovich, 112 parallax Poynting vector, 151 – astronomical, 422 pressure – heliocentric, 421 – in hydrodynamical medium, 499 parallel transport, 302, 307 primordial perturbations, 12, 82, parameter degeneracy, 142 106, 115, 129, 142, 486, 502 parsec, 422 product Pauli’s exclusion principle, 55 – dot, 270, 277, 289, 335, 338 peculiar – inner, 358 – gravitational acceleration, 122 – interior, 363 – velocity, 86, 121, 420, 447, 498, – scalar, 333 508 proper Penning trap, 221 – circumference, 257 Penrose–Terrell effect, 227 – distance, 389 phase transition, 7, 23, 40 – energy density, 497 photoionization, 75 – mass, 217 photon decoupling, 69 – motion of a star, 422 photon–electron coupling, 104 – nonorthochronous transforma- photon–nucleon fluid, 12, 487 tion, 245 photosphere – orthochronous transformation, – cosmic, 11, 195 245 pixel, 148 – time, 258, 265 Planck 534 Index

proper-time interval, 213, 261, 387 – photometric technique, 167 protogalaxy, 78, 195 – probability distribution, 475 pulsar – space, 121 – Crab, 207 redshift survey, see galaxy resonance quantum – escape process, 72 – field theory, 517 – nuclear magnetic, 237 – fields, 517 Ricci – fluctuations, 517 – identity, 308, 374 quasar, 64, 79 – lemma, 297, 371 – Cloverleaf, 444 – scalar, 312, 382 – Markarian 421, 67 – tensor, 311, 496 – spectra, 82 Riemann curvature tensor, 32, 284, 286, 303, 308, 342, 381 radiation – independent components of, 311 – driving, 513 Riemannian – era, 21, 22, 44, 194 – geometry, 238 radio – manifold, 342 – emission, 163 – space, 239, 260 – galaxy, 174 Robertson-Walker, see metric – interferometer, 163 – interferometry, 436 Sachs–Wolfe effect, 127, 507, 513 – maps, 163 – integrated, 131, 145 – telescope, 154 Sagittarius A, 163 – waves, 162 Saha equation, 75 random catalogue, 117 sample variance, 139 Rayleigh’s law, 418 Saturn rings, 183 Rayleigh–Jeans limit, 133, 152, 158 scattering rate, 44 re-ionization epoch, 16, 79, 481 Schur’s theorem, 386 reciprocity theorem, 150 Schwarzschild recombination – metric, 256, 379 – epoch of, 11, 69, 93, 127, 133, 142, – radius, 178, 448 146, 478 Shapiro time delay, 448, 459 – multi-level, 76 sidereal day, 10 – radiative, 75 Silk – rate, 77 – damping, 92, 114, 513 – temperature, 70 – mass, 92 redshift solar – at recombination, 70 – wind, 197 – extragalactic, 9, 28, 30, 390, 399 – year, 423 – gravitational, 128, 249, 261 Solar System, 17, 24, 142, 160, 164, – map, 115 187, 390, 423 Index 535

– barycenter, 141 standard sound – candle, 399, 425, 428, 439 – horizon, 93, 94, 105, 114, 129, – cosmological model, 104, 115, 143, 512 195, 487, 514 – speed of, 87, 90, 144, 182, 499, star cluster 511 – globular, 161 – wave, 91, 103, 129 – open, 162, 424 spacetime – – Hyades, 424 – asymptotically flat, 458 stars – curvature of, 35, 111, 253, 396 – black dwarf, 197 – curved, 274, 284 – brown dwarf, 198 – geodesics of, 307, 320 – Cepheid variable – interval, 388 – – period–luminosity relation, – Minkowski, 208, 239, 270 161, 425 – Newtonian, 32, 35, 286 – RR Lyrae variable, 425 – nonsingular, 339 – variable – perturbed Robertson–Walker, – – classification of, 425 458, 489 – – definition of, 425 – post-Newtonian, 447 – white dwarf, 198, 399 – Riemannian, 239, 260, 275, 295 Stefan–Boltzmann law, 8, 414 – rotations, 325 stelliferous era, 197 – Schwarzschild, 256 stochastic process, 108 – translations, 325 Stokes parameters, 481 special relativity Stokes’ theorem, 356 – accelerations and forces, 230 – generalized, 355 – causality, 212 structure formation, 103 – event Sunyaev–Zeldovich effect – – light-like, 212 – Comptonization parameter, 133 – – space-like, 212 – kinetic, 134 – – time-like, 212 – thermal, 16, 132, 133, 436 – first postulate, 203 supernova – headlight effect, 226 – SN1987A, 426 – kinematics, 208 – Type Ia, 400, 427, 438 – length contraction, 215, 242 – – light curve, 439 – mass-energy relation, 217, 218, – Type II, 434 220 Supernova Cosmology Project, – second postulate, 206 391 – simultaneity, 211 surface spectral – -brightness fluctuations, 428 – index, 111 – brightness, 151, 156, 433 spherical harmonics, 135 – density, 182 536 Index

