Existence and stability of circular orbits in time-dependent spherically symmetric spacetimes

Abstract

For a general spherically four-dimensional metric the notion of "circularity" of a family of equatorial geodesic trajectories is defined in geometrical terms. The main object turns out to be the angular-momentum function J obeying a consistency condition involving the mean extrinsic curvature of the submanifold containing the geodesics. The analysis of linear stability is reduced to a simple dynamical system formally describing a damped harmonic oscillator. For static metrics the existence of such geodesics is given when J 2 > 0, and (J 2 ) ′ > 0 for stability. The formalism is then applied to the Schwarzschild-de Sitter solution, both in its static and in its time-dependent cosmological version, as well to the Kerr-de Sitter solution. In addition we present an approximate solution to a cosmological metric containing a massive source and solving the Einstein field equation for a massless scalar. * email: wolfgang.graf@univie.ac.at mulated and proved. In section 6 the static case is analyzed in some detail. In section 7 some examples are analyzed (including time-dependent ones) before concluding with a discussion of our results in section 8.The emphasis is on timelike orbits with nonvanishing angular-momentum, J = 0. However, for completeness also the case of radial timelike trajectories, J = 0 is briefly considered, as well as lightlike trajectories. It should be stressed that we deal exclusively with one-parameter families of trajectories sweeping the whole equatorial plane and not with isolated trajectories.1 for a review, see Carrera and Giulini, 2010 [25] 2 Turyshev et al., 2012 [27] 3 Hernandez, Jimenez, Allen, 2012 [28] 4 see McGaugh, 2005 [21] 5 for an introduction, see Milgrom, 2008 [24] 6 from now on, we cite this work only by its author-name 7 more generally, contraction of V with respect to the first (from the left) free vector slot of the tensorial object W 8 the only exception to this rule is the divergence of a vector V , when expressed as ∇·V 9 the only instance where we use the , -notation 10 for a more detailed formulation, see Proposition 13 of ch. 4 in O'Neill 11 w is usually denoted by ω 12 in fact, assuming w = 0, the canonical equatorial metric is hard to achieve -if at all 13 see also Carrera and Giulini, 2010 [25]