Page:The Meaning of Relativity - Albert Einstein (1922).djvu/111

From the known numerical value of $K$, it therefore follows that

$\kappa ={\frac {8\pi K}{c^{2}}}={\frac {8\pi \cdot 6.67\cdot 10^{-8}}{9\cdot 10^{20}}}=1.86\cdot 10^{-27}.$

(105a)

From (101) we see that even in the first approximation the structure of the gravitational field differs fundamentally from that which is consistent with the Newtonian theory; this difference lies in the fact that the gravitational potential has the character of a tensor and not a scalar. This was not recognized in the past because only the component $g_{44}$, to a first approximation, enters the equations of motion of material particles.

In order now to be able to judge the behaviour of measuring rods and clocks from our results, we must observe the following. According to the principle of equivalence, the metrical relations of the Euclidean geometry are valid relatively to a Cartesian system of reference of infinitely small dimensions, and in a suitable state of motion (freely falling, and without rotation). We can make the same statement for local systems of co-ordinates which, relatively to these, have small accelerations, and therefore for such systems of co-ordinates as are at rest relatively to the one we have selected. For such a local system, we have, for two neighbouring point events,

$ds^{2}=-dX_{1}^{2}-dX_{2}^{2}-dX_{3}^{2}+dT^{2}=-dS^{2}+dT^{2}$

where $dS$ is measured directly by a measuring rod and $dT$ by a clock at rest relatively to the system: these are