Gauss's theorem in general relativity

Abstract

In a recent paper Whittaker has given a generalization of Gauss’s theorem on the Newtonian potential which is valid in Einstein’s General theory of Relativity, and a further extension of this result has been ained by Ruse. Both of these extended forms of Gauss’s theorem end upon the fact that, under the special conditions postulated by the hors, one of the components, G 44 , of the Einstein tensor G μv , can be ressed as a divergence in the 3-way, x 4 = constant. This can be seen once if the line-element is expressed in the form ds 2 = V 2 ( dx 4 ) 2 - a jk dx j dx k , ( j, k = 1, 2, 3), s can always be done without any loss of generality. Then G 44 = - V Δ 2 V + V {∂/∂ x 4 ( a jk Ω jk ) + V a jp a kq Ω jk Ω pq }, where 2VΩ jk = ∂ a jk /∂ x 4 , Δ 2 V Beltrami's second differential parameter for the 3-way x 4 = constant. Whittaker's investigations refer to the static field in which x 4 the temporal coordinate and ∂ a jk /∂ x 4 = 0. In the work of Ruse the tem of 3-ways, x 4 = constant, is chosen so that ∂ ( a jk Ω jk /∂ x 4 = - V a jp a kq Ω jk Ω pq .