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 5 The quantities and their derivatives described the metric field and gravitational field. Compared to these, the components of the electric field were essentially alien constructs. The wish to understand the gravitational field and the electromag- netic field as one fundamental entity dominated the endeavors of theoreticians dur- ing the last few years.[2] A mathematical finding that we owe to Levi-Civita and Weyl rewarded these efforts:[3] The derivative of the Riemann tensor of curvature, which is fundamental in the general theory of relativity, is based most naturally on the parallel-shift law of vectors (“affine relation”) . (2) This may be traced back to (1) by means of the postulate that a vector’s value does not change upon parallel displacement yet such a reduction is not logically neces- sary. This H. Weyl first recognized he based on this finding a generalization of Rie- mannian geometry that in his opinion yields the theory of the electromagnetic field.[4] Weyl does not assign invariance to the value of a line element or of a vector, resp., but only to the relation between the values of two line elements or vectors sharing the same starting point. The parallel displacement (2) must be designed so as to leave this relation unchanged. The basis of this theory can be described as semimetric. In my opinion, this is not the way to arrive at a physically useful theory.[5] Even from a purely logical point of view, it should surely appear more satisfying to base the theory solely on (2), provided one feels inclined to drop the invariant (1) as the theory’s basis. Eddington did this and noticed that, on the contrary, a metric invariant of type (1), whose physical existence cannot be doubted, can be based on (2). Since, from (2) follows the existence of the fourth-order Riemannian tensor[6] , and thence, by reducing the indices i and m, follows the existence of the second- order Riemannian tensor , (3) whose fundamental importance in gravitational theory is well known. is therefore an invariant of the line element that Eddington regards as a metrical invariant.[7] gμν φμν δA μ μ –Γαβ A α dxβ = [p. 33] Rk, lm i ∂Γkl i ∂xm ---------- Γτl i Γkm τ ∂Γkm i ∂xl ------------ Γτm i Γkl τ + + = Rkl ∂Γkl α ∂xα ---------- Γkβ α Γlβ α ∂Γkα α ∂xl ----------- - Γkl α Γαβ β + + = Rkldxkdxl
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