table of contents -- preface -- bibliography
the spacetime interval -- the metric -- Lorentz transformations -- spacetime diagrams -- vectors -- the tangent space -- dual vectors -- tensors -- tensor products -- the Levi-Civita tensor -- index manipulation -- electromagnetism -- differential forms -- Hodge duality -- worldlines -- proper time -- energy-momentum vector -- energy-momentum tensor -- perfect fluids -- energy-momentum conservation
examples -- non-examples -- maps -- continuity -- the chain rule -- open sets -- charts and atlases -- manifolds -- examples of charts -- differentiation -- vectors as derivatives -- coordinate bases -- the tensor transformation law -- partial derivatives are not tensors -- the metric again -- canonical form of the metric -- Riemann normal coordinates -- tensor densities -- volume forms and integration
covariant derivatives and connections -- connection coefficients -- transformation properties -- the Christoffel connection -- structures on manifolds -- parallel transport -- the parallel propagator -- geodesics -- affine parameters -- the exponential map -- the Riemann curvature tensor -- symmetries of the Riemann tensor -- the Bianchi identity -- Ricci and Einstein tensors -- Weyl tensor -- simple examples -- geodesic deviation -- tetrads and non-coordinate bases -- the spin connection -- Maurer-Cartan structure equations -- fiber bundles and gauge transformations
the Principle of Equivalence -- gravitational redshift -- gravitation as spacetime curvature -- the Newtonian limit -- physics in curved spacetime -- Einstein's equations -- the Hilbert action -- the energy-momentum tensor again -- the Weak Energy Condition -- alternative theories -- the initial value problem -- gauge invariance and harmonic gauge -- domains of dependence -- causality
pullbacks and pushforwards -- diffeomorphisms -- integral curves -- Lie derivatives -- the energy-momentum tensor one more time -- isometries and Killing vectors
the weak-field limit defined -- gauge transformations -- linearized Einstein equations -- gravitational plane waves -- transverse traceless gauge -- polarizations -- gravitational radiation by sources -- energy loss
spherical symmetry -- the Schwarzschild metric -- Birkhoff's theorem -- geodesics of Schwarzschild -- Newtonian vs. relativistic orbits -- perihelion precession -- the event horizon -- black holes -- Kruskal coordinates -- formation of black holes -- Penrose diagrams -- conformal infinity -- no hair -- charged black holes -- cosmic censorship -- extremal black holes -- rotating black holes -- Killing tensors -- the Penrose process -- irreducible mass -- black hole thermodynamics
homogeneity and isotropy -- the Robertson-Walker metric -- forms of energy and momentum -- Friedmann equations -- cosmological parameters -- evolution of the scale factor -- redshift -- Hubble's law