Index

A Brane solutions, 311 Advanced Green function(s), 284 Brane(s), viii, xi, 311 Aether, lumineferous aether, 4, 5 Bulk (field) theory(ies), xi Affine connection, 85, 140Ð145 Amaldi (Edoardo), 275 C Angular momentum, x, 105, 148, 150Ð152, Calvera, 262 159, 160, 162, 164, 165, 169, 170, 179, Canis Maior Constellation, 259 291, 299, 303 Cartan connection(s), 94 Anisotropy, anisotropies, x Carter, x Arecibo Radio Telecope, 276, 278 Causality, ix, 146 Associated bundle(s), 51, 126, 129, 194 Cavendish, 3, 229 Atlas, 37Ð43, 56, 61, 130 Centrifugal barrier, 159, 160, 180 Auxiliary field(s), 203 Cepheides, x Avogadro number, 267 Cerdonio (Massimo), 275 Azimuthal angle, 161, 162, 164, 181 CERN, 125, 126, 275 B Chandrasekhar, 238, 240 Baade, 238, 240, 261, 262 Chandrasekhar mass (limit), ix, 159, 256, 266, Base manifold, ix, 52Ð54, 56, 58Ð60, 64, 106, 267, 270, 277 120, 121, 123, 127, 128, 130, 133, 136, Charge(s), 4, 132Ð136, 158, 196, 198, 199, 193, 196, 200, 202 229, 230, 232, 273, 283, 284 Beltrami, 101, 152 Charge conjugation matrix, 314 Bessel, x Chirality matrix, 312, 314, 317 Betti (Enrico), 94, 98 Circular orbit(s), 161, 165, 166, 168Ð171, 180, Bianchi, 2, 98, 100, 101, 136 182 Bianchi classification, x Classical Lie group(s), 15 Bianchi identity(ies), ix, xi, 29, 100, 189, 197, Clifford algebra(s), 19, 21, 312Ð314, 317 209Ð211, 218, 249 Coalescence of binaries, 279 Bianchi type, x Cohomology, ix, xi, 66, 70, 82 Binary pulsar(s), 298 Cohomology group, 81Ð83 Binary system(s), 159, 160, 259, 270, 278, Compact star(s), 157, 167, 239, 262, 266, 269, 279, 300Ð302, 306, 309 277, 279 Black hole(s), viiiÐx, xii, 160, 274, 279 Compactification, 214 Boltzmann constant, 267 Compton wave length, 261, 264, 266 Born-Infeld, xi Conformal mapping (map), ix, 96 Boson(s), 19, 51, 86, 187, 226 Connection coefficients, 96, 128, 129, 141, Boundary operator, 78Ð80 191, 205

P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5361-7, 331 © Springer Science+Business Media Dordrecht 2013 332 Index

Connection one-form, 121Ð126, 129, 140, 141, E 197, 203, 212 Eccentricity, 162, 171, 175, 305Ð308, 319, Connection(s), viii, ix, 1, 2, 85Ð87, 94, 95, 320, 322, 324 106Ð108, 117Ð122, 125Ð130, 135, 136, Eddington, 238Ð240 139Ð145, 189, 190, 192Ð198, 200, Effective potential, 165, 166, 168, 169, 180, 203Ð208, 212, 213, 215Ð219, 289 182 Contorsion, 142, 143, 206 Ehresmann, 94, 106, 108, 118, 119, 121, Copernicus, 227 125 Coset manifold(s), x Ehresmann connection, 108, 118, 125 Cosmic billiard(s), x Einstein, 1, 2, 5, 8Ð10, 25, 32, 94, 98, 102, Cosmic microwave background, CMB, x, xi 106, 107, 134, 160, 163, 188, 211, 237, Cosmological parameter(s), 270 238, 273, 274 Cosmological Principle, x Einstein tensor(s), 100, 211, 212, 217, 221, Cotangent space(s), 44, 50, 51, 64, 69 235, 246, 248, 249, 324 Coulomb, 283 Electric current, 31, 197, 198, 230 Covariant derivative, 85, 96, 124, 125, 128, Electric dipole, 273 144, 196, 204, 205, 208 Electric field, 2, 4 Crab Nebula, 262, 263, 277 Electromagnetic potential, 2 Crab Pulsar, 277 