4 3 2 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 known as the “energy pseudotensor” of the gravitational field.[28] The energy law of the gravitational field results from (15a) if the are held constant. The integration of (15a) over a singularity-free region yields a surface theorem. The surface integral of A α over a (three-dimensional) hypersurface of this kind al- ways vanishes, irrespective of how the auxiliary vector (ξα) is chosen (apart from the continuity conditions apparent from the derivation). One can thus choose an ar- bitrary part of the surface, such that it is the only one on which A α does not vanish. This is what the significance of the law in examining the field in immediate prox- imity to a singular line is primarily based on. Indeed, let L be a singular line. We imagine this line wrapped along a finite distance in an infinitely narrow “sheath” M and in a sheath M′ of finite breadth. They are connected at the ends in such a way that they form the wrapping of a two-connected space, over which we inte- grate (15a). We choose the ξσ such that they themselves and their derivatives vanish everywhere on the surface, except for at a very small distance from L.[29] Then the integral of Aα over M′ vanishes, apart from the contributions that the ends reaching M make. Thus, for any choice of ξσ a state- ment results about the field immediately surrounding L, i.e., a statement about the motion of the material point. §3. Consequences of the Integral Theorem The simplest consequence that we can draw from (15a) concerns the equilibrium of the singular point in a stationary gravitational field. We first choose its singular line as the x4-axis and the ξ-vector in such a way that its first and second derivatives vanish along the inner sheath. Then the integration over the ends of the outer sheath is easily made to vanish by having both ends yield the opposites of the same amount. The integral over the inner sheath vanishes on its own and thus the integral over a spatial cross-section If we call the curly bracket in (16) , then the three-dimensional integral (17) vanishes for any σ, if integrated over a cross-section of the sheath M.[30] This is the same equilibrium condition that one would obtain by substituting the singular point by a region of material fluid flow, the way the analysis has been performed up to now in the general theory of relativity. Thus, the normal condition for the equilib- rium of a material point in a gravitational field remains the same if the material ξα [p. 9] x4 const. = tσ α tσd 1 s23 tσd 2 + s31 tσds12) 3 + ³(