– Gaussian curvature of, 31, 33 thermal – mass density, 451, 464, 474 – bremsstrahlung, 90, 147 – of last scattering, 11, 95, 129, 142, –death,19 143, 481 – electron gas, 132 – temperature, 414 – flexion of telescope, 423 – history, 193 tangent – plasma, 40, 44, 46, 69 – bundle, 333 – radiation, 6, 8, 22 – space, 333 – spectrum, 10 – vector, 332 thermodynamic telescope – equilibrium, 7, 8, 13 – Cassegrain, 153 –history,6 – Gregorian, 149 – identity, 7, 40 – made by Galileo, 161 thermodynamics – microwave, 150 – second law, 7 – radio, 154 Thomas precession, 233 temperature Thomson – effective, 414 – cross-section, 479 – freeze-out, 51 – scattering, 14, 81, 508 – of neutrinos, 46 tidal – variance, 138 – field, 284 tensor – force, 33, 247 – as multilinear map, 337 time dilation, 213 – contraction, 289 – gravitational, 251, 262 – covariant, 290 transfer function, 113 – definition of, 287 Tully–Fisher, see extragalactic dis- – density, 290 tance – differentiation, 295, 296 twin paradox, 214, 265 – dyadic multiplication, 287, 289, two-photon 338 – absorption (TPA), 222 – field, 295 – emission, 72 – lowering indices, 290 two-sphere, 257, 385, 392 – mixed, 289, 337 – raising indices, 290 units – rank of, 287 – geometrical, 34 – skew-symmetric, 288 universe – – covariant divergence of, 300 – closed, 28, 389 – symmetric, 288 – flat, 28, 389 – torsion, 370 – open, 28, 389 – trace, 289 – transformation rules, 288 vacuum, 518 tetrad (vierbein), 379 variational principle, 268, 324 Index 537 vector window function, 113 – components, 277 WMAP telescope, 13, 29, 81, 148, – field, 294, 334 149 – – curl, 299 – – divergence, 298 X-ray – – gradient, 294, 298 – cosmic background, 177 – magnitude of, 216, 270 – emission from – space, 332, 336, 341 – – active galactic nuclei, 171 – – dual (cotangent), 333 – – blazars, 174 – space-like, 303 – – clusters, 98, 133, 188 – time-like, 303 – – intracluster medium, 30 Virgocentric flow, 124 – – Markarian 421 quasar, 67 virial theorem, 98, 433 – – pulsars, 207 Vishniac effect, 132 – luminosity, 178 VLBA radio telescope, 164, 436 – spectral region, 179 volume element – spectroscopy, 437 – as a tensor density, 291 X-ray satellite – definition of, 346 – Chandra, 99 – four-dimensional, 393 Yang–Mills – invariant, 291, 347 – equations, 358 weak – field, 330 – interactions, 45 Wien approximation, 418 Zeldovich approximation, 125 Wiener-Khinchin theorem, 109 Zodiac, 423 1083

Index

abundance – electron–hole, 984 – 4He, 539 – electron–positron, 595, 960, 966 – hydrogen, 539 – operator, 985 – in Solar System, 558 – particle-antiparticle, 958 – primordial, 539 – probability, 965 accretion antimatter, 959, 985 – disk, 705 – Dirac’s prediction of, 984 – onto black hole, 703 apastron, 765, 774 – – X-ray emission, 721 aphelion, 670 acoustic Arecibo Observatory, 771 – energy, 797 astronomical unit (AU), 577, 855 – losses, 797 atomic active galactic nuclei, 727 – clock, 849 – blazars, 727 – energy levels, 849 – quasars, 727, 936 – – Balmer series, 961 – radio galaxies, 727, 936 – – Balmer’s formula, 961 – Seyfert galaxies, 727 – – conjectured by Boltzmann, – – MCG-6-30-15 galaxy, 730 1024 adiabatic – – hyperfine, 849, 962 – expansion, 1047 – – in Dirac’s theory, 961 – index, 1048, 1078 – – Lamb shift, 963 – process, 1044 – – Lyman series, 961 affine – fine-structure constant, 853, 960, – connection, 801, 1015 961, 963 – geometry, 701 – mass number, 559 – parameter, 667, 673, 683 – number, 559 alpha-decay, 565, 593 Auger effect, 579 Ampere’s` law, 851, 852, 857, 870 – modified, 856 baryon, 955 annihilation – number, 955

