Page:The principle of relativity (1920).djvu/231

 If we further assume that the gravitation-field is quasi-static, i.e., it is limited only to the case when the matter producing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiations of the positional co-ordinates on the right-hand side with respect to time, so that we get

(67) d^2x_{τ}/dt^2 = -1/2 [part]g_{4 4}/[part]x_{τ} (τ, = 1, 2, 3)

This is the equation of motion of a material point according to Newton's theory, where g_{4 4}/_{2} plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material point, only the component g_{4 4} of the fundamental tensor appears.

Let us now turn to the field-equation (53). In this case, we have to remember that the energy-tensor of matter is exclusively defined in a narrow sense by the density ρ of matter, i.e., by the second member on the right-hand side of 58 [(58a, or 58b)]. If we make the necessary approximations, then all component vanish except

Τ_{4 4} = ρ = Τ.

On the left-hand side of (53) the second term is an infinitesimal of the second order, so that the first leads to the following terms in the approximation, which are rather interesting for us;

[part]/[part]x_{1} [μν 1] + [part]/[part]x_{2} [μν 2] + [part]/[part]x_{3} [μν 3] - [part]/[part]x_{4} [μν 4].

By neglecting all differentiations with regard to time, this leads, when μ = ν =4, to the expression

-1/2 ([part]^2g_{4 4}/[part]x_{1}^2 + [part]^2g_{4 4}/[part]x_{2}^2 + [part]^2g_{4 4}/[part]x_{3}^2) = -1/2 V^2 g_{4 4}.