As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.

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.

The basic principles of general relativity are reviewed, in particular the different forms of the equivalence principle: the weak, Einstein, and strong equivalence principles. The concept of a metric is introduced within special relativity. The Einstein equations are derived in an heurisitic manner including the Christoffel symbols, the Ricci tensor, and the Ricci scalar. The Schwarzschild as the solution of Einstein‘s equation in vacuum are explicitly derived. The notion of the energy–momentum tensor, as the source term of the Einstein equations, is discussed in terms of the four-momentum of particles. For bulk matter, the definition of an ideal fluid is given. The conservation of the energy–momentum tensor in curved space-time is discussed. The Einstein equations are solved for a sphere of an ideal fluid to arrive at the Tolman–Oppenheimer–Volkoff equations, the central equations for the investigation of compact stars. Finally the analytically known solution for a sphere of an incompressible fluid, the Schwarzschild solution, is derived and used to set the Buchdahl limit on the compactness of a compact star.

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.