Relativity, Astrophysics and Cosmology Radoje Beluseviˇ c´ Copyright c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  ISBN: 978-3-527-40764-4 1084 Index

– number density, 1057 – – in M 106 spiral galaxy, 729 baryon-to-lepton ratio, 637 – – in M 87 elliptical galaxy, 728 basis – – in MCG-6-30-15 Seyfert galaxy, – orthonormal, 699, 820 730 – states, 975 – – in Milky Way, 730 Bianchi identity, 869 – surface gravity, 717, 718 binding energy – thermodynamics of, 714 – 4He, 571 – – first law, 716 – 56Fe, 561 – – second law, 716 – definition of, 559 ––thirdlaw,717 – deuterium, 572, 967 – – zeroth law, 717 – gravitational, 636, 649 blackbody, see thermal radiation – per nucleon, 632 Boltzmann Biot-Savart law, 866 – constant, 854, 1024 Birkhoff’s theorem, 665, 762 – distribution, 1026 black hole – equation, 1062 – accretion, 703 – factor, 589 – collision, 781 Boltzmann, Ludwig, 1026 – Hawking radiation, 718 Bose–Einstein – observational evidence for, 720 – distribution, 1032, 1033 – rotating (Kerr), 706 – statistics, 954 – – “naked” singularity, 709, 716 boson, 1032, 1035 – – Boyer–Lindquist coordinates, brightness 707 – of blackbody radiation, 1029 – – ergosphere, 709, 711 Buchdahl’s theorem, 692 – – frame dragging, 711, 814, 819 – – horizons, 709 Cassini spacecraft, 657, 677 – – morphology of, 711 Chandrasekhar limit, 617, 619, – Schwarzschild, 696 628, 632, 1052 – – Eddington–Finkelstein coordi- chemical nates, 699 – equilibrium, 588, 1046 – – Kruskal coordinates, 701 – potential, 1032, 1045, 1050, 1057 – – radial null geodesics, 701 Cherenkov – – space-like coordinate, 697 – detector, 582, 938 – – surface, 700 – radiation, 583 – – time-like coordinate, 697 Christoffel symbol, 663, 803 – stellar-mass, 720 Clifford algebra, 998 – – Cygnus X-1, 724 CMB, 960 – – V404 Cygni, 724 – photons, 789 – supermassive – temperature, 718 – – in galactic nuclei, 727 color index, 553 Compton Index 1085

– scattering, 602, 897, 898 – of Hilbert action, 832 – – formula, 897 – of Maxwell’s theory, 868, 870 – – inverse, 938 covariant – – Klein–Nishina formula, 900 – derivative, 1011, 1017, 1018 – wavelength, 899, 1053 – formulation, 849 conformal – integral conservation laws, 910 – mapping, 844 Crab Nebula, 721, 784, 937, 950 – tensor, 844 cross-section, 562 congruence of geodesics, 839 curvature conic section, 670 – Gaussian, 781, 841, 842 connection – invariant, 698, 709 – one-form, 686 – of spacetime, 731, 844 continuity equation, 857, 864, 907 – parameter, 836 Cooper pairs, 645 – radius of, 735, 837 coordinate transformations – scalar, 738 – background Lorentz, 736 – gauge freedom, 740 d’Alembertian, 738, 859 – infinitesimal, 736, 825, 906 de Broglie – passive, 826 – particle-wave duality, 956 coordinates – relation, 872, 957 – Cartesian, 698, 708, 766, 828 – wavelength, 567, 961, 1046, 1061 – ellipsoidal, 708 Debye – generalized, 903 – length, 546 – harmonic, 741 – shielding, 545 – orthonormal, 686, 820 – temperature, 624 cosmic degenerate – fluid, 789 – electron gas, 616, 621, 1053 – rays, 787, 985 – energy levels, 962 – time, 837 – fermion system, 645, 1049 cosmic inflation, 788 – gas pressure, 626, 959 Coulomb – neutron gas, 643, 647 – barrier, 565, 566, 570, 592, 631 Dirac – energy, 546 – “bra” and “ket” vectors, 989 – field, 866, 1076 – delta function, 918, 919, 944, 1075 – force, 964 – equation, 961, 981, 982 – gauge, 742, 873 – gamma matrices, 990 – potential, 854, 958, 961 – hole theory, 984 Coulomb’s law, 815, 851, 852, 866, 1071 – Lagrangian, 1009 covariance – matrices, 982 – Lorentz, 997 – sea, 984 – spinor, 982, 985, 1004 1086 Index

