D O C . 1 7 G R AV I TAT I O N A N D E L E C T R I C I T Y 4 1 or .. (11a) This symmetry property in the first two indices, however, is incompatible with the antisymmetry in the last two, as the following series of equations shows us: . In connection with (11a), this demands that all Δ vanish. The Γ’s are thus symmet- ric in the last two indices, just as in Riemannian geometry. Equations (10a) can then be solved in the familiar way, and one obtains . (12) Equation (12) together with (4) is the well-known law of gravity. If we had pre- sumed the symmetry of the ’s from the outset in §1, we would have arrived di- rectly at (12) and (4). This seems to me to be the simplest and most compact derivation of the gravitational equations in a vacuum. It must therefore be regarded as natural to attempt to cover the law of electromagnetism by generalizing precisely this consideration. If we had not presumed the vanishing of the , we could not have inferred the well-known law of the pure gravitational field from the assumption of symmetry for the in the manner indicated. However, if we had presumed the symmetry of the and the , the vanishing of the would have been a consequence of (9), resp. of (10a) and (7). We would then also have arrived at the law of the pure gravitational field. §3. Relation to Maxwell’s Theory. If an electromagnetic field is present, i.e., if the , or the , contain an antisymmetric component, then the equations (10a) cannot be solved for the , which makes the whole system much more complex. However, we can solve the equations if we confine ourselves to investigating the first approximation. Let us do so, and again presume that the vanish. Hence we start with the ansatz , (13) where we take the to be symmetric and the to be antisymmetric. We take the and the to be infinitely small at first order. Quantities of second or higher order are neglected. The are then likewise infinitely small at first order. Under these conditions, the equations (10a) take on the simpler form Δν μα , Δμ, αν + 0 = Δν μα , Δμ, να = Δμ να , αν –Δμ, μν –Δα, Δα νμ , Δν αμ , μα , –Δν = = = = = [p. 417] Γμν α 1 2 -- - gαβ¨ ∂xν ∂gμβ ∂xμ ∂gνβ ∂xβ ∂gμν· –+ © ¹ ¸ § = gμν φτ gμν gμν Γμν α φα gμν gμν Γμν α φμ gμν δμν – γμν φμν + + = γμν φμν γμν φμν Γμν α