Page:The Meaning of Relativity - Albert Einstein (1922).djvu/104

(95), which makes it impossible to conclude the existence of an integral equation of the form of (49). The gravitational field transfers energy and momentum to the "matter," in that it exerts forces upon it and gives it energy; this is expressed by the second term in (95).

If there is an analogue of Poisson's equation in the general theory of relativity, then this equation must be a tensor equation for the tensor $g_{\mu \nu }$ of the gravitational potential; the energy tensor of matter must appear on the right-hand side of this equation. On the left-hand side of the equation there must be a differential tensor in the $g_{\mu \nu }$. We have to find this differential tensor. It is completely determined by the following three conditions:—

1. It may contain no differential coefficients of the $g_{\mu \nu }$ higher than the second.

2. It must be linear and homogeneous in these second differential coefficients.

3. Its divergence must vanish identically.

The first two of these conditions are naturally taken from Poisson's equation. Since it may be proved mathematically that all such differential tensors can be formed algebraically (i.e. without differentiation) from Riemann's tensor, our tensor must be of the form

$R_{\mu \nu }+\alpha g_{\mu \nu }R$

in which $R_{\mu \nu }$ and $R$ are defined by (88) and (89) respectively. Further, it may be proved that the third condition requires $\alpha$ to have the value $-{\frac {1}{2}}$. For the law