D O C U M E N T 4 2 5 O N G E N E R A L R E L A T I V I T Y 3 6 9 By combining this equation with the ones obtained from reducing the indices and l, . (18) Finally, as a general result of our variation consideration, it follows that[14] . (19) These are 40 equations from which the values are computable. For this purpose we introduce the tensors and , resp., which belong to the tensor density thereby, these tensors share the same interrelationship as the one between covariant and contravariant fundamental tensors ( and ) in general relativity theory. Thus the following equations may hold. (20) . (21) Furthermore, we set (22) (23) Then we obtain from well-known calculations in general relativity theory: . (24) One must imagine these values for plugged into (11) and (12). Because from (13), s and f are expressible as g and through the choice of Hamiltonian, therefore after substitution the equations (11) and (12) suffice to determine the unknown functions. Now, in order to recognize the physical legitimacy of the Hamiltonian choice reached in (10), let us next look at the case of a missing electromagnetic field. According to (10) and (13), then where and g correspond to in the conventional relation from general rela- tivity theory. Equation (24) then takes on the well-known form , (24a) [p. 36] α 0 3s lσ 5i l + =σ 0 s kl 1 3 --δα - k i l 1 3 --δα - l i k + + =α Γ skl skl s kl gμν gμν s kl skl sik –= sαisβi δα β = il sik – il –sil = = il slτi τ .= Γkl α 1 2 --sαβ - ∂skβ ∂xl ---------- ∂slβ ∂xk --------- ∂skl ∂xβ -------- -–+ 1 2 --skliα - – 1 6 --δk - αil 1 6 --δlαik - + + = Γ φ s kl gkl g –= f kl 0, = gkl gkl Γkl α 1 2 --gαβ - ∂gkβ ∂xl ---------- - ∂glβ ∂xk ---------- ∂gkl ∂xβ --------- -–+ =