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Home , Coordinate conditions, Curvature invariant, Differentiable curve, Kasner metric, Tensor density

Index

Symbols B 1-form, 10 Betti number, 589 4-, 184 Bianchi type-I models, 448 4-force, 115 Bianchi’s differential identities, 60 4-momentum, 116 complex valued, 532–533 total, 122 consequences of in Newman-Penrose 4-, 114 formalism, 549–551 first contracted, 63 second contracted, 63 A bicharacteristic , 620 acceleration , 441 4-acceleration, 184 big-bang cosmological model, 438, 449 Newtonian, 74 Birkhoff’s theorem, 271 function or functional space, 486 (see also Lagrangian), 594, 598 , 364–433 ADM action, 606 Bondi-Metzner-Sachs group, 240 affine parameter, 77, 79 boost, 110 alternating operation, antisymmetriza- Born-Infeld (or tachyonic) scalar field, tion, 27 467–471 angle field, 44 Boyer-Lindquist coordinate chart, 334, anisotropic fluid, 218–220, 276 399 collapse, 424–431 Brinkman-Robinson-Trautman met- ric, 514 anti-de Sitter space-time, 195, 644, Buchdahl inequality, 262 663–664 bugle, 69 anti-self-dual, 671 antisymmetric oriented , 49 C , 28 canonical energy-momentum-stress ten- antisymmetrization, 27 sor, 119 parameter, 81 canonical or forms, 510 arc separation function, 85 Cartesian chart, 44, 68 arc separation parameter, 77 Casimir effect, 638 Arnowitt-Deser-Misner action , Cauchy horizon, 403, 416 606 Cauchy problem, 207 atlas, 3 Cauchy-Kowalewski theorem, 207 complete, 3 causal cone, 108 maximal, 3 causal space-time, 663

