3 7 0 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 which equation, together with (11), exactly yields the vacuum equation of the grav- itational field in the general theory of relativity with a vanishing electromagnetic field, taking the cosmological term into account. This is a strong argument in favor of our choice of Hamiltonian as well as of the usefulness of the theory overall. We now move on to the case where the electromagnetic field does not vanish. To start with, from (12) and (24) generally follows: . (25) The insight here is that at absolutely vanishing current density no electric field is possible. However, the extraordinary smallness of implies that finite ’s are only possible with tiny, practically vanishing covariant current densities. Hence, with the exception of singular spots, the current density practically vanishes. Con- sequently the equations (26) (27) hold there very approximately, the latter equation of which holds strictly, with (12) taken into account. The relation between the ’s and f ’s, with our Hamiltonian choice, is determined in that the quantities are the underdeterminants of the ’s multiplied by the root from the negatively taken determinants r of . For, if those standardized underdeterminants are called , then you get and consequently, , from which the postulate follows. The approximate computation of the ’s is thus simple in the important case that the ’s differ only infinitesimally little from the constant values (= 1 or = 0, [p. 37] 1 λ2 -----φkl 1 6 -- - ∂ik ∂xl ------ - ∂il ∂xk ------- -– = 1 λ2 ----- φkl ∂f kl ∂xl ----------- 0 . . . . = ∂φkl ∂xσ --------- - ∂φlσ ∂xk ---------- ∂φσk ∂xl ----------- + + 0 = φ rkl s kl f kl += rkl rkl gkl φkl += rkl δr rr kl δrkl = dH δ( 2 –r) 1 –r ---------δ(–r) –rrklδrkl –rrkl( δgkl δφkl) + = = = = f kl rkl δkl
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