Index

A

Advanced Green function(s), 284 Aether, lumineferous aether, 4, 5 Affine connection, 85, 140–145 Amaldi (Edoardo), 275

Angular momentum, x, 105, 148, 150–152, 159, 160, 162, 164, 165, 169, 170, 179, 291, 299, 303

Anisotropy, anisotropies, x Arecibo Radio Telecope, 276, 278

Associated bundle(s), 51, 126, 129, 194 Atlas, 37–43, 56, 61, 130

Auxiliary field(s), 203 Avogadro number, 267

Azimuthal angle, 161, 162, 164, 181

B

Baade, 238, 240, 261, 262

Base manifold, ix, 52–54, 56, 58–60, 64, 106, 120, 121, 123, 127, 128, 130, 133, 136, 193, 196, 200, 202

Beltrami, 101, 152 Bessel, x

Betti (Enrico), 94, 98 Bianchi, 2, 98, 100, 101, 136 Bianchi classification, x

Bianchi identity(ies), ix, xi, 29, 100, 189, 197, 209–211, 218, 249

Bianchi type, x Binary pulsar(s), 298

Binary system(s), 159, 160, 259, 270, 278, 279, 300–302, 306, 309

Black hole(s), viii–x, xii, 160, 274, 279 Boltzmann constant, 267

Born-Infeld, xi

Boson(s), 19, 51, 86, 187, 226 Boundary operator, 78–80

Brane solutions, 311

Brane(s), viii, xi, 311

Bulk (field) theory(ies), xi

C

Calvera, 262

Canis Maior Constellation, 259 Cartan connection(s), 94 Carter, x

Causality, ix, 146 Cavendish, 3, 229

Centrifugal barrier, 159, 160, 180 Cepheides, x

Cerdonio (Massimo), 275 CERN, 125, 126, 275 Chandrasekhar, 238, 240

Chandrasekhar mass (limit), ix, 159, 256, 266, 267, 270, 277

Charge(s), 4, 132–136, 158, 196, 198, 199, 229, 230, 232, 273, 283, 284

Charge conjugation matrix, 314 Chirality matrix, 312, 314, 317

Circular orbit(s), 161, 165, 166, 168–171, 180, 182

Classical Lie group(s), 15

Clifford algebra(s), 19, 21, 312–314, 317 Coalescence of binaries, 279 Cohomology, ix, xi, 66, 70, 82 Cohomology group, 81–83

Compact star(s), 157, 167, 239, 262, 266, 269, 277, 279

Compactification, 214

Compton wave length, 261, 264, 266 Conformal mapping (map), ix, 96 Connection coefficients, 96, 128, 129, 141,

191, 205

332

Connection one-form, 121–126, 129, 140, 141, 197, 203, 212

Connection(s), viii, ix, 1, 2, 85–87, 94, 95, 106–108, 117–122, 125–130, 135, 136, 139–145, 189, 190, 192–198, 200, 203–208, 212, 213, 215–219, 289

Contorsion, 142, 143, 206 Copernicus, 227

Coset manifold(s), x Cosmic billiard(s), x

Cosmic microwave background, CMB, x, xi Cosmological parameter(s), 270 Cosmological Principle, x

Cotangent space(s), 44, 50, 51, 64, 69 Coulomb, 283

Covariant derivative, 85, 96, 124, 125, 128, 144, 196, 204, 205, 208

Crab Nebula, 262, 263, 277 Crab Pulsar, 277

Current(s), 4, 25, 29–31, 197, 198, 211, 229, 230, 232, 233

Curtis, x

Curvature(s), xi, 2, 85, 94, 98–100, 103–107, 117, 118, 139, 141, 142, 188, 194, 195, 197–200, 202, 209–213, 221, 233, 234

D

D3-brane, xi D’Alembert, 282

De Sitter space(s), x, 146, 147 Deflection angle, 160, 184, 185

Diffeomorphism(s), 63, 108–111, 208, 212, 219, 221, 231, 233

Differentiable manifold, 2, 35–37, 39, 40, 42, 53, 55, 58, 72, 75, 77, 108, 112, 130, 136, 138–140, 157

Differentiable structure, 38, 39, 136 Differential form(s), ix, 42, 49, 50, 64, 66,