Electromagnetism, 2, 3, 273 Current(s), 4, 25, 29Ð31, 197, 198, 211, 229, Energy loss, 295, 298, 302, 304 230, 232, 233 Entropy, x Curtis, x Equation(s) of state, 247, 249, 265Ð269 Curvature(s), xi, 2, 85, 94, 98Ð100, 103Ð107, Erlangen Programme, 98 117, 118, 139, 141, 142, 188, 194, 195, Euclid, 92, 151 197Ð200, 202, 209Ð213, 221, 233, 234 Euclidian geometry, 11, 87, 88, 90, 93, 95, 152 Euclidian space(s), 11, 76, 88Ð90, 103 D Event horizon(s), x, 256 D3-brane, xi Event(s), 6, 8Ð10, 12, 15, 16, 32, 35, 36, 101, D’Alembert, 282 279, 280, 285 De Sitter space(s), x, 146, 147 Exterior derivative, 80Ð82, 106, 107, 135 Deflection angle, 160, 184, 185 Exterior form(s), 66, 67, 69 Diffeomorphism(s), 63, 108Ð111, 208, 212, 219, 221, 231, 233 F Differentiable manifold, 2, 35Ð37, 39, 40, 42, Fermi, 239, 241, 275 53, 55, 58, 72, 75, 77, 108, 112, 130, Fermi gas, 256, 258, 260Ð262, 264, 270 136, 138Ð140, 157 Fermi pressure, 262Ð265 Differentiable structure, 38, 39, 136 Fermion(s), 24, 208, 257, 264 Differential form(s), ix, 42, 49, 50, 64, 66, Feynman (Richard), 187, 227, 228, 233, 274, 68Ð70, 76, 80Ð83, 102, 106, 107, 115, 284 198, 199, 208, 219, 287 Fibre bundle, 35, 51, 55, 61, 87, 94, 98, 100, Differential geometry, ix, 1, 32, 35, 86, 94, 98, 118, 120, 121, 130, 132, 136, 142, 101, 102, 106, 144, 233 200 Dilaton, 216, 217 First integral(s), 148, 159, 164, 171, 179 Dilaton torsion, 216 First order formalism, xi, 202 Dirac, 19, 134, 228, 239 Flat metric, 139, 191, 193, 194, 232, 244 Dirac spinor(s), 19, 21, 23, 30, 208, 312 Flux compactification(s), xii Distance, x, 11, 32, 40, 87Ð90, 103, 159, 160, Four-momentum, D-momentum, 15, 26 162, 167, 172, 176, 177, 181, 182, 184, Fourier, 292 227, 238, 306 Fowler, 239, 240 Domain wall(s), xi Frascati (National Laboratories), 275 Double pulsar system, 307Ð309 Free differential algebra(s), xi Duality, 51, 198Ð200, 203 Freedman, x Duality rotations, xi Frenet, 94, 102Ð104, 106, 189 Index 333

Fundamental group, 24, 74, 75 I Fusion cycle of hydrogen, 260 Immanuel Kant, Kant, 10, 88 Impact parameter, 181Ð185 G Inertia tensor, 302, 303 Galaxy, galaxies, ix, x, 238 Inertial frames, 1, 8, 15, 32, 36, 37, 189Ð191 Galilei group, 1, 5, 8 Infeld, 274 Galileo, 1, 6, 7, 32, 36 Inflationary universe, x, 311 Gamma matrices, 312, 317 Interference fringes, 5 Gamow, x Interferometer (detectors), 279 Gauge boson(s), 51, 187, 226 Interior solution(s), 234, 245 Gauge fixing, 28, 222, 289 Irreducible representation, 19, 21, 26, 28, 57, Gauge theory, gauge theories, 189 209, 210, 226, 312 Gauge/gravity correspondence, xi Island-universe(s), x Gauss, 2, 35, 87Ð94, 98, 102, 105, 107, 134, Isometry(ies), xi, 148 136, 137, 152 Isotropy, x Gaussian coordinates, curvilinear coordinates, 87, 89Ð91, 93 K General Relativity, viiÐx, 1, 2, 35Ð37, 85, 98, κ-supersymmetry, xi 100Ð103, 106, 107, 159, 160, 180, 184, Kasner metric(s), x 191, 192, 237Ð242, 253, 254, 274, Kepler, 157Ð160, 162, 238, 300, 302, 304, 305 307Ð309, 311 Keplerian parameters, 172Ð176, 178, 298 Geodesic(s), ix, x, 144Ð154, 157Ð160, 162, Kerr-Newman