TY - JOUR
T1 - Gauss's theorem in general relativity
JF - Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences
JO - Proc R Soc Lond A Math Phys Sci
SP - 354
LP - 363
M3 - 10.1098/rspa.1936.0056
VL - 154
IS - 882
AU -
Y1 - 1936/04/01
UR - http://rspa.royalsocietypublishing.org/content/154/882/354.abstract
N2 - 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, G44, of the Einstein tensor Gμv, can be ressed as a divergence in the 3-way, x4 = constant. This can be seen once if the line-element is expressed in the form ds2 = V2 (dx4)2 - ajk dxj dxk, (j, k = 1, 2, 3), s can always be done without any loss of generality. Then G44 = - V Δ2 V + V {∂/∂x4 (ajkΩjk) + VajpakqΩjkΩpq}, where 2VΩjk = ∂ajk/∂x4, Δ2V Beltrami's second differential parameter for the 3-way x4 = constant. Whittaker's investigations refer to the static field in which x4 the temporal coordinate and ∂ajk/∂x4 = 0. In the work of Ruse the tem of 3-ways, x4 = constant, is chosen so that ∂ (ajkΩjk/∂x4 = - Vajp akq ΩjkΩpq.
ER -