68–70, 76, 80–83, 102, 106, 107, 115, 198, 199, 208, 219, 287

Differential geometry, ix, 1, 32, 35, 86, 94, 98, 101, 102, 106, 144, 233

Dilaton, 216, 217 Dilaton torsion, 216 Dirac, 19, 134, 228, 239

Dirac spinor(s), 19, 21, 23, 30, 208, 312 Distance, x, 11, 32, 40, 87–90, 103, 159, 160,

162, 167, 172, 176, 177, 181, 182, 184, 227, 238, 306

Domain wall(s), xi

Double pulsar system, 307–309 Duality, 51, 198–200, 203 Duality rotations, xi

Index

E

Eccentricity, 162, 171, 175, 305–308, 319, 320, 322, 324

Eddington, 238–240

Effective potential, 165, 166, 168, 169, 180, 182

Ehresmann, 94, 106, 108, 118, 119, 121, 125

Ehresmann connection, 108, 118, 125 Einstein, 1, 2, 5, 8–10, 25, 32, 94, 98, 102,

106, 107, 134, 160, 163, 188, 211, 237, 238, 273, 274

Einstein tensor(s), 100, 211, 212, 217, 221, 235, 246, 248, 249, 324

Electric current, 31, 197, 198, 230 Electric dipole, 273

Electric field, 2, 4 Electromagnetic potential, 2 Electromagnetism, 2, 3, 273 Energy loss, 295, 298, 302, 304 Entropy, x

Equation(s) of state, 247, 249, 265–269 Erlangen Programme, 98

Euclid, 92, 151

Euclidian geometry, 11, 87, 88, 90, 93, 95, 152

Euclidian space(s), 11, 76, 88–90, 103 Event horizon(s), x, 256

Event(s), 6, 8–10, 12, 15, 16, 32, 35, 36, 101, 279, 280, 285

Exterior derivative, 80–82, 106, 107, 135 Exterior form(s), 66, 67, 69

F

Fermi, 239, 241, 275

Fermi gas, 256, 258, 260–262, 264, 270 Fermi pressure, 262–265

Fermion(s), 24, 208, 257, 264

Feynman (Richard), 187, 227, 228, 233, 274, 284

Fibre bundle, 35, 51, 55, 61, 87, 94, 98, 100, 118, 120, 121, 130, 132, 136, 142, 200

First integral(s), 148, 159, 164, 171, 179 First order formalism, xi, 202

Flat metric, 139, 191, 193, 194, 232, 244 Flux compactification(s), xii Four-momentum, D-momentum, 15, 26 Fourier, 292

Fowler, 239, 240

Frascati (National Laboratories), 275 Free differential algebra(s), xi Freedman, x

Frenet, 94, 102–104, 106, 189

Index

Fundamental group, 24, 74, 75

Fusion cycle of hydrogen, 260

G

Galaxy, galaxies, ix, x, 238 Galilei group, 1, 5, 8 Galileo, 1, 6, 7, 32, 36 Gamma matrices, 312, 317 Gamow, x

Gauge boson(s), 51, 187, 226 Gauge fixing, 28, 222, 289 Gauge theory, gauge theories, 189 Gauge/gravity correspondence, xi

Gauss, 2, 35, 87–94, 98, 102, 105, 107, 134, 136, 137, 152

Gaussian coordinates, curvilinear coordinates, 87, 89–91, 93

General Relativity, vii–x, 1, 2, 35–37, 85, 98, 100–103, 106, 107, 159, 160, 180, 184, 191, 192, 237–242, 253, 254, 274, 307–309, 311

Geodesic(s), ix, x, 144–154, 157–160, 162, 164, 179, 180, 187

Germs of smooth functions, 43, 44, 47, 48 Giazzotto (Adalberto), 281

Göttingen, 10, 25, 30, 88, 91, 92, 100, 119, 163 Gravitational bending, 182

Gravitational binding energy, 248, 261 Gravitational wave(s), ix, 5, 222, 225, 240,

