3 5 4 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 Eddington also saw no way to bring his theory into the convenient Hamiltonian form.[10] Because I departed from Eddington’s approach to arrive at my new theory, I would like to indicate here the following consideration, which clearly illuminates the connection between my ideas and Eddington’s. I first asked myself whether an invariant volume integral existed whose invariance is based only on (2) if yes, then one can require that the first variation of this integral vanish for all possible vari- ations of the quantities . One then obtains 40 differential equations for the 40 -quantities, between which four identical relations hold. The existence of such invariants can be seen as follows: According to Riemann’s theory, within a manifold of 4 dimensions there is one tensor, if g is the determinant of the ’s and is a magnitude that vanishes, if not all four indices one antisymmetric contravariant tensor of 4th order, all of whose components have the absolute quantity . By multiplying by , the tensor density, , independent of the fundamental tensor, appears. Out of this and the Riemannian curvature tensor, nonvanishing scalar densities are formable, for ex., . If one designates I as a linear combination of such scalar densities, the Hamilto- nian equation then yields a system of 40 differential equations of 2nd order for the -quanti- ties. This system appears interesting because it is connected with manifolds of four dimensions.[11] Hence it has indeed proved possible to erect a complete theory in Eddington’s sense that is based solely on the assumption of the existence of an affine connection between the vectors or line elements and it will be necessary to check its agree- ment with experience by the centrally symmetric case.[12] I do not believe, though, that one will arrive at a theory true to reality along this path for Eddington’s inte- gral invariants have no similarity to the integral invariants Γαβ μ Γαβ μ δ iklm –g ----------- gμν δiklm 1 –g --------- - –g δiklm Ri lm , i Rι, λμ ι δ lmλμ Rι, lm i Ri, λμ ι δlmλμ δ Idτ 0= Γαβ μ [p. 5]
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