In this chapter, we concentrate on a particular endpoint of stellar evolution – the state of ongoing gravitational collapse leading to a black hole. This possibility must exist in nature because there is a maximum amount of nonrotating matter that can be supported against gravitational collapse by Fermi pressure or nuclear forces. This mass is in the neighbourhood of two solar masses. (The exact value is uncertain because our knowledge of the properties of matter above nuclear densities is uncertain.) There are many stars more massive than this upper limit, so it is likely that some must wind up in a state of ongoing collapse. This chapter also explores the properties of this state.

Cosmology is the part of science concerned with the structure and evolution of the universe on the largest scales of space and time. Gravity governs the structure of the universe on these scales and determines its evolution. General relativity is thus central to cosmology, and cosmology is one of the most important applications of general relativity. Our understanding of the universe on the largest scales of space and time has increased dramatically in recent years – both observationally and theoretically. This book concentrates on the role of relativistic gravity in cosmology, introducing only the most basic observational facts and working out the simplest theoretical models. This chapter sketches the three basic observational facts about our universe on the largest distance scales: the universe consists of stars and gas in gravitationally bound collections of matter called galaxies, diffuse radiation, dark matter of unknown character, and vacuum energy; the universe is expanding; averaged over large distance scales, the universe is isotropic and homogeneous.

The Schwarzschild black holes discussed in Chapter 12 are not the most general black hole spacetimes predicted by general relativity. They are simple objects – exactly spherically symmetric and characterized by a single parameter, the total mass. Remarkably, the most general stationary black hole solutions of the vacuum Einstein equation are not much more complicated. They are described by the family of geometries discovered by Roy Kerr in 1963, and are called Kerr black holes. Members of the family depend on just two parameters – the total mass and total angular momentum. Kerr black holes are the rotating generalizations of the Schwarzschild black hole. This chapter gives an elementary introduction to their properties.

This chapter completes the description of the Einstein equation by finding the correct measure of energy density and the more general measure of spacetime curvature. A density is a quantity per unit spatial volume, such as rest-mass density, charge density, number density, energy density, and so on. The chapter begins by discussing how densities are represented in special and general relativity – for example, the densities of energy and momentum, and their conservation.

This introductory chapter gives a brief survey of some of the phenomena for which classical general relativity is important, primarily at the largest scales, in astrophysics and cosmology. The origins of general relativity can be traced to the conceptual revolution that followed Einstein’s introduction of special relativity in 1905. Newton’s centuries-old gravitational force law is inconsistent with special relativity. Einstein’s quest for a relativistic theory of gravity resulted not in a new force law or a new theory of a relativistic gravitational field, but in a profound conceptual revolution in our views of space and time. Four facts explain a great deal about the role gravity plays in physical phenomena. Gravity is a universal interaction, in Newtonian theory, between all mass, and, in relativistic gravity, between all forms of energy. Gravity is always attractive. Gravity is a long-range interaction, with no scale length. Gravity is the weakest of the four fundamental interactions acting between individual elementary particles at accessible energy scales.