273–276, 279, 280, 285, 286, 288, 298, 301, 302, 305, 307–309, 311

Graviton(s), ix, 28, 51, 187, 189, 208, 224–227, 291, 309

Green function(s), 280–286, 291 GW detectors, 274

H

Hewish, 238, 240, 262, 263

Hilbert de Donder, 224, 288

Hodge dual, 199, 200, 212, 216

Hodge duality, 198–200, 203

Hoffmann, 274

Homeomorphism, 37–39, 55

Homogeneity, x

Homologically (non-)trivial, 71

Homology, ix, 66, 70, 75, 81–83

Homotopically (non-)trivial, 70, 73

Homotopy, 70, 72–75, 83

Hopf fibration, 130, 132

Horizon, x, 234, 256

Horizon area, x, xii

Horizontal vector fields, 124, 125

Hubble, x, 311

Hulse, 278, 279, 298, 299

333

I

Immanuel Kant, Kant, 10, 88 Impact parameter, 181–185 Inertia tensor, 302, 303

Inertial frames, 1, 8, 15, 32, 36, 37, 189–191 Infeld, 274

Inflationary universe, x, 311 Interference fringes, 5 Interferometer (detectors), 279 Interior solution(s), 234, 245

Irreducible representation, 19, 21, 26, 28, 57, 209, 210, 226, 312

Island-universe(s), x Isometry(ies), xi, 148 Isotropy, x

K

κ-supersymmetry, xi Kasner metric(s), x

Kepler, 157–160, 162, 238, 300, 302, 304, 305 Keplerian parameters, 172–176, 178, 298 Kerr-Newman metric, ix

Killing, 16, 105

Killing vector, 105, 148, 245, 246 Kinetic energy, 9, 15, 243, 249, 267 Klein, 25, 98, 100

Kronecker, Kronecker delta, 221, 243 Kruskal, ix

L

Lagrange, 292

Lagrangian(s), xi, 23, 24, 145, 148, 153, 163, 164, 179, 197, 208, 219, 220, 229, 231–233

Lane-Emden, 239, 268–270 Laplace, ix, 187, 292 Leavitt, x, 25

Left-handed, 316

Left-invariant vector field, one-form, 110–117, 124, 125

Levi Civita, 2, 87, 94, 97–102, 106, 136, 137, 144, 188, 189

Levi Civita connection, 85, 87, 94, 96, 100, 125, 139, 142–145, 189, 190, 205–207, 212, 324, 325, 327

Lie bracket, 63, 111

Lie group, x, 6, 15, 16, 19, 23, 25, 30, 52, 55–57, 63, 70, 102, 103, 105, 106, 108, 110–112, 114–116

Light-cone, 223, 224, 285, 286, 288, 289 Ligo (I, II), 280

Line bundle, 58

Little group, 23, 26–28, 225, 226 Lobachevskij, 152

334

Lobachevskij-Poincaré plane, 151

Local trivialization, 54–56, 58–60, 62, 120, 122–124, 127, 129–131, 133, 135, 193, 196, 197

Lorentz, 1, 4, 6–8 Lorentz algebra, 21, 317

Lorentz bundle, 200, 204, 205, 207, 215, 218, 221

Lorentz group, 1, 13, 15, 16, 18–24, 27, 32, 194, 209, 210, 216, 217, 221, 226

Lorentz transformations, 6–8, 10, 17, 18, 23, 26, 27, 36, 208, 212, 213, 244, 287, 289

Lorentzian manifold(s), 145

M

Magnetic field, 2, 4, 133, 135, 197 Magnetic monopole, 130, 132–134 Majorana spinor(s), 312, 316 Majorana-Weyl spinor(s), 312, 316

Manifold, ix–xii, 35–48, 50–56, 58–66, 69–78, 82, 83, 93, 94, 98–100, 106–108, 110–112, 114–116, 120–124, 136–140, 142–146, 190–193, 198–202

Mass, ix, x, 9, 26, 162, 165–167, 181–185, 187, 188, 213, 229, 237–242, 247–249, 251–253, 255–257, 259–261, 263–270, 305, 306

Maurer, Maurer Cartan forms, 102, 115 Maurer Cartan equations, 117, 118 Maxwell, 2–4, 7

Maxwell equations, 1–4, 6, 8, 197–199, 273 Metric(s), ix, x, 85–87, 93–96, 136–139,

143–148, 153, 154, 159, 187–194, 199–203, 205, 206, 210–213, 223–225, 232–237, 244–250, 286–289, 324–327

Michelson and Morley, 1, 5, 279 Minkowski, 8–10, 12, 16, 163

Minkowski metric, 12, 27, 31, 221, 225, 235, 244, 287

Minkowski space, 9, 10, 12, 31, 139, 146, 200, 208, 223, 229, 230, 288

Momentum, x, 14, 15, 23, 24, 26–28, 150–152, 159, 160, 164, 165, 169, 170, 225, 229, 230, 232, 233, 242, 243, 257, 258, 291, 299, 303

