Appendix B - Spherically symmetric solutions and Birkhoff's theorem

Summary

We wish to consider Einstein's equations in the case of a spherically symmetric space–time. One might regard the essential feature of a spherically symmetric space–time as the existence of a world-line ℒ such that the space–time is spherically symmetric about ℒ. Then all points on each spacelike two-sphere _d centred on any point p of ℒ, defined by going a constant distance d along all geodesies through p orthogonal to ℒ, are equivalent. If one permutes directions at p by use of the orthogonal group SO(3) leaving ℒ invariant, the space–time is, by definition, unchanged, and the corresponding points of _d are mapped into themselves; so the space–time admits the group SO(3) as a group of isometries, with the orbits of the group the spheres _d. (There could be particular values of d such that the surface _d was just a point p′; then p′ would be another centre of symmetry. There can be at most two points (p′ and p itself) related in this way.)

However, there might not exist a world-line like ℒ in some of the space–times one would wish to regard as spherically symmetric. In the Schwarzschild and Reissner–Nordström solutions, for example, space–time is singular at the points for which r = 0, which might otherwise have been centres of symmetry. We shall therefore take the existence of the group SO(3) of isometries acting on two-surfaces like _d as the characteristic feature of a spherically symmetric space–time. Thus we shall say that space–time is spherically symmetric if it admits the group SO(3) as a group of isometries, with the group orbits spacelike two-surfaces.