698 Index 699 causality violation, 663 contravariant index, 51 characteristic hypersurface, 623 contravariant order, 18 characteristic , 625 coordinate chart or system characteristic ordinary differential equa- Boyer-Lindquist, 334, 399 tions, 612 Cartesian, 44, 68 characteristic polynomial equation, 177 comoving, 198 characteristic surface, 619 Doran, 415 charge-current, 125 doubly-null, 130 charged dust, 125, 134, 226, 317 Eddington-Finkelstein, 371, 383 chart, 1 Gaussian normal, 67, 202 Christoffel symbol, 57 normal, 67, 102, 202 first kind, 57 Kerr-Schild, 414 second kind, 57 Kruskal-Szekeres, 379–380 closed form, 33 local, 1 Codazzi-Mainardi equations, 100 local Minkowskian, 153 commutator or Lie bracket, 40 Minkowskian, 105 comoving coordinate chart, 198 normal or hypersurface orthog- complete atlas, 3 onal, 67 complex conjugate coordinates, 621 orthogonal, 61 complex electromagnetic potential, 356 Painlev´e-Gullstrand,370 complex gravo-electromagnetic poten- pseudo-Cartesian, 45, 68 tial, 356 Regge-Wheeler tortoise, 371 complex null tetrad field, 485 Riemann normal, 67 complex potential, 331–332, 348 Synge’s doubly-null, 372 components Weyl’s, 289 coordinate, 21 Weyl-Lewis- Papapetrou (WLP), covariant, 11 330 orthonormal, 47 coordinate components, 21 physical, 120 coordinate conditions, 165 conformal group, 640 cosmological constant, 163, 233, 435 conformal mapping, 70 cosmological principle, 434 conformal tensor, 71 cotangent vector field, 9 conformally flat domain, 71 cotangent , 9 conformally flat space-time, 149, 639– Cotton-Schouten-York tensor, 309 646 covariant, 52 conformastat metric, 309 covariant components, 11 conformastationary metric, 358 covariant , 51 conjugate holomorphic function, 622 of relative and oriented relative conjugate points, 81 tensor fields, 63 conjugate, contravariant, or inverse covariant index, 51 , 44 covariant order, 18 constant , space, 68, 73 covariant vector field, 9 continuity equation, 122, 126, 186, critical energy density, 448 232 critical point, 593 , 117 curvature invariant, 63 contraction operation, 21 curvature scalar, 63 700 Index curvature tensor, 55 doubly-null coordinates, 130 dual vector space, 9 integral, 35 dubiosity, 82 non-null, 81 dust, 122, 125, 134, 185, 186, 216, null, 112 226, 317, 435, 437, 438, 441, parameterized, 11 448, 452, 470 profile, 92 collapse, 383–386 spacelike, 111 timelike, 112 E curved domain, 64 eccentricity, 245 Curzon-Chazy metric, 299 Eddington-Finkelstein coordinates, 371, cusp, 16 383 Eguchi-Hansen gravitational instan- D tons, 671 dark energy, 461 Einstein space, 69 de Sitter space-time, 431, 450, 644 Einstein static , 434 deceleration function, 448 Einstein summation convention, 7 deceleration parameter, 448 , 63 deDonder gauge, 649 Einstein’s field equations, 164 deformable solid, 220–222, 276 complex-valued form, 532 degree (system of p.d.e.s), 611 Einstein-Hilbert Lagrangian density, delta 168 Kronecker, 8 Einstein-Maxwell equations, 224, 263 derivative Einstein-Maxwell-Klein-Gordon equa- covariant, 51, 63 tions, 560 covariant directional, 52 orthonormal form, 565 directional, 7, 52 Einstein-Rosen bridge, 655 exterior, 31 electrical charge density, 125 Fermi, 147 electro-vac universe, 263 gauge covariant, 561 electromagnetic duality-rotation, 233 Lie, 35–38 electromagnetic field tensor, 33, 50 variational, 595, 599 electromagnetic four-potential, 34, 225, derivative mapping, 22 233, 504, 561 determinate system, 166 electromagneto-vac domain, 224 differentiable , 3 electrostatic potential, 319 differential conservation of energy- mo- elementary divisors, 632 mentum, 120 elliptic p.d.e., 618 differential form, 31 embedded manifold, 94 , 7 Emden equation, 575 domain energy conditions, 180 curved, 64 energy-momentum-stress tensor, 119 flat, 64 equation of state, 216 dominant energy condition, 181 equilibrium of a deformable body, 221 Doran coordinate chart, 415 , 139 or inner-product, 42 ergosphere, 402 double Hodge-dual, 84 , 333 Index 701 eternal black holes, 418 closed, 33 eternal white holes, 418 differential, 31 Euler equations for a perfect fluid, exact, 33 232 Frenet-Serret formulas, 83 Euler-Lagrange equations, 78, 159, generalized, 82 596 frequency of a wave, 504 classical mechanics, 596 Friedmann-Lemaˆıtre-Robertson-Walker for particle in curved space-time, metrics (or F-L-R-W), 436– 157 471 for particle in Schwarzschild space- time, 242, 243 G for scalar field, 599 G¨odeluniverse, 664–665 horizon, 367 gauge covariant , 561 exact form, 33 gauge transformation, 34 expansion scalar, 184 Gauss’ equations, 100 expansion tensor, 184 Gauss’ theorem, generalized, 65 extended extrinsic curvature, 604 Gauss-Bonnet term, 608 extended real numbers, 409 , 91 , 31 Gaussian normal coordinate chart, 67, exterior product, 28 202 extrinsic curvature tensor, 89–104, 167 , 131 extended, 604 , 168 F generalized D’Alembertian, 222 F-L-R-W metrics (or Friedmann-Lemaˆıtre- generalized Frenet-Serret formulas, 82 Robertson-Walker), 436–471 generalized Kronecker tensor, 29 F-W transport, 147 generalized wave operator, 222 Fermi derivative, 147 geodesic, 76 Fermi-Walker transport, 147 non-null, 77 fifth force, 643 null, 77 fine-structure constant (electrodynamic), geodesic deviation equations, 80 583 geodesic equations, 77 fine-structure parameter (eigenvalue for dust, 186 of charged scalar-wave grav- for particle in T -domain, 369 itational condensate), 583 for particle in Schwarzschild space- Finsler metric, 157 time, 242 first contracted Bianchi’s identities, for the Euclidean E2 , 78 63 geodesic normal coordinate chart, 67, first curvature, 147 102, 202 first fundamental form, 91 geodesically complete manifold, 170 Flamm’s paraboloid, 241 geometrized units, 164 flat domain, 64 geons, 591 flat manifold, 67 Global Positioning System (G.P.S.), Fock-deDonder gauge, 649 336 form gravitational constant, G, 66, 162 1-form, 10 gravitational field equations, 163 702 Index gravitational instantons, 351, 667– holomorphic function, 622 671 holonomic constraint, 158 gravitational mass, 138 homeomorphism, 1 gravitational , 250 homogeneity, 442 astronomy, 647 homogeneous p.d.e., 611 gravitational waves, 167, 647–653 homogeneous state of the complex gravitons, 649 wave function, 570 gravo-electromagnetic potential, 356 Hopf’s theorem, 307 group Hubble function, 448 Bondi-Metzner-Sachs, 240 Hubble parameter, 448 conformal, 640 hybrid tensor field, 96–98 Lie, 45 hyperbolic p.d.e., 618 Lorentz, 45, 109 hypersurface, 367 M¨obius,356 null, 131 Poincar´e,109 hypersurface orthogonal coordinate, symplectic, 173 67 group of motion, 73 I H I-S-L-D jump conditions, 167 H-K-S-D metric (or Hawking-Kloster- identity tensor, 46 Som-Das), 670 immersion, 94 Hamilton-Jacobi equation, 615 incoherent dust, 185 Hamiltonian index for particle in a static gravita- contravariant, 51 tional potential, 326 covariant, 51 for particle in an electromagnetic index lowering, 48 field, 160 index raising, 48 relativistic, 158 inertial mass, 138 relativistic equations of motion, inertial observer, 113 160 inflationary era, 446 , 158 initial value problem, 207 super, 158 inner-product or dot product, 42 harmonic coordinate conditions, 174 instanton, 351, 667–671 harmonic function, 307, 622 instanton-horizon, 668 harmonic gauge, 174, 649 integrability conditions, 99 Hausdorff manifold, 1 integral conservation, 120 Hawking-Kloster-Som-Das metric (or integral curve, 35 H-K-S-D), 670 intrinsic curvature heat flow vector, 232 extended, 604 Heaviside-Lorentz units, 123 intrinsic metric, 90 Hessian, 616 invariant eigenvalue problem, 91 Higgs fields, 471 invariant eigenvalues, 177 Hilbert-Palatini approach for deriv- irrotational motions, 184 ing field equations, 601 isomorphism, 501 Hodge star operation, 50 on tensor space, 48 Hodge-dual operation, 496 isotropic , 441 Index 703 isotropy, 441 for particle in electromagnetic field, isotropy equation of static spherical 160 symmetry, 260 for particle in Schwarzschild space- time, 242, 243 J for scalar field, 599 Jacobian, 3 for static field equations, 303 Jacobian mapping, 22 for stationary field equations, 353, Jordan decomposition theorem, 630 354 junction conditions for tachyonic scalar field, 468 I-S-L-D, 167 relation to super-Hamiltonian, 159 Synge’s, 166 square-root particle, 468 K Lagrangian density, 168 Kaluza-Klein theory, 472, 479 augmented gravitational, 603 , 212, 306, 310, 323 Einstein-Hilbert, 168, 601 , 334–339, 399–417 Einstein-Maxwell, 360 higher dimensional generalization, most general curvature yielding 343 second-order equations, 608 W Kerr-Newman solution, 337, 415 Lambert -Function, 374, 375 Kerr-Schild coordinate chart, 414 Lane-Emden equation, 575 Killing tensor, 414 Laplacian operator, 84 Killing vector, 72 Legendre transformation, 158 Klein-Gordon field Lemaˆıtremetric, 280 (see also scalar field, massive), lemma of Poincar´e,32 560–591 length or norm, 43 Kottler solution, 252 Levi-Civita permutation symbol, 30 Kretschmann invariant, 86, 400 Levi-Civita tensor, 49 , 8 Lichnerowicz-Synge lemma, 208 Kronecker tensor, generalized, 29 Lie bracket or commutator, 40 Kruskal-Szekeres coordinates, 379–380 , 35–38 Kruskal-Szekeres metrics, 379–380 Lie group, 45 , 108 L linear equation, 611 , 159 linear p.d.e., 611 Lagrange’s solution method, 612 linearized theory of , 647– Lagrangian, 594 653 alternative Lagrangians, 161 Liouville equation, 621 augmented gravitational, 603 Lipschitz condition, 35 classical mechanics, 596 local , 1 Einstein-Hilbert, 601 local Minkowskian coordinates, 153 Einstein-Maxwell, 225 London equation, 571 Einstein-Maxwell-Klein-Gordon, Lorentz gauge condition, 174, 222, 564 649 first-order gravitational, 605 Lorentz group, 45, 109 for particle in T domain, 369 Lorentz matrix, 133 for particle in curved space-time, Lorentz metric, 45, 64 156, 161 Lorentz signature, 43 704 Index