– – “small” & “large” components, – – multipole expansion of, 947 987 – – retarded, 945 – – adjoint, 991 electromagnetic field – – charge-conjugated, 1001 – as a collection of photons, 956 – – Lorentz-transformed, 992 – equations of motion, 905 Dirac, Paul, 983 – from scalar and vector poten- Dirac–Pauli representation, 982, tials, 918 997 – integral conservation law, 914 dispersion relation, 743, 902 – interaction with, 979, 988 distribution function, 1032, 1039, – invariants of, 891 1061 – Lagrangian description of, 900 Doppler effect, 725, 729, 771, 777, – minimal coupling prescription, 873 999 Dulong-Petit law, 624 – momentum, 879 – of a point charge, 919 Eotv¨ os¨ experiment, 677 – quantized, 963 Eddington – tensor, 863 – equation, 599 electromagnetic radiation – limit, 727 – by moving charges, 922 – measurement of light deflection, – cyclotron, 929 655 – electric dipole, 949 Einstein – electric quadrupole, 949 – field equations, 731 – magnetic dipole, 949 – – for stellar structure, 689 – quadrupole moment tensor, 948 – – linear, 735, 738 – synchrotron, 929 – manifolds, 837 – wave zone, 945 – tensor, 689, 732, 837 electromagnetic waves, 860 electromagnetic – angular momentum of, 876 – energy density, 882, 894 – discovery by Hertz, 861 – energy flux vector, 894 – plane, 884 – energy–momentum four-vector, – polarization of, 874 894, 914 – polarization vector, 872 – four-potential, 864 – traveling, 886 – interaction, 960, 977 – wave equation, 861, 872 – momentum density, 880, 886, – wavevector, 872, 884 895 electroweak – scalar potential, 860 – mixing angle, 955, 977, 979 – – retarded, 945 – symmetry, 1020 – stress tensor, 878 – theory, 974, 1014, 1020 – stress–energy–momentum ten- ellipse, 671, 765, 775 sor, 889, 908 ensemble – vector potential, 860 Index 1087

– canonical, 1026 flux density entropy – of electromagnetic radiation, 882 – conservation of, 1040 – of gravitational radiation, 755 – definition of, 1024 – of solar radiation, 547 – physical meaning of, 1042 Fourier equivalence principle – integral, 933 – weak, 677, 1073 frame dragging, see Kerr black ergosphere, see Kerr black hole hole ether, 895 Friedmann-Lemaˆıtre Euclidean – model, 837 – space, 708 fusion Euler’s Γ-function, 1064 – barrier penetration, 565 Euler–Lagrange equations, 662, – carbon burning, 591, 631 862, 902 – Gamow – – energy, 565, 576 Fermi – – peak, 568, 589, 591 – “sea”, 645 – helium burning, 587 – constant, 971, 1007 – – helium flash, 590, 626 – derivative, 684 – – triple-alpha process, 588, 627, – energy, 1050 630 – momentum, 635 – hydrogen burning, 570 – surface, 645, 1049 – – CNO cycle, 575, 628 – temperature, 1052 – – proton–proton chain, 571 Fermi–Dirac – interaction rate, 562 – distribution, 1032, 1033 – neon burning, 592 – statistics, 954, 1003 – oxygen burning, 592, 632 Fermi–Walker transport, 684, 819 – penetration probability, 557, 565, fermion, 561, 954, 1032, 1035 567 – doublet, 975 – silicon burning, 592, 632 Feynman – thermonuclear, 559 – diagram, 959, 999, 1019 – – rate, 569 – path integral approach, 1000 – tunneling probability, 566 – rules, 959, 999 Feynman, Richard, 1000 gamma-ray, 1003 flux – emission from – electric, 856 – – excited state of argon, 579 – magnetic, 856, 937 – – supernova SN1987A, 639 – of gravitational radiation, 759, – spectrum of Crab Nebula, 938 797 gauge – of solar neutrinos, 581, 586 – bosons, 954, 1020 – of stellar radiation, 600 – – masses of, 970, 980 – Poynting, 733 – Coulomb, 873 1088 Index

