Page:The Meaning of Relativity - Albert Einstein (1922).djvu/83

single atoms due to their previous history if the mass and frequencies of the single atoms of the same element were always the same.

Space-time regions of finite extent are, in general, not Galilean, so that a gravitational field cannot be done away with by any choice of co-ordinates in a finite region. There is, therefore, no choice of co-ordinates for which the metrical relations of the special theory of relativity hold in a finite region. But the invariant $ds$ always exists for two neighbouring points (events) of the continuum. This invariant $ds$ may be expressed in arbitrary co-ordinates. If one observes that the local $dX_{\nu }$ may be expressed linearly in terms of the co-ordinate differentials $dx_{\nu },ds^{2}$ may be expressed in the form

$ds^{2}=g_{\mu \nu }dx_{\mu }dx_{\nu }$

(55)

The functions $g_{\mu \nu }$ describe, with respect to the arbitrarily chosen system of co-ordinates, the metrical relations of the space-time continuum and also the gravitational field. As in the special theory of relativity, we have to discriminate between time-like and space-like line elements in the four-dimensional continuum; owing to the change of sign introduced, time-like line elements have a real, space-like line elements an imaginary $ds$. The time-like $ds$ can be measured directly by a suitably chosen clock.

According to what has been said, it is evident that the formulation of the general theory of relativity assumes a generalization of the theory of invariants and the theory of tensors; the question is raised as to the