Exact Formulation of the General Principle of Relativity

We are now in a position to replace the provisional formulation of the general principle of relativity given in [chapter:XVIII]Section XVIII by an exact formulation. The form there used, "All bodies of reference $K$ , $K^{'}$ , etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion," cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature."

We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time

variables $x$ , $y$ , $z$ , $t$ , of a (Galileian) reference-body $K$ by the space-time variables $x^{'}$ , $y^{'}$ , $z^{'}$ , $t^{'}$ , of a new reference-body $K^{'}$ . According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ ,

the equations must pass over into equations of the same form; for every transformation (not only the Lorentz

transformation) corresponds to the transition of one Gauss co-ordinate system into another.

If we desire to adhere to our "old-time" three-dimensional

view of things, then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains, i.e. to those in which no gravitational field exists. In this connection a Galileian reference-body

serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of "isolated" material points holds relatively to it.

Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies

also. A gravitational field of a special kind is

then present with respect to these bodies (cf. Sections [chapter:XX]XX and [chapter:XXIII]XXIII).

In gravitational fields there are no such things as rigid

bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by