3 5 2 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 , (2) where according to Riemannian geometry the ’s are given by (3) The quantities seem to have been first introduced by Christoffel.[2] However, its geometric interpretation as the coefficient of the law of parallel displacement originally goes back to Levi-Civita and Weyl.[3] The fundamental meaning of the ’s is primarily that they alone determine the Riemannian curvature tensor as well as its important contraction in gravitational field theory. The dependence (3) of the parallel displacement law (2) on the fundamental invariant (1) arises in Riemannian geometry, as well as in the original form of the general theory of relativity (with the exception of the independently postulated symmetry condition ),[4] because the parallel displacement law demands that the quantity of a contravariant vector not vary upon dis- placement according to (2).[5] Analytically this amounts to the condition that the tensor formed out of the fundamental tensor ( ) or ( ), resp., through differ- entiation (“extension”) (4) vanishes identically.[6] Because the newer theories by Weyl and Eddington[7] modify or eliminate this connection, they arrive at modifications of the general theory of relativity that we will look at briefly. Ultimately, we want to construct a new theory that, although related to Eddington’s theory, follows more naturally and simply than it does from the original general theory of relativity. §2. The Theories by Weyl and Eddington H. Weyl sets out from the idea that a more fundamental importance be attached to the elementary law of light propagation, or the “light cone” , than to the ds itself. According to him, only the relations of the ’s not of the ’s have their own real meaning (for an established system of coordinates). In accordance with this interpretation, he requires of the displacement law (2) that it Aμ δ ΓαβAαdxβ μ –= Γαβ μ Γαβ μ 1 2 --gμσ - ∂gσα ∂xβ ----------- ∂gσβ ∂xα ----------- ∂gαβ ∂xσ ----------- –+ = Γαβ μ Γ Rκ, lm i Rκl Γαβ μ Γβα μ = gμνAμAν [p. 2] gμν gμν gμν σ ∂gμν ∂xσ ---------- - gανΓμσ α – gαμΓνσ α – = gμνdxμdxν 0= gμν g[ μν]