– coupling constant, 1012, 1017– – binding energy, 636 1020 – collapse, 635, 637, 648, 651, 695, – de Donder, 738 702, 706, 724, 937 – field theory, 1008 – constant, 953, 1071 – fields, 1012, 1017 – energy – – self-coupling of, 1019 – – carried by neutrinos, 637 – group, 1020 – – of a polytrope, 614 – harmonic, 741, 805 – energy–momentum pseudoten- – Lorentz, 871 sor, 754 – principle, 1014 – energy–momentum tensor, 752 – theory, 953, 1020 – flux, 1073 – transverse-traceless, 742, 744 – light deflection, 655, 673 gauge invariance, 871, 1020 – potential, 1073 – and charge conservation, 905, – – of the Sun, 1081 1013 – potential energy, 649, 1071, 1076 – of Maxwell’s electrodynamics, – self-energy, 1080 871 gravitational field gauge transformation – multipole expansion of, 1080 – global, 1009 – static, 833 – in general relativity, 737 – stationary, 813, 814 – in Maxwell’s electrodynamics, gravitational lensing, 660 871 gravitational radiation, 657, 731 – local, 1011 – by the Jupiter–Sun system, 731 – of the second kind, 872 – from a binary system, 762 Gauss’s law, 1073 – from PSR 1913+16, 772 Gauss’s theorem, 857, 902, 911 – quadrupole, 762 Gauss, Carl Friedrich, 841 – quadrupole moment tensor, 759 Gaussian gravitational wave equation, 739, – curvature, 839, 841, 842 742 – system of units, 854 – dispersion relation, 743 generalized momentum, 863, 986 – plane-wave solutions, 742 geodesic – polarization tensor, 743 – motion, 703 gravitational waves, 731 – null, 839 – effect on free particles, 747 – space-like, 839 – from collisions, 779 – time-like, 667, 839 – from spinning masses, 782 Gibbs distribution, 1036 – interferometers, 789 gluon, 954, 969 – resonant-mass antennas, 796 gravitational – sources of, 786 – “Cherenkov radiation”, 734 – tidal effect, 750 – acceleration, 1073 gravitomagnetism, 815 Index 1089

– “gravitoelectromagnetic” fields, helioseismology, 578 816 Hertzsprung–Russell diagram, graviton, 734, 740 542, 552 gravity – evolutionary tracks, 625 – linearized theory of, 735 – Hertzsprung gap, 556, 629 – Newtonian, 606, 649, 815, 1071 – horizontal branch, 555 – post-Newtonian, 800 – instability strip, 630 – – asymptotic fields, 812 – turn-off point, 555 – – conservation laws, 809 Higgs – – field equations, 801, 806 – field, 955, 979 – – gauge conditions, 809 – mass, 1020 – – orbiting gyroscope, 819 – mechanism, 1020 – – particles and photons, 810 – – spontaneous symmetry break- Gravity Probe B, 661, 823 ing, 1020 Green’s function, 942, 1075 Hilbert group – action, 832 – Abelian, 1014 – space, 976 – generators, 1018 Hipparcos satellite, 618 – non-Abelian, 1015 horizon – Poincare,´ 1008 – angular velocity, 714 – rotation, 834 – common, 781 gyroscope, 659, 685, 815, 819 – event, 695, 696, 707, 709, 720, 834 – Kerr, 709 hadrons, 969 – outer, 714 Hamiltonian, 863, 989 – surface area, 710, 715, 718 Hawking Hubble – area theorem, 716 – Space Telescope, 728 – radiation, 696, 718 hydrostatic equilibrium, 540, 557, heat 690 – bath, 1025, 1035 – capacity, 623, 1047, 1048 ideal-gas equation, 540 – specific, 541, 1047 ideal-gas law, 604, 1037, 1047 – transfer inertial – – by convection, 597, 603 – frame, 681 – – by radiative diffusion, 597 – mass, 677 – – by thermal conduction, 597 – observer, 910 ––instars,597 – torque, 678 Heisenberg’s uncertainty princi- interferometer ple, 957 – double-slit, 956 helicity, 746, 1004 – gravitational-wave, 789 – neutrino, 1004 – – Fabry–Perot, 790, 795 – operator, 989 – – Michelson, 789, 793, 795 1090 Index

– radio, 655 – momentum operator, 986 inverse β-decay, 635 – term for gauge field, 1018 ionization Klein–Gordon equation, 902, 957, – energy, 545 981 – fraction, 1057 – near stellar surface, 626 Lagrangian isometry, see manifold – density, 901, 979 isospin, 975 – – of QED, 1012 – introduced by Heisenberg, 1015 – Dirac, 1009 – invariance, 977 – field theory, 900 – matrices, 1018 – Yang-Mills, 1019 – nuclear, 975 Laplace’s equation, 942, 958, 1073 – space, 975, 1015 Laplacian, 859 – – rotation in, 1017 Larmor formula, 924, 949 – weak, 975, 978 – relativistic generalization of, 927 isotope Leibniz rule, 830 – argon, 579 Lense–Thirring effect, 660, 688, – carbon, 971 713, 815, 817, 819, 822, 823, 825 – definition of, 560 lepton, 953 – oxygen, 583 – doublet, 975 – unstable to α-decay, 593 – families, 953 – number, 955 Jacobian, 564 lepton-number conservation, 974 jansky, 855 Levi–Civita – (pseudo)tensor, 891 Kepler’s – symbol, 859, 878 – first law, 670 Lienard–Wiechert´ potentials, 917 – second law, 774 Lie – third law, 618, 687, 725, 763, 776 – bracket, 831 Killing – derivative, 830 – equation, 752, 827, 838 light cone, 696 – vector, 828 – forward, 701, 703 – – for Kerr metric, 834 – in Minkowski spacetime, 696 – – for Schwarzschild metric, 833 – past, 757 – – space-like, 835 LIGO gravitational-wave detector, – – time-like, 829 793, 795 – vector field, 835 Liouville’s theorem, 1061 kinetic LISA gravitational-wave detector, – energy 787 – – of bulk motion, 1079 local conservation law, 870 – – of fermion gas, 1050 Lorentz – – of rotating object, 950 – boost, 821, 994 Index 1091

