What are far fields?

The linearized theory of gravitation is based on the presumption that over whole regions of space, at any rate in the vicinity of the sources of the field, the gravitational field is weak, and the metric deviates only slightly from that of a Minkowski space. In nature we often meet a situation in which a distribution of matter (a satellite near the Earth, the Earth, the planetary system, our Galaxy) is surrounded by vacuum, and the closest matter is so far away that the gravitational field is weak in an intermediate region. In the neighbourhood of the sources, however, the field can be strong.

If such an intermediate region exists, and far away sources are not present or their influence can be neglected, then we speak of the far field of the configuration in question (Fig. 28.1). Notice that here, by contrast, for example, to most problems in electrodynamics, we may not always assume an isolated matter distribution which is surrounded only by a vacuum. The assumption of a void (the ‘infinite empty space’) into which waves pass and disappear contradicts the basic conception of General Relativity; also the fact that we orient our local inertial system towards the fixed stars indicates that we must always in principle take into account the existence of the whole Universe whenever we examine the properties of a part of the Universe.

In the examples and applications considered up until now we have always correctly taken into account the non-linearity of the Einstein equations, but most of the properties and effects discussed do not differ qualitatively from those of other classical (linear) fields. Now, in the discussion of black holes and of cosmological models, we are going to encounter properties of the gravitational field which deviate clearly from those of a linear field. The structure of the space-time is essentially changed by comparison with that of Minkowski space, and essentially new types of questions arise.

How does one find the fundamental physical laws?

Before turning in the next chapter to the laws governing the gravitational field, that is, to the question of how the matter existing in the universe determines the structure of the Riemannian space, we shall enquire into the physical laws which hold in a given Riemannian space; that is to say, how a given gravitational field influences other physical processes. How can one transcribe a basic physical equation, formulated in Minkowski space without regard to the gravitational force, into the Riemannian space, and thereby take account of the gravitational force?

In this formulation the word ‘transcribe’ somewhat conceals the fact that it is really a matter of searching for entirely new physical laws, which are very similar to the old laws only because of the especially simple way in which the gravitational field acts. It is clear that we shall not be forced to the new form of the laws by logical or mathematical considerations, but that we can attain the answer only by observation and experiment. In searching for a transcription principle we therefore want our experience to be summarized in the simplest possible formulae.

In the history of relativity theory the principle of covariance plays a large rôle in this connection. There is no clear and unique formulation of this principle; the opinions of different authors diverge here.

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. […]