The theory of special relativity incorporates a modification of Newtonian mechanics together with electromagnetism. A natural question to consider is how gravitation fits into the framework of relativity. In this chapter we focus our analysis of this question along two main ideas, that of the equivalence between uniform acceleration and a uniform gravitational field, and that of the gravitational redshift. These will lead us to the Einstein equation, which we then show can be given a variational formulation. We present some solutions of the Einstein equation, with particular attention given to the Schwarzschild spacetime and its Kruskal extension.

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.