– covariance, 997 – homogeneous, 869 – force, 862, 888 – in integral form, 856 – gauge, 871 – inhomogeneous, 869 luminosity Maxwell’s velocity distribution, – class, 550 557, 563, 1039 – Eddington limit, 727 Maxwell, James Clerk, 857 – stellar, 541, 546 Maxwell–Boltzmann distribution, lunar laser ranging, 688 588, 1033 mesons, 955 magnetic metric – moment, 987 – axially symmetric, 707, 713, 715 ––Lande´ g-factor, 985 – isometries of, 825 – permeability, 852 – Kerr, 707, 834 magnitude – Schwarzschild, 661, 667 – absolute, 553 – static, 833 main sequence, 541 – stationary, 707, 713, 715, 833, 834 – mass–luminosity relation, 549 Minkowski – mass-radius relation, 545 – four-vector, 858 – properties of stars, 543 – spacetime, 828, 844 – zero-age (ZAMS), 543 – volume element, 901 main-sequence stars, 541 manifold neutrino – isometry of, 825 – bremsstrahlung, 595, 632 maser – emission by hot stars, 595 – water-vapor, 729 – in supernova explosion, 636 mass – left-handed, 980 – deficit, 559 – oscillations, 586 – density, 1034 neutrinos, see solar – dipole moment, 1081 neutron – fraction – β-decay, 570, 958 – – of nucleus, 1060 – composition of, 954 – function, 691 – discovery by Chadwick, 643 – of gauge bosons, 980 – drip, 645 – of Higgs particle, 1020 neutron star, 636, 643 – quadrupole moment tensor, 1081 – as a pulsar, 643, 936 maximally symmetric space, 836 – Landau model, 643 Maxwell’s electrodynamics, 856 – Oppenheimer-Volkoff theory, – conservation laws, 906 643 – – integral form of, 910 – superconducting core, 645 – gauge invariance of, 871 – superfluid interior, 643 Maxwell’s equations, 856, 861 neutronization effect, 621, 635, 650 – displacement current, 870 Newtonian 1092 Index

– field equation, 661 – – inverse bremsstrahlung, 602 – gravitational force density, 810 – – photoionization, 602 – gravity, 606, 649, 815, 1071 – Rosseland mean, 600 – limit, 688, 750, 762, 774 – stellar, 600 – potential, 800, 808 Oppenheimer-Volkoff Noether – equation, 648, 692 – current, 906, 907, 1010 – limit, 706 – theorem, 906, 907, 909, 1010 nuclear parity – β-decay, 1003 – “intrinsic”, 997 – binding energy, 560 – violation, 1004 – cross-section, 565 Pauli – fission – equation, 986, 987 – – energy release, 561 – matrices, 976, 982, 1016 – force, 967 – spinor, 988 – isospin, 975, 1015 Pauli’s exclusion principle, 561, – photodisintegration, 592 626, 959, 969, 985, 1035, 1037, – resonance, 967 1049, 1051, 1052, 1054 – size, 565, 855, 967 periastron, 765, 771, 774, 777 – spin-orbit interaction, 963 – shift, 658, 672, 772, 776, 778 – statistical equilibrium, 637, 1059 perihelion, 670, 765 nuclear fusion, see fusion photodisintegration of iron, 633, nucleosynthesis 636 – n-capture reactions photoneutrino process, 595, 632 – – r-process, 595 photosphere – – s-process, 593 – opacity of, 552 – and stellar evolution, 558 – solar, 551 – Bethe’s theory, 557 – stellar, 550 – explosive, 595 photosynthesis, 587 – p-process, 593 Planck – stellar, 539, 559, 594, 595 – constant, 897, 1029 – valley of stability, 593 – distribution, 1028 – formula, 597 observer – law, 1027 – inertial, 910, 914 – mass, 720 – static, 712 – spacecraft, 789 – stationary, 712 plasma opacity, 548, 599 – jet, 727 – Kramers’ law, 548, 603 Poincare’s´ lemma – radiative, 601 – for irrotational fields, 1072 – – bound-bound absorption, 603 Poisson’s equation, 612, 808, 942, – – electron–photon scattering, 602 1074 Index 1093 positronium – volume, 888 – disintegration rate, 965 protostar, 540 – eigenstate of C-operator, 1002 pulsar post-Newtonian approximation, – binary, 644, 658 see gravity – Crab, 643 Poynting vector, 879, 886, 922, 923, – lighthouse model of, 937 948 – PSR 1913+16, 658, 762, 769 – “gravitoelectromagnetic”, 815 – – discovery of, 770 Poynting’s theorem, 881, 882, 893, – – gravitational radiation from, 914 772 precession – radio, 643 – geodetic, 659, 685, 688, 779, 822, 823 quantum – of Earth’s equinoxes, 657 – concentration, 1046 – of Mercury’s perihelion, 656, – electrodynamics, 955, 963, 985, 672, 771 1006 – of orbital perihelia, 667, 1081 – expectation value, 975 – of orbiting gyroscope, 659, 822 – field theory, 985, 1008 – of spin in gravitational field, 660, – – CPT theorem, 1008 681 – – spin-statistics theorem, 1008 pressure – fields, 981 – definition of, 1040 – observable, 975 – in terms of occupancy function, – probability 1038 – – amplitude, 975 – of electromagnetic radiation, ––density,957 880, 887 – state, 975 – of fermion gas, 1051 – theory, 956 Proca equation, 1014 – transition product – – probability, 964 – cross, 860 – – rate, 965 – dot, 1004 – zero-point energy, 957 – scalar, 835, 860, 975 quark, 969 proper – “color charge”, 969 – circumference, 662 – asymptotic freedom, 970, 1015 – density, 888 – confinement, 969, 1015 – mass, 649 – doublet, 975 – number density, 888 – families, 953 – radius, 693 – fragmentation, 969 – time, 810, 926 quasar, 727, 936 – – of Schwarzschild black hole, – redshift, 936 698 radar echo delay, 675 1094 Index

