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Extra resources for Relativity on curved manifolds
58) is restricted to satisfy χ := ∇μ ∇ μ χ = 0. 47b) as ∇ν ∇ ν Aμ − ∇ν ∇ μ Aν = −4πj μ . 61) Let us impose the Lorentz condition. We can use this in the second term with the help of the Ricci identity for the commutator of two covariant derivatives (DG, Eq. 92)) (∇μ ∇ν − ∇ν ∇μ )Aα = R αβμν Aβ . 4 Non-gravitational Laws in External Gravitational Fields 31 This leads to ∇ν ∇ ν Aμ − R μν Aν = −4πj μ . 63) reduces to the inhomogeneous wave equation ∂ν ∂ ν Aμ = −4πj μ . 63). This example illustrates possible ambiguities in applying the ∂ −→ ∇ rule to second order differential equations, because covariant derivatives do not commute.
The pair (M, g) is called a Lorentz manifold and g is called a Lorentzian metric. Remark At this point, readers who are not yet familiar with (pseudo-) Riemannian geometry should study the following sections of the differential geometric part at the end of the book: All of Chaps. 11 and 12, and Sects. 6. These form a self-contained subset and suffice for most of the basic material covered in the first two chapters (at least for a first reading). References to the differential geometric Part III will be indicated by DG.
99) to obtain 40 2 μ;ν ;ν 0 = −A = Re Physics in External Gravitational Fields + R μν Aν 1 ν i k kν a μ + εbμ + · · · − 2 k ν a μ + εbμ + · · · ε ε2 i − k ν;ν a μ + εbμ + · · · − a μ + · · · ε ν ;ν + R μν a ν + · · · ;ν eiψ/ε . 103) telling us that the wave vector is null. Using kμ = ∂μ ψ we obtain the general relativistic eikonal equation g μν ∂μ ψ∂ν ψ = 0. 104) The terms of order ε −1 give 1 μ k ν kν bμ − 2i k ν a ;ν + k ν;ν a μ = 0. 103), this implies 1 μ k ν a ;ν = − k ν;ν a μ . 105) As a consequence of these equations, we obtain the geodesic law for the propagation of light rays: Eq.