Lorentz transformation, 45, 109 components, 44 orthochronus, 192 conformastat, 309 Lorentz-Heaviside units, 123 conformastationary, 358 conjugate, contravariant, or in- M verse, 44 magnetic potential, 315 Curzon-Chazy, 299 manifold de Sitter, 431, 450, 644 constant curvature, 68, 73 Einstein static universe, 434 differentiable, 3 Finsler, 157 Einstein, 69 Friedmann-Lemaˆıtre-Robertson- embedded, 94 Walker (or F-L-R-W), 436– flat, 67 471 geodesically complete, 170 G¨odel,664 Hausdorff, 1 Hawking-Kloster-Som-Das (or H- orientable, 3 K-S-D), 670 paracompact, 1 Kasner, 212, 306, 310, 323 pseudo-Riemannian, 64 Kerr, 334–339, 399–417 Ricci flat, 70, 164 higher dimensional generaliza- Riemannian, 64 tion, 343 Riemannian or pseudo-Riemannian, Kerr-Newman, 337, 415 56, 64 Kottler, 252 topological, 3 Kruskal-Szekeres, 379–380 Maple(TM), 672 Lemaˆıtre,280 mass Lorentz, 45, 64 gravitational, 138 positive definite, 43, 64 inertial, 138 pseudo-Schwarzschild, 311 mass density, 118 Reissner-Nordstr¨om-Jeffery, 265, mass-shell, 117 394 Mathematica(TM), 672 Robinson-Trautman, 549 matrix Schwarzschild, 239–251, 311, 350, characteristic, 625 364–382 Lorentz, 133 higher dimensional generaliza- nilpotent, 629 tion, 253 rank of, 18 Schwarzschild’s interior, 263 unit, 8 signature, 43 -dominated era, 445 Taub-NUT, 671 maximal atlas, 3 tensor, 42 maximum-minimum principle, 307 Tolman-Bondi-Lemaˆıtre,276 Maxwell’s equations, 34, 123 Vaidya, 284 Maxwell-Lorentz equations, 124 Van Stockum, 665 , 91 warp-drive, 660 Meissner effect, 571 Weyl-Majumdar-Papapetrou-Das metric (or W-M-P-D), 319 anti-de Sitter, 195, 644, 664 , 656 Bianchi type-I, 448 metric equations, 321 Brinkman-Robinson-Trautman, 514 metric tensor, 42 Index 705