metric, ix 164, 179, 180, 187 Killing, 16, 105 Germs of smooth functions, 43, 44, 47, 48 Killing vector, 105, 148, 245, 246 Giazzotto (Adalberto), 281 Kinetic energy, 9, 15, 243, 249, 267 Göttingen, 10, 25, 30, 88, 91, 92, 100, 119, 163 Klein, 25, 98, 100 Gravitational bending, 182 Kronecker, Kronecker delta, 221, 243 Gravitational binding energy, 248, 261 Kruskal, ix Gravitational wave(s), ix, 5, 222, 225, 240, 273Ð276, 279, 280, 285, 286, 288, L 298, 301, 302, 305, 307Ð309, 311 Lagrange, 292 Graviton(s), ix, 28, 51, 187, 189, 208, Lagrangian(s), xi, 23, 24, 145, 148, 153, 163, 224Ð227, 291, 309 164, 179, 197, 208, 219, 220, 229, Green function(s), 280Ð286, 291 231Ð233 GW detectors, 274 Lane-Emden, 239, 268Ð270 Laplace, ix, 187, 292 H Leavitt, x, 25 Hewish, 238, 240, 262, 263 Left-handed, 316 Hilbert de Donder, 224, 288 Left-invariant vector field, one-form, 110Ð117, Hodge dual, 199, 200, 212, 216 124, 125 Hodge duality, 198Ð200, 203 Levi Civita, 2, 87, 94, 97Ð102, 106, 136, 137, Hoffmann, 274 144, 188, 189 Homeomorphism, 37Ð39, 55 Levi Civita connection, 85, 87, 94, 96, 100, Homogeneity, x 125, 139, 142Ð145, 189, 190, 205Ð207, Homologically (non-)trivial, 71 212, 324, 325, 327 Homology, ix, 66, 70, 75, 81Ð83 Lie bracket, 63, 111 Homotopically (non-)trivial, 70, 73 Lie group, x, 6, 15, 16, 19, 23, 25, 30, 52, Homotopy, 70, 72Ð75, 83 55Ð57, 63, 70, 102, 103, 105, 106, 108, Hopf fibration, 130, 132 110Ð112, 114Ð116 Horizon, x, 234, 256 Light-cone, 223, 224, 285, 286, 288, 289 Horizon area, x, xii Ligo (I, II), 280 Horizontal vector fields, 124, 125 Line bundle, 58 Hubble, x, 311 Little group, 23, 26Ð28, 225, 226 Hulse, 278, 279, 298, 299 Lobachevskij, 152 334 Index

Lobachevskij-Poincaré plane, 151 Noether’s theorem, 23, 24, 29Ð31, 229, Local trivialization, 54Ð56, 58Ð60, 62, 120, 232 122Ð124, 127, 129Ð131, 133, 135, 193, Null geodesic(s), 147Ð149 196, 197 Null-like, 12, 13, 27, 146Ð148, 187, 226 Lorentz, 1, 4, 6Ð8 Lorentz algebra, 21, 317 O Lorentz bundle, 200, 204, 205, 207, 215, 218, Observer(s), x, 1, 6, 8, 36, 37, 213 221 Olbers, Olbers paradox, x Lorentz group, 1, 13, 15, 16, 18Ð24, 27, 32, Open chart, 37Ð43, 46, 47, 54, 56, 58, 60, 61, 194, 209, 210, 216, 217, 221, 226 108Ð110, 130, 131, 141 Lorentz transformations, 6Ð8, 10, 17, 18, 23, Oppenheimer, ix, 238, 241, 250, 254 26, 27, 36, 208, 212, 213, 244, 287, 289 Lorentzian manifold(s), 145 P p-chain(s), 78 M Parallax, x, 158 Magnetic field, 2, 4, 133, 135, 197 Parallel transport, 35, 86, 94Ð96, 98, 99, Magnetic monopole, 130, 132Ð134 142Ð144, 207 Majorana spinor(s), 312, 316 Particle horizon(s), x Majorana-Weyl spinor(s), 312, 316 Penrose diagram(s), ix, 311 Manifold, ixÐxii, 35Ð48, 50Ð56, 58Ð66, 69Ð78, Perfect fluid, 237, 242Ð245 82, 83, 93, 94, 98Ð100, 106Ð108, Periastron, 160, 170, 172Ð174, 176Ð178, 180, 110Ð112, 114Ð116, 120Ð124, 136Ð140, 298Ð300, 307, 322 142Ð146, 190Ð193, 198Ð202 Pesando, xiii Mass, ix, x, 9, 26, 162, 165Ð167, 181Ð185, Pithagora’s theorem, 90, 91 187, 188, 213, 229, 237Ð242, 247Ð249, Pizzella (Guido), 275 251Ð253, 255Ð257, 