radiation Schrodinger¨ equation, 565, 957, – cyclotron, 929 981 – electric dipole, 950 Schur’s theorem, 837 – electric quadrupole, 950 Schwarzschild – gravitational, 657, 731 – black hole, 696, 698, 703, 707, – magnetic dipole, 950 713, 718, 719, 780 – pattern, 925 – – accretion by, 706 – pressure, 1079 – metric, 661, 667 – synchrotron, 929 – radius, 695 – – astronomical sources of, 934 – singularity, 667, 699, 709 – – critical frequency, 934 Shapiro time delay, 657 – – discovery of, 929 sidereal day, 935 – zone, 757 solar radio – neutrino unit (SNU), 580 – galaxy, 936 – neutrinos, 576 – interferometry, 655, 688 – – flux on Earth, 577 Rayleigh–Jeans limit, 1031 – – GALLEX experiment, 581 redshift – – Homestake experiment, 578 – gravitational, 618, 655, 677, 730, – – oscillation, 586 771, 777 – – SNO experiment, 582 resonance – – Super-Kamiokande experi- – Δ++, 967 ment, 582 – Breit–Wigner formula, 568, 968 – properties, 542 – definition of, 967 – wind, 787, 881 – nuclear, 568 solar model, see standard Ricci Solar System, 655, 668, 855 – curvature, 844 – barycenter, 770 – scalar, 837 – nuclear abundances, 558, 576, – tensor, 664, 837, 842 590 Riemann curvature tensor, 746, sound 837, 842, 844 – speed of, 797 Riemann’s ζ-function, 1065, 1067 – wave, 797 Riemann, Bernhard, 841 spacetime Roll–Krotkov–Dicke experiment, – around a rotating black hole, 707 678 – continuum, 903 – curvature of, 732, 844 Sagittarius A, 935 – curved, 659, 675 Saha equation, 545, 551, 1057 – diagram, 701 scattering rate, 586 – geodesics of, 667 Schonberg–Chandrasekhar¨ limit, – geometry of, 842 558, 629 – isometric mapping of, 832 Index 1095

