Page:The Meaning of Relativity - Albert Einstein (1922).djvu/109

we put

$\left.{\begin{aligned}T^{\mu \nu }&=\sigma {\frac {dx_{\mu }}{ds}}{\frac {dx_{\nu }}{ds}}\\ds^{2}&=g_{\mu \nu }dx_{\mu }dx_{\nu }\end{aligned}}\right\}.$

(102)

In this, $\sigma$ is the density at rest, that is, the density of the ponderable matter, in the ordinary sense, measured with the aid of a unit measuring rod, and referred to a Galilean system of co-ordinates moving with the matter.

We observe, further, that in the co-ordinates we have chosen, we shall make only a relatively small error if we replace the $g_{\mu \nu }$ by $-\delta _{\mu \nu }$, so that we put

$ds^{2}=-\sum dx_{\mu }^{2}.$

(102a)

The previous developments are valid however rapidly the masses which generate the field may move relatively to our chosen system of quasi-Galilean co-ordinates. But in astronomy we have to do with masses whose velocities, relatively to the co-ordinate system employed, are always small compared to the velocity of light, that is, small compared to 1, with our choice of the unit of time. We therefore get an approximation which is sufficient for nearly all practical purposes if in (101) we replace the retarded potential by the ordinary (non-retarded) potential, and if, for the masses which generate the field, we put

${\frac {dx_{1}}{ds}}={\frac {dx_{2}}{ds}}={\frac {dx_{3}}{ds}}=0,{\frac {dx_{4}}{ds}}={\frac {{\sqrt {-1}}dl}{dl}}={\sqrt {-1}}.$

(103a)

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