Minkowskian chart, 105 oriented relative tensor field, 26 Minkowskian chronometry, 113 oriented tensor field, 50 Monge surface, 102 oriented volume, 65 monogenic forces, 156 orthochronus , most general solution, 609 192 multilinearity conditions, 18 orthogonal coordinate chart, 61 M¨obiusgroup, 356 orthogonal vector fields, 42 orthonormal , 42 N orthonormal components, 47 Newman-Penrose equations, 538, 545– outer ergosurface, 402 547 outer product, 19 Newtonian acceleration, 74 overdetermined system, 166 Newtonian approximation, 245 Newtonian gravitational constant, G, P 66, 162 Painlev´e-Gullstrandcoordinate chart, Newtonian gravitational potential, 66, 370 138, 142, 162, 169, 203, 280, Palatini approach for deriving field 291–294, 299, 304, 319 equations, 601 Newtonian mass, 66 Papapetrou-Ehlers class, 350 nilpotent matrix, 629 parabola, semi-cubical, 16 nilpotent operator, 33 parabolic p.d.e., 618 no hair theorem, 412 paracompact manifold, 1 non-degeneracy conditions, 11 parallelly propagated, 74 non-linear eigenvalue problem for a parameterized curve, 11 theoretical fine-structure con- parametric surface, 89 stant, 582 parametrized hypersurface, 94 non-null differentiable curve, 81 Penrose-Carter compactification, 409 non-null geodesic, 77 perfect fluid, 180, 216–218 norm or length, 43 perihelion, 245 normal coordinate, 67 perihelion shift, 246 normal forms, 619 permutation symbol, 30 normal or canonical forms, 510 Petrov-types-I, D, II, N and III, 511 null cone, 108 physical components, 120 null curve, 112 Planck length, 473 null electromagnetic field, 231, 503 Planck mass, 591 null energy condition, 180 Planck’s constant, 561 null hypersurface, 131 Poincar´egroup, 109 null vectors, 106 Poincar´e’slemma, 32 numerical tensor, 29, 111 Poisson’s equation, 162 NUT potential, 669 polarization states of gravitational waves, 652 O polarization tensor, 651 Oppenheimer-Volkoff equation, 261 polytropic equation of state, 431 order (system of p.d.e.s), 611 positive definite metric, 43, 64 order of a tensor, 18 potential orientable manifold, 3 complex, 331–332, 348 706 Index

