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 7 for determining these quantities. Hamilton’s principle offers the most comfortable method for this. Let H be a scalar density dependent on the ’s and their first deriv- atives, then for each continuously vanishing variation of the ’s at the edge of the integration region (8) should hold. The field equations that, because of the tensorial character of are likewise tensorial in character, then read , (9) where . Thereby it is assumed that H is an (algebraic) function of the ’s. Our main task involves choosing this function. Tensor densities exist that are rational second-order functions of the ’s they can be obtained by means of the tensor density , the components of which are equal to 1 or , depending on whether iklm is an even or odd permuta- tion of 1, 2, 3, 4. One such tensor density is, e.g., . I consider it correct, however, to confine ourselves to the tensor densities formed out of the reduced tensor , or, resp., out of and ,[10] because we are only inclined to ascribe physical meaning to these quantities. Then we have to permit irrational functions, as we are already accustomed to doing from the previously developed general theory of relativity (e.g., ). Then, too, there are various options, among which the following seems the most interesting to me: , (10) which is analogous to the volume’s tensor density and is formed from without being broken down into the symmetric and antisymmetric parts. This Hamiltonian proves to be usable, consequently the theory achieves in an ideal way the unifica- tion of gravitation and electricity under a single concept not only the same ’s determine both kinds of fields, but also the Hamiltonian is a thoroughly uniform one, whereas it had hitherto been composed of logically independent summands.[11] In the following the usefulness of the theory is made plausible. Γ Γμν α δ Hdτ 0= δΓμνα 0 Hα μν ∂H ∂Γμν α ------------ ∂ ∂xσ -------- ∂H ∂Γμν σ , α ---------------- - – = = ∂Γμν α ∂xσ ------------ Γμν σ , α = Rk, lm i Rk, lm i δ iklm 1– Rk, lm i Rik, στ δlmστ Rkl Skl φkl –g H 2 Rkl –= Rkl [p. 35] Γ