Index accretion torus, 177 and energy-momentum tensor, action 89 of a free particle, 47 covariant derivative, 33 of general relativity, 86 of a dual vector, 36 adapted coordinates, 115 of a tensor, 38 algebra, 70 covariant differentiation, 32 alternation operator, 69 Cramer’s rule, 54, 85 atlas, 15 curvature, 39 differentiable, 16 and tidal field, 81 curvature tensor, 40 Bianchi identities in arbitrary basis, 114 contracted, 45 symmetries, 44 first and second, 40 curve, 23 for Riemannian connection, 44 derivation, 20 Birkhoff’s theorem, 125 diffeomorphism generalised, 186 invariance, 93 one-parameter group, 23, 63 Cartan’s equation, 72 differential forms, 69 Cartan’s structure equations, 113 exact and closed, 72 chart, 15 differential of a function, 24 chirp mass, 108 Doppler shift, 8 Christoffel symbols, 32 and redshift, 54 for linearised gravity, 91 dual vector, 24 of Riemannian connection, 43 transformation, 33 Eddington-Finkelstein coordinates, codifferential, 74 154 coordinate expression, 74 eikonal equation, 52 cofactor, 54 Einstein commutator, 24 elevator, 7 connection field equations, 83 affine, 32 linearised in Hilbert gauge, linear, 32 94 metric, 42 tensor, 45 Riemannian or Levi-Civita, 42 energy non-conservation, 58 symmetric, 33 energy-momentum tensor, 88 connection forms, 111 of electromagnetic field, 88 contraction, 26 equivalence principle convergence, 202 Einstein’s, 5 cosmological constant, 84 weak, 5 231 232 INDEX exponential map, 35 interior product, 71 exterior derivative, 71 ISCO, 133 coordinate expression, 72 isometry, 65 exterior product, 70 Jacobi equation, 79 Faraday 2-form, 75 Jacobian matrix, 22 Fermat’s principle, 96 frames Kerr spacetime freely-falling, 7 angular velocity, 169 inertial, 6 ergosphere, 171 Friedmann’s equations, 193 Kepler’s third law, 172 Frobenius condition, 116 Killing horizon, 171 redshift near static limit, 170 gauge invariance, 93 static limit, 169 Gauss’ theorem, 76 Kerr(-Newman) solution, 166 generator of a transformation, 23 gyromagnetic moment, 167 geodesic, 34 radial magnetic field, 167 and free fall, 48 Killing and light rays, 53 equation, 68 congruence, 77 vector field, 68 deviation, 80 vector fields and conservation equation, 34 laws, 69 in Newtonian limit, 58 Koszul formula, 42 geometrical optics, 51 Kruskal(-Szekeres) change of amplitude, 53 continuation, 151 polarisation transport, 51 coordinates, 151 Gibbons-Hawking-York boundary term, 86 Laplace expansion, 55 ff Grassmann algebra, 71 Lense-Thirring e ect, 99 Grassmann identity, 101 Lie derivative, 65 gravitational wave and commutator, 66 equation, 100 coordinate expressions, 67 polarisation tensor, 102, 103 Lie transport, 78 gravitomagnetic field, 98 light deflection, 9 equation of motion in, 99 by the universe, 205 Lense-Thirring effect, 99 in Newtonian limit, 59 Gravity Probe B, 100 light speed in weak gravitational field, 96 Hilbert gauge, 94 Lorentz force, 50 Hilbert-Palatini action, 89 Lorentzian spacetime, 28 Hodge ∗ operator, 73 Lovelock’s theorem, 84 index of refraction in weak gravita- manifold, 15 tional field, 96 differentiable, 16 inhomogeneous wave equation for Maxwell’s equations electromagnetic field, 51 in general relativity, 50 innermost stable circular orbit, 133 in special relativity, 49 integration of an n-form, 75 metric INDEX 233 (pseudo-)Riemannian, 28 Schwarzschild solution, 122 Friedmann-Lemaître-Robert- Schwarzschild spacetime son-Walker, 194, 198 Eddington-Finkelstein coordi- Kerr(-Newman), 166 nates, 154 of static spacetime, 116 effective potential, 129 Reissner-Nordström, 162 equation of motion in, 130 Schwarzschild, 122 Kruskal(-Szekeres) continuation, tensor, 27 151 Lagrangian of, 127 neutron stars last stable orbit, 133 mass limit, 210 light deflection, 137 optical tidal matrix, 202 perihelion shift, 135 spin precession, 140 Palatini identity, 85 Schwarzschild tetrad, 118 parallel transport, 34 curvature forms of, 120 Penrose-Carter diagram, 152, 227 Einstein tensor of, 121 perihelion shift Ricci scalar of, 121 in scalar theory of gravity, 13 Ricci tensor of, 120 in Schwarzschild spacetime, 135 Shapiro delay, 97 Planck shear temperature, 181 components, 202 units, 4 matrix, 202 polarisation simultaneous events, 6 and parallel transport, 53 spacetimes with constant curvature tensor, 102 metric, 190 product manifolds, 18 Ricci scalar, 189 pull-back, 64 Ricci tensor, 189 push-forward, 64 spatially homogeneous spacetime, 188 quadrupole spatially isotropic spacetime, 188 formula, 106 spherically symmetric spacetime, tensor, 105 183 redshift connection forms, 185 cosmological, 197 curvature forms, 185 gravitational, 8, 10 Einstein tensor, 186 near Schwarzschild radius, 159 Ricci scalar, 185 Reissner-Nordström solution, 162 Ricci tensor, 185 Ricci static spacetime, 116 identity, 42 stationary spacetime, 115 scalar, 45 axially symmetric, 164 for linearised gravity, 92 Stokes’ theorem, 76 tensor, 40 structure equations, Cartan’s, 113 for linearised gravity, 92 Riemann tensor, 40 tangent space, 20 Robertson-Walker metric, 191, 198 basis, 22 tensor, 25 scalar product, 27 transformation of, 26 234 INDEX tensor field, 25 Tolman-Oppenheimer-Volkoff equation, 208 torsion, 38 forms, 112 tensor, 39 in arbitrary basis, 114 two-sphere, 16 atlas, 16 variational principle of general rel- ativity, 86, 89 vector field, 22 flow of, 63 global flow of, 63 integral curve of, 63 parallel, 34 vector space, 19 vectors and infinitesimal transfor- mations, 23 volume form, 73 canonical, 73 weakness of gravity, 4 Weyl tensor, 200.