We define Penrose diagrams, which keep the causal and topological properties of gravitational spacetimes, while moving infinity to a finite distance on the diagram. We use the examples of Minkowski space, in two dimensions and dimensions greater than two, then describe Anti-de Sitter spacetime in Poincaré coordinates (the Poincaré patch), and finally consider the Schwarzschild black hole.

The vielbein–spin connection formulation of general relativity is described, and this being the one that appears in supergravity. Anti-de Sitter space, as a Lorentzian version of Lobachevsky space, is described. It is a symmetric space solution for the case of a cosmological constant. Black holes, as objects with event horizons and singularities at the center, are described.

Solutions of the Einstein and Einstein–Maxwell equations for spherically symmetric metrics (those of Schwarzschild and Reissner–Nordstr\“{o}m) are derived and discussed in detail. The equations of orbits of planets and of bending of light rays in a weak field are derived and discussed. Two methods to measure the bending of rays are presented. Properties of gravitational lenses are described. The proof (by Kruskal) that the singularity of the Schwarzschild metric at r = 2m is spurious is given. The relation of the r = 2m surface to black holes is discussed. Embedding of the Schwarzschild spacetime in a 6-dimensional flat Riemann space is presented. The maximal extension of the Reissner–Nordstr\“{o}m metric (by the method of Brill, Graves and Carter) is derived. Motion of charged and uncharged particles in the Reissner–Nordstr\“{o}m spacetime is described.

The Lemaitre–Tolman class of cosmological models (spherically symmetric inhomogeneous metrics obeying the Einstein equations with a dust source) is derived and discussed in much detail, from the point of view of its geometry and its applications to cosmology. It is shown that these metrics can be used to describe the formation of cosmic voids and of galaxy clusters out of small perturbations of homogeneity at the time of emission of the cosmic microwave background radiation. Apparent horizons for central and noncentral observers, the formation of black holes, the existence and avoidance of shell crossings, the equations of redshift and the generation and meaning of blueshift are discussed. A simple example of a shell focussing singularity is derived. Among the cosmological applications are: solving the horizon problem without inflation, mimicking the accelerating expansion of the Universe by mass-density inhomogeneities in a decelerating model, drift of light rays, lagging cores of Big Bang, misleading conclusions drawn from observed mass distribution in redshift space.

Geodesics are introduced and the geodesic equation analysed for the geometries introduced in chapter 2, using variation principles of classical mechanics. Geodesic motino on a sphere is described as well as the Coriolis effect and the Sagnac effect. Newtonian gravity is derived as the non-relativistic limit of geodesic motion in space-time. Geodesics in an expanding universe and heat death is described. Geodesics in Schwarzschild space-time are treated in detail: the precession of the perihelion of Mercury; the bending of light by the Sun; Shapiro time delay; black holes and the event horizon. Gravitational waves and gravitational lensing are also covered.

Thermal x-rays from neutron stars are mainly radiated by accretion discs originating in the flux of material from a companion star. The companions are white dwarf stars with a range of masses, and some black holes. X-ray bursts are attributed to catastrophic nuclear events on the neutron star surface following accretion from the companion. Structure in the rotating accretion disc is observed as quasi-periodic oscillations (QPOs).

The post-main-sequence evolution of stars with higher initial mass (>8 solar masses) has some distinct differences from those of solar and intermediate-mass stars. We show how multiple-shell burning can lead to core-collapse supernovae, which are important in generating elements heavier than iron. Some supernovae can lead to the curious stellar endpoints of neutron stars and black holes.