complex electromagnetic, 356 tensor, 18 complex gravo-electromagnetic po- Raychaudhuri-Landau equation, 185 tential, 356 redshift, 250 electromagnetic four, 34, 225, 233, Regge-Wheeler tortoise coordinate, 371 504, 561 region, 409 electrostatic, 319 regularity conditions, 11 equation, 321, 323 Reissner-Nordstr¨om-Jefferymetric, 265, function, 307 394 gravo-electromagnetic potential, relative tensor field, 26, 63 356 relativistic canonical equations of mo- magnetic, 315 tion, 160 Newtonian gravitational, 66, 138, relativistic Hamiltonian equations of 142, 162, 169, 203, 280, 291– motion, 160 294, 299, 304, 319 relativistic Hamiltonian mechanics, 158 NUT, 669 relativistic Lorentz equations of mo- scalar field, 462, 464 tion, 126 twist, 347, 353, 356 relativistic total , potential equation, 321, 323 122 potential function, 307 reparameterization of a curve, 13 pressure, 180 reparametrization, 95 , 147 Ricci flat space, 70, 164 principal pressures, 182 Ricci identities, 61 principal tensions, 182 Ricci rotation coefficients, 58 principle of equivalence, 139 complex valued, 531 Proca equation, 571 Ricci tensor, 62 profile curve, 92 , 55 projection tensor, 183 Riemann normal coordinate chart, 67 , 112 Riemann-Christoffel curvature tensor, pseudo-angle, 44 59, 62 pseudo-Cartesian chart, 45, 68 algebraic identities, 59 pseudo-Riemannian manifold, 64 differential identities, 60 pseudo-, 311 Riemannian manifold, 64 Riemannian or pseudo-Riemannian man- Q ifold, 56, 64 quantum field theory in curved space- rigid motion, 40 times, 561 rigid motions, 184 quantum , 607, 654, 671 Robinson-Trautman metric, 549 quasi-linear p.d.e., 611, 612 Routh’s theorem, 243 quintessence scalar field, 461–466 ruled surface, 85, 97

R S radiation era, 446 scalar field, 20, 456–471 Rainich problem, 566 Born-Infeld or tachyonic, 467– raising and lowering of indices, 48 471 rank massive, 458–461 matrix, 18 massless, 456–458, 472–479, 599 Index 707

