By Norbert Straumann
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There it is also shown that gμν is really the physical metric. Conclusion The consequent development of the theory shows that it is possible to eliminate the flat Minkowski metric, leading to a description in terms of a curved metric which has a direct physical meaning. The originally postulated Lorentz invariance turns out to be physically meaningless and plays no useful role. The flat Minkowski spacetime becomes a kind of unobservable ether. The conclusion is inevitable that spacetime is a pseudo-Riemannian (Lorentzian) manifold, whereby the metric is a dynamical field, subjected to field equations.
The Lorentz equation of motion for a charged test mass becomes in GR m α β d 2xμ μ dx dx + Γ αβ dτ dτ dτ 2 = eF μν dx ν . 43) allows us also in GR to introduce vector potentials, at least locally. By Poincaré’s Lemma (DG, Sect. 4), F is locally exact F = dA. 56) In components, with A = Aμ dx μ , we have Fμν = ∂μ Aν − ∂ν Aμ (≡ ∇μ Aν − ∇ν Aμ ). 58) As in SR there is a gauge freedom A −→ A + dχ or where χ is any smooth function. This can be used to impose gauge conditions, for instance the Lorentz condition ∇μ Aμ = 0 (or δA = 0).
122). 122) is generally invariant and so holds in any coordinate system. 116) in terms of local coordinates. We want to demonstrate that such a derivation can be faster. 115) in components Kμ Kν,λ + Kν Kλ,μ + Kλ Kμ,ν = 0. The left hand side does not change if partial derivatives are replaced by covariant derivatives. 122), we obtain −Kμ K λ Kλ ;ν + Kν K λ Kλ ;μ + K λ Kλ (Kμ;ν − Kν;μ ) = 0. 116). 2 The Redshift Revisited The discussion of the gravitational redshift in Sect. 6 was mathematically a bit ugly.