The observations described in Chapter 17 show our universe to be approximately homogeneous and isotropic on spatial distance scales above several hundred megaparsecs. The simplest cosmological models enforce these symmetries exactly as a first approximation. For instance, the matter in galaxies and the radiation are approximated by smooth density distributions that are exactly uniform in space. Similarly, the geometry of spacetime incorporates the homogeneity and isotropy of space exactly. These simplifying assumptions define the Friedman–Robertson–Walker (FRW) family of cosmological models, which are the subject of this chapter.

Gravitational waves provide a window on the universe of astronomical phenomena that is different from any in the electromagnetic spectrum. Mass in many different varieties of motion is a source of propagating ripples in spacetime curvature. In order to interpret the observations of gravitational wave detectors on Earth and in space, it is necessary to solve the Einstein equation for the gravitational radiation produced by given sources. Predicting the gravitational radiation from strong-curvature, rapidly varying sources is a problem generally tractable only by numerical simulation of the fully nonlinear Einstein equation – a subject well beyond the scope of this book. However, some insight into the production of gravitational waves can be obtained from examining the more tractable problem of the small ripples in spacetime emitted by weak, nonrelativistic sources.

This chapter discusses the geometry of space and the notion of time assumed in Newtonian mechanics. This discussion will also serve to review aspects of mechanics and special relativity that will be important for later developments. Newtonian mechanics assumes a geometry for space and a particular idea for time. The laws of Newtonian mechanics take their standard and simplest forms in inertial frames. Using the laws of mechanics, an observer in an inertial frame can construct a clock that measures the time. Coordinate transformations can make the connection between different inertial frames. Newtonian mechanics assumes there is a single notion of time for all inertial observers. We explore Newtonian gravity and the Principle of Relativity: that identical experiments carried out in different inertial frames give identical results.

The relation between local spacetime curvature and matter energy density is given by the Einstein equation – it is the field equation of general relativity in the way that Maxwell’s equations are the field equations of electromagnetism. Maxwell’s equations relate the electromagnetic field to its sources – charges and currents. Einstein’s equation relates spacetime curvature to its source – the mass-energy of matter. This chapter gives a very brief introduction to the Einstein equation; we consider the equation in the absence of matter sources (the vacuum Einstein equation) and will include matter sources in Chapter 22. Even the vacuum Einstein equation has important implications. Just as the field of a static point charge and electromagnetic waves are solutions of the source-free Maxwell’s equations, the Schwarzschild geometry and gravitational waves are solutions of the vacuum Einstein equation.

Einstein’s 1905 special theory of relativity requires a profound revision of the Newtonian ideas of space and time that were reviewed in the previous chapter. In special relativity, the Newtonian ideas of Euclidean space and a separate absolute time are subsumed into a single four-dimensional union of space and time, called spacetime. This chapter reviews the basic principles of special relativity, starting from the non-Euclidean geometry of its spacetime. Einstein’s 1905 successful modification of Newtonian mechanics, which we call special relativity, assumed that the velocity of light had the same value, c, in all inertial frames, which requires a reexamination, and ultimately the abandonment, of the Newtonian idea of absolute time. Instead, he found a new connection between inertial frames that is consistent with the same value of the velocity of light in all of them. The defining assumption of special relativity is a geometry for four-dimensional spacetime.

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.

Mass produces spacetime curvature – that is a central lesson of general relativity. The static spherical mass of the Sun produces the Schwarzschild geometry outside it. Mass in (nonspherical, nonuniform) motion is the source of ripples of curved spacetime, which propagate away at the speed of light. These propagating ripples in spacetime curvature are called gravitational waves. Their free propagation will be discussed in this chapter. There are many important sources of gravitational waves in the universe – binary star systems, supernova explosions, collapse to black holes, and the Big Bang are all examples. Gravitational waves provide a window for exploring these astronomical phenomena that is qualitatively different from any band of the electromagnetic spectrum. However, the weakness of the gravitational interaction in everyday circumstances means that gravitational waves are not easily detected.