D O C . 4 4 3 G E N E R A L R E L AT I V I T Y A N D M O T I O N 4 3 3 point is replaced by a singularity.[31] By adding electromagnetic terms, it could eas- ily be shown that this also applies to an electrically charged mass point under the influence of a gravitational and an electromagnetic field. One then only has to in- clude the components of the electromagnetic energy tensor in addition to the in (17). In order to obtain the forces acting on the singular point in terms of the mass and the external field strength, we must consider a question that is important to the en- tire problem. By help of our present tools we can only obtain very few exact solu- tions to the gravitational field equations but solutions to first approximation are easy, because the relevant differential equations are linear. This approximation is characterized by setting , (18) where the ’s are small compared to 1 and the squares and products of the γ’s (and their derivatives) are neglected: we shall call the ’s small quantities of first order. We just saw that not nearly all solutions of those linear differential equations correspond to exact solutions. For example, a solution of the linear equations cor- responding to a pointlike singularity at rest in a homogeneous gravitational field does exist, whereas there is no exact solution of this kind[32] because the equilibrium conditions, which we have just derived from the exact equations, are violated in this case. With this state of affairs, the question arises: What additional conditions must an approximate solution satisfy to correspond to an exact solution? We must at least demand that the second and higher approximations of the will not appear, or if they appear, that they are not of the same order as the .[33] This is the reason why we rearranged things such that we obtained equation (16) we had to do so to arrive at a suitable equilibrium condition. For the ’s as well as the are small at first order, and therefore the are small at second order. If we added second-order terms to the , then the would obtain third-order terms, which we may neglect. Thus, even though (15)–(16) refer to second-order quanti- ties, it is in the special case of equilibrium permissible to neglect quantities of second order in the . Then we can insert for the in (18) solutions of the lin- earized field equations (10). Within the domain of this approximation, it is permis- sible to regard the field ( ) in the vicinity of a singularity as the sum of two terms: an “inner” field and an “outer” field .[34] is regular at the sin- gular point. For , we can insert the static solution, which we write in the form: tσ ν gμν δμν – γμν += γμν γμν [p. 10] γμν γμν Γμν α gσ μν tν α gμν tσ ν gμν γμν γμν γμν γμν γμν γμν