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Even before Newton had written down the laws of motion, Galileo had observed that it is impossible to detect uniform motion in an enclosed space. If you do experiments in the cabin of a ship on a calm sea—for example, by dripping water into a bucket or by observing the flight of insects—then you will get the same results whether the ship is moving uniformly or at rest. The common motion of the ship and the objects of the experiment has no detectable effect.

Before we take further the development of the relativistic theory of gravity, we need to establish an appropriate mathematical framework for special relativity. This must survive in the general theory as the formalism for describing local observations made by observers in free-fall in a gravitational field. In this chapter, familiarity with special relativity is assumed: the purpose is not to derive special relativity, but to introduce the language in which it will be extended to general relativity.

Einstein’s general theory has at its heart an equation that, like Poisson’s equation, relates the gravitational field of a distribution of matter to its energy density. The quantity that encodes energy density in special relativity is a symmetric two-index tensor called the energy-momentum tensor . We introduce it first in the simplest case of a noninteracting distribution of particles, and then extend the definition to fluids and to electromagnetic fields.

We are now ready to make the transition from Minkowski’s space—time of special relativity to the curved space—time of general relativity. We build on two foundations: first, the equivalence principle, the local equivalence of the effects of acceleration and gravity, and second, the well-established apparatus of special relativity theory, applied over short times and small distances in free-fall. Our starting point is the following.

We have seen that the space—time of general relativity is a four-dimensional manifold and that gravity is encoded in the metric tensor. It manifests itself in the relative acceleration of local inertial frames, and thus in variations in the metric from event to event.

The relative acceleration of two nearby particles in free-fall is determined by the equation of geodesic deviation

In this chapter, we find the gravitational field outside a spherical body of mass m . That is, we find the solution of the vacuum equation analogous to the Newtonian potential

We now look at particle motion in the Schwarzschild background. Our main aim is to derive corrections to Kepler’s laws, so we think of the gravitational field as that of the sun. By a ‘particle’, we mean a very small body, such as a planet, whose own gravitational field can be ignored.

We now look more closely at what happens at the Schwarzschild radius, r = 2 m . It is clear that something goes wrong there in the formula (7.6) for the metric coefficients. We show, however, that the singularity is not in the space—time geometry itself, but simply in the coordinates in which it is expressed. The singular behaviour at r = 2 m goes away when we make an appropriate change of coordinates.

The Schwarzschild metric gives us some of the classic tests of relativity: the bending of light, Mercury’s perihelion precession, and other predictions from the analysis of geodesic motion. It also allows us to make some dramatic predictions about the end states of the gravitational collapse of stars to black holes. To find deeper tests, we have to look for more subtle effects of general relativity, which cannot be seen in the Schwarzschild space—time. One is the ‘dragging of inertial frames’ by a rotating body. The predictions here allow the testing of Einstein’s equations as well as of the geometric model of space—time. They can be observed in the effect of the earth’s rotation on an orbiting gyroscope.