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 3 9 . (2) Independently of this affine connection we introduce a contravariant tensor density , for which we likewise do not assume any symmetries.[3] From both we form the scalar density (3) and postulate that the integral vanishes for variations of the and . These variations are carried out inde- pendently (we do not vary at the boundaries). Variation after yields the 16 equations[4] , (4) variation after initially yields the 64 equations . (5) Let us now make some considerations that allow us to substitute equations (5) by simpler ones. On the left-hand side of (5), we contract the indices ν, α, resp. μ, α, and thus obtain the equations (6) . (7) Furthermore, we introduce the , which are the normed subdeterminants of the . Thus, they satisfy the equations . We now multiply (5) by , and thus obtain an equation that, after raising an in- dex, can be written as follows , (8) if one denotes the determinant of the as g. We write equations (6) and (8) in the form , (9) Rμν Rμ να , α α ∂Γμν ∂xα ----------- -– Γμβ α Γαν β α ∂Γμν ∂xν ----------- - Γμν α Γαβ β – + + = = [p. 415] gμν H gμνRμν = ℑ x1d d x2d x3dx4 ³H = gμν Γμν α gμν Rμν 0= Γμν α ∂gμν ∂xα ---------- gβνΓβα μ gμβΓαβ– ν δα ν ∂gμβ ∂xβ ---------- gσβΓσβ¹ μ + © § · gμνΓαβ β – + + 0 = 3© ∂gμα ∂xα ---------- - gαβΓαβ¹ μ + § · gμα( Γαβ β Γαβ β – ) + 0 = ∂gνα ∂xα ---------- - ∂gαν ∂xα ---------- - – 0 = gμν gμν gμαgνα gαμgαν δμ ν = = gμν 2gμα© ∂xα ∂ lg g Γαβ¹ β + § · Γαβ β Γβα β – ( ) δβ μ§ ∂gβα ∂xα ---------- - gσβΓσβ¹ β + © · + + 0 = gμν fμ 1 3 --gμα( - Γαβ β Γβα β – ) ∂gμα ∂xα ---------- - gαβΓαβ¹ μ + © · –§ g ∂lg ∂xα -------------- - Γαβ¹ β + © · –gμα§ = = =