D O C U M E N T 4 1 7 O N G E N E R A L R E L A T I V I T Y 3 5 7 ,[18] (11) where means the extension . With the aid of (11) it is easy to prove (by forming both contractions) that equation (10) is therefore equivalent to the equation . (10a) Equations (3) and (9) together, however, are precisely the equations of the gravita- tional field which the general theory of relativity had reached earlier under the pre- condition that relations (3) or (4), resp., or (10a) hold a priori. §5. Law of the Electromagnetic Field In the general case that an electromagnetic field exists besides the gravitational field, the Hamiltonian function must also depend on tensor (7). Pursuant to the ear- lier results of the general theory of relativity, one should put [20] (11) where is given by (8) and (12) is set. Executing the variation yields (13) ,[21] (14) where, to abbreviate, (15) is set. The first system of field equations (5a) yields the equations of the gravita- tional field, taking into account the field excitation influence of the electromag- netic field. The second system of (5a), however, in combination with (11) and (14), yields a relation between the current density, the metric and gravitational fields. One obtains after a simple transformation:[22] 1 –g --------- -I1 αβ μ gαβ μ 1 2 --δμ - αgβσ σ – 1 2 --δμ - βgασ σ – gαβgστgστ α – = gαβ μ ∂g αβ ∂xμ ----------- - gαβΓμσ α gασΓμσ β + + gαβ μ 0= [p. 8] I I1 I2 += I1 I2 1 2 --gμσgντ - ∂Γμα α ∂xν ------------ - ∂Γνα α ∂xμ ------------ – ∂Γσβ β ∂xτ ------------ ∂Γτβ β ∂xσ ----------- -– –g = 1 –g --------- -I2μν 1 4 --ϕσαϕτβgστgαβgμν - – ϕμαϕνβgαβ + = 1 –g --------- -I2 αβ μ δ2 αiβ – δμ βiα += iα –g ∂ ϕστgσαgτβ –g) ( ∂xβ -------------------------------------------- - =