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 x1, y1, z1, t1, of a new reference-body K1.  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 20 and 23).

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 gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity.

For this reason non-rigid reference-bodies are used, which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion.  Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time.  We have to imagine each of these clocks fixed at a point on the non-rigid reference-body.  These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount.  This non-rigid reference-body, which might appropriately be termed a “reference-mollusc”, is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily.  That which