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90) where the spatial endpoints are kept fixed. Instead of the energy functional for gF we can, of course, also use the length functional. This result is useful for calculating the propagation of light rays in gravitational fields. 76) for the metric. 91) dx , 1 + 2φ with dx 2 = (dx 1 )2 + (dx 2 )2 + (dx 3 )2 . Fermat’s principle becomes δ ˙ dλ = 0, (1 − 2φ) x(λ) 38 2 Physics in External Gravitational Fields ˙ denotes the Euclidean norm of dx/dλ. This agrees with the Fermat prinwhere |x| ciple of geometrical optics ˙ n x(λ) x(λ) dλ = 0 δ for the refraction index n = 1 − 2φ.

In spite of this failure we add some further instructive remarks. First, we want to emphasize that the interaction is necessarily attractive, independent of the matter content. 8) where φ is proportional to ϕ and g is a coupling constant. Note first that only g 2 is significant: Setting φ˜ = gφ, we have L=− 1 ˜ μ φ˜ + T · φ˜ + Lmat , ∂μ φ∂ 2g 2 involving only g 2 . Next, it has to be emphasized that it is not allowed to replace g 2 by −g 2 , otherwise the field energy of the gravitational field would be negative.

This has nothing to do with coordinates or frames. For instance, instead of saying: “. . let Ric be the Ricci tensor, u a vector field, and consider Ric(u, u) . ”, one says: “. . let Rμν be the Ricci 5 It is not permitted to introduce in addition to g μν other “external” (absolute) elements such as a flat metric which is independent of g. 22 2 Physics in External Gravitational Fields tensor, uμ a vector field, and consider Rμν uμ uν , . . ” In particular, the number and positions of indices specify which type of tensor is considered, and repeated indices indicate the type of contraction that is performed.

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