Stationary axisymmetric exteriors for perturbations of isolated bodies in general relativity, to second order

Abstract

Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by a matter-filled interior and an asymptotically flat vacuum exterior joined at a surface where the Darmois matching conditions are satisfied, are considered. The initial state is assumed to be static. The perturbations of the matching conditions are derived and used as boundary conditions for the perturbed Ernst equations in the exterior region. The perturbations are calculated to second order. The boundary conditions are overdetermined: necessary and sufficient conditions for their compatibility are derived. The special case of perturbations of spherical bodies is given in detail.

Revisiting Hartle's model using perturbed matching theory to second order: amending the change in mass

“…3 Proposition 2 Let (V 0 , g) with Σ 0 be the static and spherically symmetric background matched spacetime as described in Proposition 1, and assume that (20) is satisfied. Let it be perturbed to first order by K (1)± plus Q ± 1 and T ± 1 so that (18), (19), (21), (22) hold. Consider the second order metric perturbation tensor K (2)± as defined in (9) at either side, plus two unknown functionsQ ± 2 (τ, ϑ) and two unknown vectors…”

“…As discussed, the first order matching conditions are invariant under such spacetime gauges (at either or both sides, with corresponding C + and C − ), that is, the first order matching conditions (18), (19), (21) and (22) transform to just the same expressions with g superscripts.…”

“…Theorem 1 Let (V 0 , g) with Σ 0 be the static and spherically symmetric background matched spacetime configuration, perturbed at either side to first order by the functions ω ± (r ± , θ ± ) through K (1)± as defined in (8) plus the unknowns Q ± 1 (τ, ϑ) and T ± 1 (τ, ϑ), as described in Proposition 1, so that the first order matching conditions (18) and (19) plus (21) and (22) hold. Let the configuration be perturbed to second order by K (2)± as defined in (9), plus the unknownsQ ± 2 (τ, ϑ) and T ± 2 (τ, ϑ) on Σ 0 , and assume that the interior region (+) satisfies the field equations for a perfect fluid with barotropic equation of state and that the exterior (−) region is asymptotically flat and satisfies the vacuum field equations up to second order.…”

“…Some aspects which are shown in [14] by using the continuity of functions, mainly the structure of the metric functions (no l > 2 sector), are still to be proven given the present state of things. That work, done in parallel, started in [26] by using the framework construted in [18], consisting of a completely general perturbative approach to second order around static configurations of the exterior (asymptotically flat) vacuum problem of stationary and axisymmetric bodies with arbitrary matter content. All in all, given that the matching conditions may be needed in more general situations in future works, we have preferred to keep some generality in the results by focusing first on a purely geometric setting in order to impose the field equations later.…”

Phenomenological relations for axial quasinormal modes of neutron stars with realistic equations of state

“…The junction conditions between two space-times have been considered in classical works by Darmois [54], Lichnerowicz [55], O'Brien and Synge [56], and Israel [57], and, more recently, in [58][59][60]. In the context of relativistic rotating stars, it has been considered in [61][62][63].…”