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.

This chapter introduces compact objects, white dwarfs, neutron stars, and black holes. The properties of compact objects are summarized as the typical radius, mass, and compactness. These compact objects are the final end point in stellar evolution. The stellar evolution in terms of the mass of the star is outlined, focusing on the burning stages and the final collapse of the star, either as a white dwarf or in a core-collapse supernova. Historical notes are given for the discovery of white dwarfs and neutron stars.

I distinguish between two versions of the black hole information-loss paradox. The first arises from apparent failure of unitarity on the spacetime of a completely evaporating black hole, which appears to be non-globally hyperbolic; this is the most commonly discussed version of the paradox in the foundational and semipopular literature, and the case for calling it `paradoxical' is less than compelling. But the second arises from a clash between a fully statistical-mechanical interpretation of black hole evaporation and the quantum-field-theoretic description used in derivations of the Hawking effect. This version of the paradox arises long before a black hole completely evaporates, seems to be the version that has played a central role in quantum gravity, and is genuinely paradoxical. After explicating the paradox, I discuss the implications of more recent work on AdS/CFT duality and on the `Firewall paradox', and conclude that the paradox is if anything now sharper. The article is written at a (relatively) introductory level and does not assume advanced knowledge of quantum gravity.

After many years of effort, Einstein discovered the general theory of relativity, the extension of special relativity to include the force of gravity. To do this, he extended the four-vector structure of special relativity to more general Riemannian geometries in which space-time is bent under the influence of gravity. The theory has successfully passed all the high precision tests now available. Predictions include the existence of black holes and gravitational waves. Both have now been convincingly observed. In this chapter, the basic concepts behind the theory are described and then illustrated by analysis of the Schwarzschild metric.

We begin with a brief description of the work on (a) the regularization of the stress-energy tensor of quantum fields in Schwarzschild spacetime in the 80s and (b) the black hole end-state and information-loss issues in the 80s, the ‘black hole complementarity principle’ of the 90s and the recent ‘firewall’ conjecture and its controversies. We then treat two classes of problems: (1) the backreaction of Hawking radiation on a black hole in the quasi-stationary regime, which occupies the longest span of a black hole’s life, and (2) the metric fluctuations of the event horizon of an evaporating black hole. In (1) the far field case can be solved analytically via the influence functional, highlighting nonlocal dissipation and colored noise; for the near horizon case we describe a strategy by Sinha et al. for treating the backreaction and fluctuations. In (2) we describe Bardeen’s model and discuss the results of Hu and Roura, who reached the same conclusion as Bekenstein, namely, that even for states regular on the horizon the accumulated fluctuations become significant by the time the black hole mass has changed substantially, well before reaching the Planckian regime. These results have direct implications for the end-state issue.