This last part of the book introduces the Einstein equation – the basic equation of general relativity, in much the same way that Maxwell’s equations are the basic equations of electromagnetism. Geometries such as the Schwarzschild geometry, or those of the FRW cosmological models, are particular solutions of the Einstein equation. Just three new mathematical ideas are needed to give an efficient and standard discussion of the Einstein equation: a more precise definition of vectors in terms of directional derivatives; the notion of dual vectors as a linear map from vectors to real numbers; and the covariant derivative of a vector field in curved spacetime. These mathematical concepts are introduced in this chapter.

This chapter (and the next one) covers some basic mathematics needed to describe four-dimensional curved spacetime geometry. Much of this is a generalization of the concepts introduced in Chapter 5 for flat spacetime. Coordinates are a systematic way of labeling the points of spacetime. The choice of coordinates is arbitrary as long as they supply a unique set of labels for each point in the region they cover, but for a particular problem, one coordinate system may be more useful than another. We then define the metric for a general geometry and explain common conventions. We show how to compute lengths of curves, areas, three-volumes, and four-volumes for a given metric. Concepts such as wormholes, extra dimensions, the Lorentz hyperboloid, and null spaces are introduced.

The orbits of test particles and light rays in the Schwarzschild geometry that were worked out in Chapter 9 are not only important for the delicate tests of general relativity in the solar system discussed in Chapter 10. They are also central to a number of astrophysical applications. This chapter introduces three of these applications – gravitational lensing, relativistic frequency shifts from accretion disks, and weighing stars in binary pulsars, which act both as a laboratory for general relativity and and a tool for astronomy. Some tests of Einstein’s theory were the subject of the previous chapter; some of its applications are the subject of this.

The stellar endpoint leading to its collapse to a black hole was described in Chapter 12. This chapter explores the other possibility, where a star is supported against gravity by a nonthermal source of pressure. This is realized in nature by white dwarf stars and neutron stars. Unlike black holes, which can be understood entirely in the context of general relativity, an understanding of the stars at the endstate of stellar evolution requires almost all of the rest of physics in some way. We cannot hope to review the range of physics necessary for a complete understanding of the equilibrium endstates of stellar evolution, but we can isolate the essential role of gravitational physics and discuss the overall structure of these stars. To an excellent approximation, the properties of the matter relevant for the gross structure of neutron stars and white dwarfs can be summarized by an equation of state relating the pressure of an ideal matter fluid to its energy density.

The simplest curved spacetimes of general relativity are the ones with the most symmetry, and the most useful of these is the geometry of empty space outside a spherically symmetric source of curvature – for example, a spherical star. This is called the Schwarzschild geometry. To an excellent approximation, this is the curved spacetime outside the Sun and therefore leads to the predictions of Einstein’s theory most accessible to experimental test. In this chapter, we explore the geometry of Schwarzschild’s solution, assuming it’s given. We will concentrate on predicting the orbits of test particles and light rays in the curved spacetime of a spherical star that exhibit some of the famous effects of general relativity – the gravitational redshift, the precession of the perihelion of a planet, the gravitational bending of light, and the time delay of light.

Black holes are the outcome of unhalted gravitational collapse. Gravitational collapse to a black hole occurs on a wide range of mass scales in the universe because gravity is an attractive and universal force. This chapter describes black holes of three different origins, with three different mass scales, how they have or could be identified, and sketches how they are at the heart of some of the most energetic phenomena in astrophysics. These are black holes in X-ray binaries, black holes in galaxy centers, and exploding primordial black holes. Black holes are not only interesting because they check general relativity, they also contribute to the explanation of frontier astrophysical phenomena.

Chapter 9’s analysis of the orbits of test particles and light rays in the Schwarzschild geometry identified four effects of general relativity that can be tested in the solar system: the gravitational redshift, the deflection of light by the Sun, the precession of the perihelion of a planetary orbit, and the time delay of light. This list does not exhaust the tests that can be carried out in the solar system, but describes some of the more important ones. Experiments that measure these effects confirm the predictions of general relativity in the solar system to a typical accuracy of a fraction of 1 percent. The discussion in this chapter is not a review of the experimental situation in general relativity either in the past or at the time of writing. Instead, it presents a discussion of representative experiments that are currently among the most accurate, but are not necessarily the most accurate.

The Schwarzschild geometry that underlies much of the physics in previous chapters is exactly spherically symmetric. It is an excellent approximation to the geometry outside a nonrotating star, and is the exact geometry outside a nonrotating black hole. However, no body in nature is exactly nonrotating. The Sun, for example, is rotating at the equator with a period of approximately 27 days, and it is not exactly spherically symmetric, but is slightly squashed along the rotation axis (it is less than 1 part in 100,000 longer than a diameter along the rotation axis). The small value of that difference is why the Schwarzschild geometry is an excellent approximation to the curved spacetime geometry outside the Sun. The curved spacetimes produced by rotating bodies have a richer and more complex structure than the Schwarzschild geometry. This chapter explores one simple example of a gravitomagnetic effect – the dragging of inertial frames by a slowly rotating body.

We will trace out some parts of the path that led Einstein to a new theory of gravity that is, unlike Newtonian gravity, consistent with the principle of relativity. The result will be general relativity, a theory that is qualitatively different from Newtonian gravity. In general relativity, gravitational phenomena arise not from forces and fields, but from the curvature of four-dimensional spacetime. The starting point for these considerations is the equality of gravitational and inertial mass, one of the most accurately tested principles in all physics. This leads to Einstein’s equivalence principle, the idea that there is no experiment that can distinguish a uniform acceleration from a uniform gravitational field – the two are fully equivalent.

Which of the four-parameter family of Friedman–Robertson–Walker (FRW) cosmological models best fits our universe and why? This chapter addresses these two central questions for observation and theory in cosmology. Of the four parameters that define an FRW model, only two are determined by observations so far: the Hubble constant; and the ratio of energy density in radiation to the critical density. To determine the others, the spacetime geometry of the universe must be measured on large scales through a study of how matter moves through it. We describe two illustrative ways of doing that – one based on observations of distant supernovae, and the other on observations of the cosmic background radiation. Remarkably, the best cosmological parameter values are consistent with the universe being spatially flat – right on the borderline between positive and negative spatial curvature.

The laws of Newtonian mechanics have to be changed to be consistent with the principles of special relativity introduced in the previous chapter. This chapter describes special relativistic mechanics from a four-dimensional, spacetime point of view. Newtonian mechanics is an approximation to this mechanics of special relativity that is appropriate when motion is at speeds much less than the velocity of light in a particular inertial frame. We begin with the central idea of four-vectors, defined as a directed line segment in four-dimensional flat spacetime, and how to manipulate them. Special relativistic kinematics shows how four-vectors are used for describing the motion of a particle in spacetime terms. Concepts such as four-velocity and four-momentum are introduced. We will posit the principle of extremal proper time for a free particle in curved spacetime, and use it to derive the free particle equation of motion.