3 6 8 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 § 2. The New Theory’s Relationship with the Earlier Results of General Relativity Theory First a comment about equation (5). represents the metric invariant for a “cosmic” measuring rod. If is supposed to describe a squared unit length of human dimensions, then one has to set , (5a) where is a very large number.[12] One therefore has, according to (3), (11) . (12) We now execute the variation indicated in (8) under the general assumption that H is a function of and , left undetermined for the time being. Then[13] , (13) where signifies a symmetric tensor density, and an antisymmetric one. Taking (11), (12), and (13) into account, (8) assumes the form . (14) Since we interpret as the electromagnetic field’s covariant tensor, we should re- gard as the electromagnetic field’s contravariant tensor density and (15) as the density of the current. In (14), means the covariant extension of according to the formula . (16) From (14) follows . (17) gkldxkdxl gkldxkdxl λ2Rkl gkl φkl += λ 1 λ2 -----gkl ∂Γkl α ∂xα ---------- 1 2 -- - ∂Γkα α ∂xl ----------- - ∂Γlα α ∂xk ----------- + Γkβ α Γlβ α Γkl α Γαβ β – + + = 1 λ 2 -----φkl 1 2 -- - ∂Γkα α ∂xl ------------ ∂Γlα α ∂xk ----------- – = gkl φkl δH ∂H ∂gkl ---------δgkl - ∂H ∂φkl ---------δφkl - + s kl δgkl f kl δφkl + = = s kl f kl 0 α s kl 1 2 --δα - k s lσ 1 2 --δα - l s kσ 1 2 --δα - k ∂f lσ ∂xσ ---------- 1 2 --δα - l ∂f kσ ∂xσ ---------- - –– σ – σ – α dτδΓkl = φkl f kl il ∂f lσ ∂xσ ---------- -= s kl α s kl s kl α ∂s kl ∂xα ------------ s σl Γσα k s kσ Γσα l s kl Γασ σ – + + = 0 s kl 1 2 --δα - k s lσ 1 2 --δα - l s kσ 1 2 --δα - k il 1 2 --δα - l ik –– σ – σ – α =