– locally Lorentzian, 648 – canonical ensemble, 1026 – mapping onto itself, 826 – equipartition of energy, 1038 – Minkowski, 800, 828, 844 – grand partition function, 1036 – Schwarzschild, 667, 696, 701, 833 – macrostate, 1023 – singularity, 695 – mean occupation number, 1032 – static, 833 – microstate, 1023 – symmetries of, 715, 835 – partition function, 1026 – translations, 906 – phase space, 1032 – with a Killing vector, 829 – quantum concentration, 1046 spectral – selected integrals, 1063 – type, 550, 553 – two basic postulates, 1024 standard Stefan–Boltzmann law, 1031 – model, 955, 979, 1014, 1020 stellar – solar model, 543, 576, 581, 586, – evolution 610 ––1M stars, 625 star cluster – – 25M stars, 631 – globular, 554, 629 ––5M stars, 628 – – giant branch, 555 – – beyond main sequence, 624 – – horizontal branch, 555, 626 – – core collapse, 633 – – Population II stars, 555 – – planetary nebula, 628, 631 – open, 629 – – pre-supernova state, 632 – – Hertzsprung gap, 556 – – red-giant branch, 629 – – Hyades, 554 – – subgiant phase, 625 – – Population I stars, 555 – opacity, 600 stars – photosphere, 550 – black dwarf, 622, 628 – spectra, 549 – blue giant, 552 – structure, 606 – Cepheid variable, 630 – – Clayton model, 606 – polytrope, 610 – – fundamental equations of, 606 – – Lane-Emden equation, 613 – – polytropic model, 610 – – white dwarf, 611 – winds, 627 – red dwarf, 552 STEP experiment, 680 – red giant, 553, 626 Stokes’ theorem, 857, 911, 1072 – red supergiant, 631 strong interactions, 967 – RR Lyrae variable, 626 superconductor, 645 – white dwarf, 552, 616, 628 – Josephson junction, 824 – – cooling of, 621 – London moment, 824 – – mass-radius relation, 620 – SQUID device, 680, 824 – – Sirius B, 617 – theory of, 1020 – – surface temperature, 623 – transducer, 796 statistical physics superfluid 1096 Index

– “crust”, 645 – spectrum, 717, 720 – helium, 823 thermodynamic – neutron–proton “sea”, 646 – equilibrium, 598, 1024, 1027, supernova 1040 – explosion, 633 – identity, 1040 – remnant, 636 thermodynamics – shock wave, 635 – first law, 1044 – SN1987A, 638 – of black holes, 696, 714 – – light curve, 640 – reversible processes, 1044 – – neutrino emission, 641 – second law, 547, 720, 1045 – Type II, 636 – zeroth law, 717 surface Thomas precession, 682, 683, 822 – chemical composition, 549 Thomson – Gaussian curvature of, 841 – cross-section, 547, 602, 899, 931 – of infinite redshift, 700 – limit, 939 – of white dwarf, 621 – scattering, 602, 899 – pressure, 607 tidal – temperature, 541, 553 –field – – of neutron star, 644 – – of gravitational wave, 750 – – of white dwarf, 623 Trouton–Noble experiment, 895 two-sphere, 662 temperature – absolute zero, 643, 1038 units – definition of, 1040 – CGSE (Gaussian) system, 853 – effective, 541, 547, 637 – Heaviside-Lorentz system, 856, tensor 861 – canonical, 908 – miscellaneous, 854 – symmetric – MKSA (SI) system, 851, 862 – – covariant divergence of, 753 – natural system, 850, 953 – Weyl, 842 Theorema Egregium, 841 vacuum, 984 thermal – polarization, 963 – conductivity, 598, 601 van der Waals force, 560, 967 – energy, 615, 623, 1078 variational principle, 662, 667, 901 – equilibrium, 1038 vector – evolution, 617 – calculus, 858 – motion, 600 – cross product, 860 – noise, 796, 798 – dual, 975 – photons, 598, 633 – field, 684 – pressure, 546 – – curl, 859 – radiation, 597, 720, 939, 1027, – – divergence, 859 1028 – – gradient, 859 Index 1097

– – nabla operator, 858 Wien’s displacement law, 546, – scalar product, 860 1031 – space-like, 683 VIRGO gravitational-wave detec- X-ray tor, 795 – astronomy, 720 virial theorem, 540, 615, 961, 1078, – emission from 1080 – – active galactic nuclei, 727 VLBA radio telescope, 729 – – binary source, 706 volume element – – black-hole accretion disk, 721, – coordinate, 649 725 – Minkowski, 901 – – Crab nebula, 644 – proper, 649 – – Cygnus X-1, 725 ––Kα fluorescent transition, 730 weak – – pulsars, 644, 721 – interactions, 637, 970 – – quasars, 936 – – charge-changing, 975 – – supernova SN1987A, 639 – – charge-preserving, 976 – – surface of neutron star, 646 – – V-A theory, 1007 – in Compton’s experiment, 898 – isospin, 975, 978 – intergalactic, 938 – – current, 1020 X-ray satellite – – doublet, 1015 – Chandra, 722 – – rotations, 976 – ROSAT, 721 – – space, 976, 1015 – Uhuru, 721 – isotriplet, 976 Weyl Yang-Mills – conformal tensor, 842 – Lagrangian, 1019 – curvature, 844 – theories, 1020 Wien approximation, 1031 Yukawa theory, 958, 967