quintessence, 461–466 Sylvester’s law, 623 scalar field potential, 462, 464 , 27 Schwarzschild metric, 239–251, 311, symmetrization operation, 27 350, 364–382 symmetrized curvature tensor, 155, higher dimensional generalization, 172 253 symplectic group, 173 Schwarzschild radius, 240, 364 Synge’s doubly-null coordinates, 372 Schwarzschild’s interior solution, 263 Synge’s junction conditions, 166 second contracted Bianchi’s identi- Synge’s world-function, 86 ties, 63 , 91 T Segre characteristic, 632–634 T-domain, 280 of energy-momentum-stress ten- tachyon condensation, 468 sors, 177–180 tachyonic (or Born-Infeld) scalar field, of Ricci tensor, 516 467–471 self-dual, 670 vector, 6, 7 self-duality, 497 space, 7 semi-cubical parabola, 16 Taub-NUT metric, 671 semi-linear p.d.e., 611 tensor semiclassical gravity, 561 antisymmetric, 28 separation of a vector field, 42 canonical energy-momentum-stress, shear tensor, 184 119 shock waves, 610 conformal, 71 signature of the metric, 43 conjugate, contravariant, or in- singularity theorems, 432–433, 449 verse metric, 44 solid particle, 228 Cotton-Schouten-York, 309 space of constant curvature, 68, 73 curvature space-time foam, 654, 655 Riemann, 55 spacelike curve, 111 extended extrinsic, 604 spacelike vectors, 106 extrinsic, 89–104, 167 spin coefficients, 538 Ricci, 62 algebra, 547 Riemann-Christoffel, 59, 62 spinor analysis, 547 symmetrized, 155, 172 spinor fields, 111 density, 26 stationary limit surface, 402 Einstein, 63 Stokes’ theorem, generalized, 34 electromagnetic field, 33, 50 stress density, 118 energy-momentum-stress, 119 stress tensor field, 119 expansion, 184 string theory, 468, 472, 479, 560 extended extrinsic curvature, 604 strong energy condition, 181 extrinsic curvature, 89–104, 167 summation convention, 7 field, 20 surface generalized Kronecker, 29 Monge, 102 hybrid field, 96–98 parametric, 89 identity, 46 ruled, 85, 97 Killing, 414 surface of revolution, 92 Levi-Civita, 49 708 Index

metric, 42 totally antisymmetric tensor, 27 numerical, 29, 111 totally differentiable function, 10 order of, 18 totally geodesic hypersurface, 325 oriented, 26, 50, 63 trace-reversed perturbation tensor, 648 oriented antisymmetric, 49 traceless-transverse gauge, 651 polarization, 651 twist potential, 347, 353, 356 projection, 183 rank of, 18 U relative, 26, 63 underdetermined system, 166 Ricci, 62 unit matrix, 8 Riemann, 55 unit vector field, 42 Riemann-Christoffel, 59, 62 upper triangular form, 629 shear, 184 stress, 119 V symmetric, 27 vacuum field equations, 164 symmetrized curvature, 155, 172 Vaidya metric, 284 torsion, 54 Van Stockum space-time, 665–666 totally antisymmetric, 27, 49 variational derivative, 595, 599 trace-reversed perturbation, 648 variationally permissible boundary con- vorticity, 184 ditions, 596, 599 Weyl conformal, 71 vector Weyl’s projective, 71 covariant, 9, 11 field, 26 heat flow, 232 tensor field, 20 Killing, 72 , 19 norm or length, 43 tetrad, 106, 481 null, 106 complex null, 485 separation, 42 Theorema Egregium, 101 spacelike, 106 tidal forces, 143 tangent, 6, 7 time machine, 662–666 timelike, 106 time orientable space-times, 662 unit, 42 timelike curve, 112 vector space timelike vectors, 106 cotangent, 9 Tolman-Bondi-Lemaˆıtremetric, 276 dual, 9 Tomimatsu-Sato solution, 342 tangent, 7 topological manifold, 3 volume topology, 1 oriented, 65 , 54 vorticity tensor, 184 total 4-momentum, 122 total charge, 134 W total conservation of energy-momentum, warp-drive, 659–662 120 wave number, 504 total mass, 134 weak energy condition, 180 total mass function, 272, 273 weak solution, 610 totally antisymmetric oriented ten- wedge product, 28 sor, 49 weight of relative tensor field, 26 Index 709

Weingarten’s equations, 98 Weyl conformal tensor, 71 Weyl’s coordinate chart, 289 Weyl’s projective tensor, 71 Weyl-Lewis-Papapetrou (WLP) charts, 330 Weyl-Majumdar-Papapetrou-Das (or W-M-P-D) metric, 319 , 379 Willmore’s theorem, 640 winding number, 14 , 113 world tube, 118 world-function (Synge’s), 86 wormhole, 171, 265, 654–659

Y Yukawa equation, 573