Page:The principle of relativity (1920).djvu/174

 measurements with rods and clocks. The g_{τσ}'s are here to be so chosen, that g_{τσ} = g_{στ}; the summation is to be extended over all values of σ and τ, so that the sum is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.

From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviour of g_{στ} in a finite region it is possible to choose the system of co-ordinates in such a way that g_{στ} assumes constant values—

{ -1, 0, 0, 0 (4)  {  0 -1  0  0      {  0  0 -1  0      {  0  0  0 +1

We would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible.

From the considerations in § 2 and § 3 it is clear, that from the physical stand-point the quantities g_{στ} are to be looked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. The g_{στ}'s then have the values given in (4). A free material point moves with reference to such a system uniformly in a straight-*line. If we now introduce, by any substitution, the space-time co-ordinates x_{1}x_{4} then in the new system g_{μν}'s are no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the new