The Einstein field equations

As we have already indicated more than once, the basic idea of Einstein's theory of gravitation consists of geometrizing the gravitational force, that is, mapping all properties of the gravitational force and its influence upon physical processes onto the properties of a Riemannian space. While up until the present we have concerned ourselves only with the mathematical structure of such a space and the influence of a given Riemannian space upon physical laws, we want now to turn to the essential physical question. Gravitational fields are produced by masses – so how are the properties of the Riemannian space calculated from the distribution of matter? Here, in the context of General Relativity, ‘matter’ means everything that can produce a gravitational field (i.e. that contributes to the energy-momentum tensor), for example, not only atomic nuclei and electrons, but also the electromagnetic field.

Of course one cannot derive logically the required new fundamental physical law from the laws already known; however, one can set up several very plausible requirements. We shall do this in the following and discover, surprisingly, that once one accepts the Riemannian space, the Einstein field equations follow almost directly.

The following requirements appear reasonable.

1. (a) The field equations should be tensor equations (independence of coordinate systems of the laws of nature).

2. (b) Like all other field equations of physics they should be partial differential equations of at most second order for the functions to be determined (the components of the metric tensor grrmn), which are linear in the highest derivatives.

3. […]

The Kerr solution

Most known stars are rotating relative to their local inertial system (relative to the stars) and are therefore not spherically symmetric; their gravitational field is not described by the Schwarzschild solution. In Newtonian gravitational theory, although the field certainly changes because of the rotational flattening of the star, it still remains static, while in the Einstein theory, on the other hand, the flow of matter acts to produce fields. The metric will still be time-independent (for a timeindependent rotation of the star), but not invariant under time reversal. We therefore expect that the gravitational field of a rotating star will be described by an axisymmetric stationary vacuum solution which goes over to a flat space at great distance from the source. Depending on the distribution of matter within the star there will be different types of vacuum fields which, in the language of the Newtonian gravitational theory, differ, for example, in the multipole moments of the matter distribution. One of these solutions is the Kerr (1963) solution, found almost fifty years after the discovery of the Schwarzschild metric. It proves to be especially important for understanding the gravitational collapse of a rotating star. To avoid misunderstanding we emphasize that the Kerr solution is not the gravitational field of an arbitrary axisymmetric rotating star, but rather only the exterior field of a very special source.

We shall now discuss the Kerr solution and its properties. Since its mathematical structure is rather complicated, we shall not construct a derivation from the Einstein field equations.

Gravitational forces are the only forces presently known which are long range (in contrast to the nuclear forces, for example) and which cannot be compensated (there are no negative masses). It is therefore to be expected that, for large quantities of matter distributed over wide regions of space, they will be the decisive forces, and hence the gravitational forces will determine the evolution and dynamics of the universe.

Physical laws get their importance from the fact that a single law describes many very different situations. Technically this comes out by writing the laws as differential equations (usually of second order), which admit a multitude of initial or boundary conditions. The law itself has often been found by extracting some common rules from the observed variety of effects. All these features are also present in the theory of gravitation.

In cosmology, however, we encounter a very different situation. There is only one realization of a cosmos, that which we are living in. And if there was an extra physical law for this cosmos, we could not find and prove it the usual way. That is to say, if we find a surprising new phenomenon, we cannot easily decide which of its properties are a new law, and which are due to initial conditions. Sometimes it is claimed that, in a proper theory, initial conditions should be excluded, the cosmos must not depend on them (this was one of the assumptions of the ‘inflation’ theory). Or one claims that the new law can be obtained from other principles, such us the anthropic principle.