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 3 leave unchanged not the amount of one vector, but the ratio of the values of two vectors impinging on the same point. Thus he succeeds in having a linear form enter into the geometric theory besides the quadratic form (1). By equating these ’s with the electromagnetic potentials, one obtains a mathematical theory with invariants in which the potentials for gravitation and electromagnetism appear in a mutually dependent way. In my opinion, the great importance of this theory by Weyl is not that it be phys- ically accurate but, much rather, that it first demonstrated the independence of the law (2) from its original metric basis (1). Its weakness, however, lies in its point of departure. The concept of parallel displacement originates, like all concepts of Euclidean geometry, its emergence from the consideration of positioning laws or laws of the relative displacements of rigid bodies. That is how the assertion stip- ulation that upon parallel displacement a path does not change its value acquires its cogency. The transition to four-dimensionality does not essentially change any- thing in this situation. Two (infinitesimally small) bodies of reference originally with equal measuring rods and clocks always measure the same ds between two events the lengths of two measuring rods or, resp., the running speeds of two clocks that are brought together, remain the same if they had ever been the same. Now, one certainly can eliminate from the theory those elements that concern the rigid body and the clock. One can also assume that just the equation has real meaning. One can introduce a law (2) of affine connection without any physical interpretation by means of the rigid body. But it is then pretty arbitrary to demand that this affine law leave unchanged the ratio between the values of two vectors upon displacement if the interpretation of (2) is .[8] Eddington consistently continues down the path that Weyl had blazed.[9] Because (1) does not form a sufficient basis for (2), yet the Riemannian tensor so fundamental for the physics is grounded solely on (2), one must try to make do with (2) alone. Eddington does not doubt that a metric invariant of type (1) exists in nature however, his goal is to derive a type-(1) invariant from (2) that has metri- cally physical meaning. This easily works. The Riemannian curvature tensor derived just from (2) yields, after contracting i and m, the symmetric covariant ten- sor , which with the line element’s component yields the invariant . Because this is the simplest invariant that can be attributed to the line element on the basis of (2), it is regarded as the fundamental metric invariant. The weakness in Eddington’s theory lies in that it did not lead to all the necessary equa- tions for determining the 40 quantities . Furthermore, Eddington did not suc- ceed in linking it to the existing secure results of the general theory of relativity. ϕμdxμ ϕμ [p. 3] ds2 0= Rκ, lm i Rκ, l dxκ Rκldxκdxl Γαβ μ [p. 4]
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