N

Neutron star(s), ix, 240, 241, 256, 258, 261–264, 266, 269, 270, 277–279, 306–308

New first order formalism, xi

Newtonian potential, 160, 170, 226, 249, 299 Newton’s law, 2, 8, 32, 159, 187, 251 Noether, 25, 228

Index

Noether’s theorem, 23, 24, 29–31, 229, 232

Null geodesic(s), 147–149

Null-like, 12, 13, 27, 146–148, 187, 226

O

Observer(s), x, 1, 6, 8, 36, 37, 213 Olbers, Olbers paradox, x

Open chart, 37–43, 46, 47, 54, 56, 58, 60, 61, 108–110, 130, 131, 141

Oppenheimer, ix, 238, 241, 250, 254

P

p-chain(s), 78 Parallax, x, 158

Parallel transport, 35, 86, 94–96, 98, 99, 142–144, 207

Particle horizon(s), x Penrose diagram(s), ix, 311 Perfect fluid, 237, 242–245

Periastron, 160, 170, 172–174, 176–178, 180, 298–300, 307, 322

Pesando, xiii

Pithagora’s theorem, 90, 91 Pizzella (Guido), 275

Plane gravitational wave, 288 Poincaré bundle, 194, 195, 208, 218

Poincaré group, Poincaré algebra, 23, 26, 194, 219

Polar coordinate(s), 131, 134, 160, 162, 283, 285, 302

Polytrope(s), 267

Pressure, 163, 227, 237–240, 243, 247, 249–256, 258, 259, 261–265, 267, 270, 275

Pressure equation, 250, 253, 265

Principal bundle(s), 2, 51, 57, 58, 85, 108, 123, 129, 130, 132, 141

Principal connection, 2, 94, 108, 129, 190, 195, 208, 218

Propagator(s), 197, 201, 283, 284 Proper time, 163

Pseudo-Riemannian metric(s), 136, 157 Pseudo-sphere(s), 152

PSR 1913+16, 278, 300, 305–308 Pull-back, 108, 110, 123, 146 Push-forward, 108–111, 121

Q

Quadrupole moment, 273, 293, 295, 298, 304

Quadrupole radiation, 295, 298, 302 Quantum chromodynamics, 228 Quartic (symplectic) invariant, xii

Index

R

Radiation region, 294

Reference frame, 5–8, 13, 32, 106, 183, 191, 243

Regge, vii, 274

Reissner Nordström (solution, black hole, metric), 326

Repère mobile, 103, 188, 189, 191, 200 Representations of Lorentz group, algebra, 20,

21, 23, 24, 27, 216, 217 Rest energy, 9, 15, 249 Restricted holonomy, xii

Retarded Green function(s), 284–286 Rheonomic, xi

Rheonomy (principle), xi

Ricci, Ricci Curbastro, 2, 94, 96–102, 106, 136, 137, 144, 188, 189

Riemann, 2, 88, 91–96, 98, 101, 102, 105, 107, 136, 137, 141, 188

Riemann curvature, 85, 188, 212, 324 Riemann tensor, ix, 99, 107, 140, 209–212,

218, 235, 325, 328

Right invariant vector field, one-form, 115 Right-handed, 316

Rindler space time, ix

Root(s), 86, 105, 121, 144, 166, 168, 189, 278 Rosen, 274

Rubbia (Carlo), 125, 126

S

Saccheri, 152

Salam (Abdus), 125, 126

Scalar field(s), xi, 25–27, 30, 31, 216, 220, 237 Scalar manifold(s), xi

Scalar product, 11, 12, 16, 32, 36, 95, 143 Schwarzschild emiradius, 165, 167, 169, 171,

174, 319

Schwarzschild (metric), ix, 157, 159, 162, 163, 179, 187, 233, 234, 236, 247, 318, 320

Schwarzschild radius, 241, 249, 254–256, 324 Section(s) of (a) fibre bundle(s), 52, 134 Semi simple Lie algebra(s), 201

