4 3 0 D O C . 4 4 3 G E N E R A L R E L A T I V I T Y A N D M O T I O N Thus, in order for the metric to remain regular in the presence of an external field in the vicinity of a singular point, the field strength of the external field must vanish at the singular point itself. In this sense, the equilibrium condition is contained in the field equations.[22] Already on the basis of this result, one will come to the con- clusion that in all generality the law of motion of the singularities is contained in the field equations. This will be shown more generally in the following.[23] §2. A Surface Theorem Equivalent to the Field Equations The general idea at the root of the following considerations and calculations is as follows. It is well known that the gravitational field equations are related to lin- ear differential equations whose solutions differ very little from the solutions to the exact equations in the cases that really matter. However, we saw that not all solu- tions to the approximative equations correspond to exact solutions. For example, according to the approximative equations, there is a solution that corresponds to a mass point at rest in a homogeneous gravitational field.[24] According to the exact equations, such an exact solution does not exist, as we saw—at least if we require of the metric field to have no singularities outside of the mass point. We must there- fore look for additional conditions which solutions to the approximative equations must adhere to, in order to approximate exact solutions. These conditions, to be de- rived from the exact field equations, must refer to the field in the immediate vicin- ity of a singular world line. For this we need a surface theorem, as already put forward by Hilbert and Klein in a similar vein.[25] We set out from the Hamilton function , (9) and derive from it the field equations, in that we vary the and the independently.[26] The field equations then read , (10) , (11) where signifies the derivative . If one multiplies (10) by and (11) by , one obtains after simple rearrangement the equation . (12) [p. 7] H gμν© α ∂Γμν ∂xα ----------- -– α ∂Γμα ∂xν ------------ Γμβ α Γνα β Γμν α Γαβ¹ β + + + § · gμνRμν = = gμν Γμνα ∂gμν ∂H 0= α ∂Γμν ∂H ∂xτ© ∂ τ , α ∂Γμν ∂H ¹ § · – 0 = Γμν τ , α ∂Γμν ∂xτ ----------- - δgμν α δΓμν ∂xτ© ∂ τ , α ∂Γμν ∂H α δΓμν¹ § · – δH 0=