3 5 6 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 is set. On the choice of the Hamiltonian function I, the following remark: If the Rie- mannian curvature tensor is contracted by the indices i and m, one obtains the tensor , (6) which in the general theory of relativity together with the Riemann-Christoffel con- straint (3) or (4) produced the theory of the pure gravitational field. If the Rieman- nian relations between the ’s and g’s are given up, the Riemannian tensor has a second, generally vanishing contraction (by the indices i, ):[15] . (7) We want to interpret this tensor as the expression of the electromagnetic field, as Eddington has already done.[16] We want to introduce the obvious assumption that the sought Hamiltonian function contain the -quantities only in combinations (6) and (7). §4. Law of the Pure Gravitational Field In the case of the pure gravitational field, the Hamiltonian function will depend only on (6). One arrives at the goal by assuming the simplest dependence, namely, the linear one: . (8) Equations (5a) then take on the form (9) (10) Form (10), which must, of course, have a tensor character—as an Italian mathema- tician already discovered on another occasion—can be brought into the form Rκ, lm i ∂Γκl i ∂xm ---------- -– Γαl i Γκm α ∂Γκm i ∂xl ------------ - Γαm i Γκlα – + + = Rκl α ∂Γκl ∂xα ---------- -– Γκβ α Γlβ α α ∂Γκα ∂xl ------------ Γκl α Γαβ β – + + = Γ κ ϕμν ∂Γμα α ∂xν ------------ - ∂Γναα ∂xμ ------------ –= [p. 7] Γ I1 gκlRκl g –= 1 –g --------- -I1μν Rμν 1 2 --gμνR - – 0 = = 1 –g --------- -I1 αβ μ 1 –g --------- - ∂gαβ –g ∂xμ ---------------------- 1∂gασ 2 ------------------------δμ - –g ∂xσ β – 1∂gβσ 2 ----------------------- - –g ∂xσ -δμα – = 1 2 --gασΓμσ - β 1 2 --gβσΓμσ - α 1 2 --gστΓστ - α δμ β – 1 2 --gστΓστ - β δμ α – gαβΓμσ σ – + + 0 =