Semilatus rectum, 162, 171, 175, 301, 305–308, 319, 320, 322

Serret, 94, 102–104, 106, 189 Shapley, x

Signature, ix, 12, 137–139, 143, 147, 153, 157, 190, 194, 199, 202, 283

Simplex, standard symplex, 76–80, 83 Simply connected, 75, 83

Sirius (A, B), 238, 239, 259 Slow rolling, xi

Smooth manifold, 52, 60, 61, 82, 83, 85 SO(1, 3), 7, 22, 23, 221

335

SO(9), 226

Soldering, 94, 107, 195, 200, 204, 205, 207–213, 215, 218, 219, 221

Space-like, 12, 13, 146, 147, 150–152, 157, 159, 226, 246

Special Kähler, xii

Special Lorentz transformations, 6, 17, 18, 244 Special Relativity, viii, 1, 2, 5, 8–10, 12, 13,

15, 19, 23, 24, 31, 32, 36, 102, 124, 229, 242, 244

Spectral index, xi

Speed of light, velocity of light, 4–8, 10, 13, 14, 27, 146, 159, 165, 213, 223, 225, 229, 265, 286, 298

Sphere(s), 40, 41, 44, 46, 61, 62, 75, 130–135, 152, 160, 161, 247–249, 251, 267

Spin, spin of a particle, 26, 27, 226

Spin connection(s), ix, 203, 204, 206–208, 212, 213, 215–219, 221, 234, 250, 287, 289

Spin-statistics, 19, 20

Spinor(s), xii, 19, 21, 23, 30, 94, 102, 105, 125, 189, 195, 208, 312, 316, 317

Spinor representations, 19–21, 312 Standard cosmological model, x

Standard fibre, 53–55, 57, 58, 128, 130, 196 Standard model, 125

Static limit, x

Stellar equilibrium, ix, 234, 237, 241, 245, 250, 251, 270, 311

Stellar mass, ix, 167, 253, 270 Stereographic projection, 40, 41, 130, 132 Stokes lemma, 83

Stress energy tensor(s), 31, 218, 242, 245, 295 Sullivan’s theorem(s), xi

Super-gauge completion, xii

Supergravity, viii, x–xii, 70, 201, 226, 237, 311, 312

Supermultiplet(s), xi Supernova(e), 261, 262 Supernova Ia, 158, 159, 270 SuperPoincaré, xi

Superstring(s), superstring theory, xi, 226, 237, 317

Supersymmetry, viii, xii, 311, 312 Sylvester, 12, 138, 139 Symmetric spaces(s), xi, 102, 106 Symplectic embedding(s), xi

T

Tangent bundle(s), ix, 58–64, 85, 94, 98, 106, 108, 111, 112, 116, 127, 136, 140–142, 190, 200, 204, 205, 207

336

Tangent space(s), 46–48, 50, 58, 59, 62–64, 66, 69, 111, 120–122, 145

Taylor, 278, 279, 298, 299

Time-like, 12–15, 27, 146, 147, 149–151, 157–159, 179, 180, 187, 226, 245

Tolman Oppenheimer Volkoff, 254

Torsion, 99, 104–107, 139–142, 194, 195, 200, 205–207, 211, 212, 214–216, 218

Torsion equation, 206, 214, 234, 287, 289 Torsionful connection(s), 215 Torsionless connection, 142, 144

Total differential, 49, 50, 65, 66, 70 Total manifold, 53

Tycho Brahe, 158, 159

U

U(1) group, factor, bundle, 87, 125, 133–135, 196, 198, 200

UIR (unitary irreducible representations), 26, 226

Universal recession, x

V

Vector bundle(s), 51, 57, 58, 60, 65, 125–129, 140, 141, 193, 196, 202, 204, 205, 207

Vector field(s), ix, xi, 27, 28, 42, 60–65, 69, 105, 108, 110–116, 119–122, 124–129, 136, 137, 140, 143, 238, 245, 246

Index

Vielbein(s), ix, 31, 94, 188, 189, 191–195, 200–209, 211–218, 220, 221, 234, 245, 289

Virgo, 280, 281 Volkoff, 241, 250, 254

W

Wave length(s), 261, 264, 266 Weak field limit, 220

Weber, 274, 275

Weight(s), 252, 256, 260, 275

Weinberg (Steven), 125, 126 Weyl, 85

Weyl spinor(s), 23, 312, 316 Weyl transformation, 217 Wheeler, 228, 274

White dwarf, 159, 256, 259–261, 264, 269, 270 Wilson, 261

WMAP, 311 World line, 191

Y

Yang (C.N.), 86, 87, 94, 124, 125 Yang-Mills theories, 189, 195, 200, 203, 212,

219

Young tableau(x), 209

Z

Zwicky, 238, 240, 261