259Ð261, 263Ð270, Plane gravitational wave, 288 305, 306 Poincaré bundle, 194, 195, 208, 218 Maurer, Maurer Cartan forms, 102, 115 Poincaré group, Poincaré algebra, 23, 26, 194, Maurer Cartan equations, 117, 118 219 Maxwell, 2Ð4, 7 Polar coordinate(s), 131, 134, 160, 162, 283, Maxwell equations, 1Ð4, 6, 8, 197Ð199, 273 285, 302 Metric(s), ix, x, 85Ð87, 93Ð96, 136Ð139, Polytrope(s), 267 143Ð148, 153, 154, 159, 187Ð194, Pressure, 163, 227, 237Ð240, 243, 247, 199Ð203, 205, 206, 210Ð213, 223Ð225, 249Ð256, 258, 259, 261Ð265, 267, 270, 232Ð237, 244Ð250, 286Ð289, 324Ð327 275 Michelson and Morley, 1, 5, 279 Pressure equation, 250, 253, 265 Minkowski, 8Ð10, 12, 16, 163 Principal bundle(s), 2, 51, 57, 58, 85, 108, 123, Minkowski metric, 12, 27, 31, 221, 225, 235, 129, 130, 132, 141 244, 287 Principal connection, 2, 94, 108, 129, 190, Minkowski space, 9, 10, 12, 31, 139, 146, 200, 195, 208, 218 208, 223, 229, 230, 288 Propagator(s), 197, 201, 283, 284 Momentum, x, 14, 15, 23, 24, 26Ð28, 150Ð152, Proper time, 163 159, 160, 164, 165, 169, 170, 225, 229, Pseudo-Riemannian metric(s), 136, 157 230, 232, 233, 242, 243, 257, 258, 291, Pseudo-sphere(s), 152 299, 303 PSR 1913+16, 278, 300, 305Ð308 Pull-back, 108, 110, 123, 146 N Push-forward, 108Ð111, 121 Neutron star(s), ix, 240, 241, 256, 258, 261Ð264, 266, 269, 270, 277Ð279, Q 306Ð308 Quadrupole moment, 273, 293, 295, 298, New first order formalism, xi 304 Newtonian potential, 160, 170, 226, 249, 299 Quadrupole radiation, 295, 298, 302 Newton’s law, 2, 8, 32, 159, 187, 251 Quantum chromodynamics, 228 Noether, 25, 228 Quartic (symplectic) invariant, xii Index 335

R SO(9), 226 Radiation region, 294 Soldering, 94, 107, 195, 200, 204, 205, Reference frame, 5Ð8, 13, 32, 106, 183, 191, 207Ð213, 215, 218, 219, 221 243 Space-like, 12, 13, 146, 147, 150Ð152, 157, Regge, vii, 274 159, 226, 246 Reissner Nordström (solution, black hole, Special Kähler, xii metric), 326 Special Lorentz transformations, 6, 17, 18, 244 Repère mobile, 103, 188, 189, 191, 200 Special Relativity, viii, 1, 2, 5, 8Ð10, 12, 13, Representations of Lorentz group, algebra, 20, 15, 19, 23, 24, 31, 32, 36, 102, 124, 21, 23, 24, 27, 216, 217 229, 242, 244 Rest energy, 9, 15, 249 Spectral index, xi Restricted holonomy, xii Speed of light, velocity of light, 4Ð8, 10, 13, Retarded Green function(s), 284Ð286 14, 27, 146, 159, 165, 213, 223, 225, Rheonomic, xi 229, 265, 286, 298 Rheonomy (principle), xi Sphere(s), 40, 41, 44, 46, 61, 62, 75, 130Ð135, Ricci, Ricci Curbastro, 2, 94, 96Ð102, 106, 152, 160, 161, 247Ð249, 251, 267 136, 137, 144, 188, 189 Spin, spin of a particle, 26, 27, 226 Riemann, 2, 88, 91Ð96, 98, 101, 102, 105, 107, Spin connection(s), ix, 203, 204, 206Ð208, 136, 137, 141, 188 212, 213, 215Ð219, 221, 234, 250, 287, Riemann curvature, 85, 188, 212, 324 289 Riemann tensor, ix, 99, 107, 140, 209Ð212, Spin-statistics, 19, 20 218, 235, 325, 328 Spinor(s), xii, 19, 21, 23, 30, 94, 102, 105, Right invariant vector field, one-form, 115 125, 189, 195, 208, 312, 316, 317 Right-handed, 316 Spinor representations, 