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Symbols 1-Form, 10 4-Acceleration, 184 4-Force, 115 4-Momentum, 116

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Symbols 1-Form, 10 4-Acceleration, 184 4-Force, 115 4-Momentum, 116 Index Symbols B 1-form, 10 Betti number, 589 4-acceleration, 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-velocity, 114 formalism, 549–551 first contracted, 63 second contracted, 63 A bicharacteristic curves, 620 acceleration big crunch, 441 4-acceleration, 184 big-bang cosmological model, 438, 449 Newtonian, 74 Birkhoff’s theorem, 271 action function or functional bivector space, 486 (see also Lagrangian), 594, 598 black hole, 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 tensor, 49 C antisymmetric tensor, 28 canonical energy-momentum-stress ten- antisymmetrization, 27 sor, 119 arc length parameter, 81 canonical or normal forms, 510 arc separation function, 85 Cartesian chart, 44, 68 arc separation parameter, 77 Casimir effect, 638 Arnowitt-Deser-Misner action integral, 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 matrix, 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 geodesic 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 vector space, 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 derivative, 51 conjugate holomorphic function, 622 of relative and oriented relative conjugate points, 81 tensor fields, 63 conjugate, contravariant, or inverse covariant index, 51 metric tensor, 44 covariant order, 18 constant curvature, space, 68, 73 covariant vector field, 9 continuity equation, 122, 126, 186, critical energy density, 448 232 critical point, 593 continuum mechanics, 117 curvature invariant, 63 contraction operation, 21 curvature scalar, 63 700 Index curvature tensor, 55 doubly-null coordinates, 130 curve 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 universe, 434 deceleration function, 448 Einstein summation convention, 7 deceleration parameter, 448 Einstein tensor, 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 manifold, 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 directional derivative, 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 equivalence principle, 139 dot product or inner-product, 42 ergosphere, 402 double Hodge-dual, 84 Ernst equation, 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 event horizon, 367 gauge covariant derivatives, 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 Gaussian curvature, 91 exterior derivative, 31 Gaussian normal coordinate chart, 67, exterior product, 28 202 extrinsic curvature tensor, 89–104, 167 general covariance, 131 extended, 604 general relativity, 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 plane 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 redshift, 250 homogeneity, 442 gravitational wave 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 relativistic mechanics, 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 sectional curvature,
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