Worldine deviations and epicycles

In general relativity, only relative acceleration has an observer-independend meaning: curvature and non-gravitational forces determine the rate at which world lines of test bodies diverge or converge. We derive the equations governing both in the conventional geometric formalism as well as using the background field method. This allows us to generalize the results to test bodies with charge and/or spin. The application of the equations to the motion of particles in a central field results in an elegant, fully relativistic version of the Ptolemaean epicycle scheme. 1. World line deviation equations According to the equivalence principle, structureless test bodies (sometimes referred to as point masses) in a gravitational field move on geodesics of spacetime. Their worldline xμ(τ) is a solution of the geodesic equation D2xμ Dτ = d2xμ dτ + Γ μ λν dxλ dτ dxν dτ = 0, (1) where the world-line parameter τ is to be taken as proper time. Introducing the four-velocity as the time-like tangent unit vector to the worldline: uμ = dxμ/dτ , the equation can be written in geometrical language as u · ∇u = 0, u = −1. (2) with ∇ the covariant derivative. It is easily observed from eq.(1), that the proper acceleration aμ = d2xμ/dτ2 is not a covariant object. In particular, its vanishing or non-vanishing has no observer-independent meaning. In contrast, the relative acceleration between worldlines is a covariant quantity, and its vanishing or non-vanishing does not depend on the frame of reference [1]. We recall the argument. Consider a one-parameter congruence of geodesics xμ(τ ;λ), where λ labels the geodesics and τ is the proper-time parameter along the geodesic. We suppose the parametrization to be smooth, hence we can construct the tangent vector fields uμ = ∂xμ/∂τ , and nμ = ∂xμ/∂λ. It is straightfowardly established that (u · ∇n) = ∂2xμ ∂τ∂λ + Γ μ λν ∂xλ ∂τ ∂xν ∂λ = (n · ∇u). (3) As a corrolary, we obtain u · ∇(u · ∇n) = u · ∇(n · ∇u) = (u · ∇n) · ∇u+ un(∇μ∇νu) = (n · ∇u) · ∇u+ un(∇μ∇νu) = n · ∇(u · ∇u) + uμnν [∇μ,∇ν ]u = u nRμν [u, ·]. (4) In component notation this reads D2nμ Dτ = R μ κλν u un. (5) The interest in the deviation vector nμ obviously derives from the fact that, if x 0 (τ) = xμ(τ ;λ0) is a solution of the geodesic equation (1), then to first order x 1 = x 0 + nμ∆λ is a solution as well: x(τ ;λ1) = x (τ ;λ0) + ∆λ ∂xμ ∂λ (τ, λ0) ≈ x (τ ;λ0 +∆λ). (6) It follows, that eq.(5) describes the covariant relative acceleration between these world lines. Of course, nμ is only a first approximation to the neighboring geodesic at λ1 = λ0 + ∆λ. To increase the precision of the approximation,