Spherical symmetry

Abstract

This chapter serves two purposes. Firstly, a large isometry group simplifies the problem of solving Einstein’s equation considerably. Virtually all explicitly known solutions of Einstein’s equations for physically plausible matter fields have a high degree of symmetry. Secondly, a spherically symmetric spacetimes are very good descriptions of non-rotating, isolated stars and therefore of astrophysical interest. (If the star rotates the rotation axis breaks the symmetry). In Sect. 7.2 we will see that there is a unique 1-parameter family of spherically symmetric solutions to Einsteins’s equation for vacuum with vanishing cosmological constant. The parameter can be interpreted as the mass of the isolated star. If the mass of the sun is chosen, one obtains an excellent model of the gravitational field in our solar system. Some aspects of this model have been verified experimentally. These solutions also form the basis for much of our intuition of black holes.