19Ð21, 312 Rindler space time, ix Standard cosmological model, x Root(s), 86, 105, 121, 144, 166, 168, 189, 278 Standard fibre, 53Ð55, 57, 58, 128, 130, 196 Rosen, 274 Standard model, 125 Rubbia (Carlo), 125, 126 Static limit, x Stellar equilibrium, ix, 234, 237, 241, 245, S 250, 251, 270, 311 Saccheri, 152 Stellar mass, ix, 167, 253, 270 Salam (Abdus), 125, 126 Stereographic projection, 40, 41, 130, 132 Scalar field(s), xi, 25Ð27, 30, 31, 216, 220, 237 Stokes lemma, 83 Scalar manifold(s), xi Scalar product, 11, 12, 16, 32, 36, 95, 143 Stress energy tensor(s), 31, 218, 242, 245, 295 Schwarzschild emiradius, 165, 167, 169, 171, Sullivan’s theorem(s), xi 174, 319 Super-gauge completion, xii Schwarzschild (metric), ix, 157, 159, 162, 163, Supergravity, viii, xÐxii, 70, 201, 226, 237, 179, 187, 233, 234, 236, 247, 318, 320 311, 312 Schwarzschild radius, 241, 249, 254Ð256, 324 Supermultiplet(s), xi Section(s) of (a) fibre bundle(s), 52, 134 Supernova(e), 261, 262 Semi simple Lie algebra(s), 201 Supernova Ia, 158, 159, 270 Semilatus rectum, 162, 171, 175, 301, SuperPoincaré, xi 305Ð308, 319, 320, 322 Superstring(s), superstring theory, xi, 226, 237, Serret, 94, 102Ð104, 106, 189 317 Shapley, x Supersymmetry, viii, xii, 311, 312 Signature, ix, 12, 137Ð139, 143, 147, 153, 157, Sylvester, 12, 138, 139 190, 194, 199, 202, 283 Symmetric spaces(s), xi, 102, 106 Simplex, standard symplex, 76Ð80, 83 Symplectic embedding(s), xi Simply connected, 75, 83 Sirius (A, B), 238, 239, 259 T Slow rolling, xi Tangent bundle(s), ix, 58Ð64, 85, 94, 98, 106, Smooth manifold, 52, 60, 61, 82, 83, 85 108, 111, 112, 116, 127, 136, 140Ð142, SO(1, 3), 7, 22, 23, 221 190, 200, 204, 205, 207 336 Index

Tangent space(s), 46Ð48, 50, 58, 59, 62Ð64, Vielbein(s), ix, 31, 94, 188, 189, 191Ð195, 66, 69, 111, 120Ð122, 145 200Ð209, 211Ð218, 220, 221, 234, 245, Taylor, 278, 279, 298, 299 289 Time-like, 12Ð15, 27, 146, 147, 149Ð151, Virgo, 280, 281 157Ð159, 179, 180, 187, 226, 245 Volkoff, 241, 250, 254 Tolman Oppenheimer Volkoff, 254 Torsion, 99, 104Ð107, 139Ð142, 194, 195, 200, W 205Ð207, 211, 212, 214Ð216, 218 Wave length(s), 261, 264, 266 Torsion equation, 206, 214, 234, 287, 289 Weak field limit, 220 Torsionful connection(s), 215 Weber, 274, 275 Torsionless connection, 142, 144 Weight(s), 252, 256, 260, 275 Total differential, 49, 50, 65, 66, 70 Weinberg (Steven), 125, 126 Total manifold, 53 Weyl, 85 Tycho Brahe, 158, 159 Weyl spinor(s), 23, 312, 316 Weyl transformation, 217 Wheeler, 228, 274 U White dwarf, 159, 256, 259Ð261, 264, 269, 270 U(1) group, factor, bundle, 87, 125, 133Ð135, Wilson, 261 196, 198, 200 WMAP, 311 UIR (unitary irreducible representations), 26, World line, 191 226 Universal recession, x Y Yang (C.N.), 86, 87, 94, 124, 125 V Yang-Mills theories, 189, 195, 200, 203, 212, Vector bundle(s), 51, 57, 58, 60, 65, 125Ð129, 219 140, 141, 193, 196, 202, 204, 205, 207 Young tableau(x), 209 Vector field(s), ix, xi, 27, 28, 42, 60Ð65, 69, 105, 108, 110Ð116, 119Ð122, 124Ð129, Z 136, 137, 140, 143, 238, 245, 246 Zwicky, 238, 240, 261