Introduction

In Euclidean geometry or in the pseudo-Euclidean spacetime of special relativity, the geometrical properties are invariant under translations and rotations. The same is not necessarily true of the non-Euclidean spacetimes of general relativity. As we shall see in Chapter 8, the spacetime geometry is intimately related to the distribution of gravitating matter (and energy). A completely general spacetime arising from an arbitrary distribution of gravitating objects will not have any symmetries at all. Such cases are difficult to solve as solutions of Einstein's gravitational equations. It is, however, easier to solve problems where mass distributions have certain symmetries. For example, a point mass in an otherwise empty space is expected to generate a solution that has spherical symmetry about that point. Cases like these may be looked upon as approximations to reality. A similar approach is adopted in Newtonian gravitation. For example, as a first approximation the gravitating masses in the Solar System (the Sun and the planets) are treated as spherical distributions. In this chapter we will look at certain symmetric spacetimes that will be of use in solving specific problems in general relativity. The main question that we shall begin with is that of how to identify a symmetry in a given spacetime. How do we discover an intrinsic property like symmetry, when given the spacetime metric?

We will have occasion to use symmetric and antisymmetric tensors.