The linear stability of the Schwarzschild solution to gravitational perturbations

Abstract

Note that the mode analysis corresponding to formulation 1. described above yields necessary but not sufficient conditions for either statements (2a) and (2b) of true linear stability.( 1 ) In the case of the linear scalar wave equation g ϕ = 0, which can be thought of as a "poor man's" version of linearised gravity, the analogue of (2a) for Schwarzschild was proven by , and the analogue of (2a) and ( 2b) are shown now for the full subextremal Kerr family in [26], following a host of recent activity [24], [73], [1], [21]. See [25], [23] for a survey. See [6], [2] for generalisations to the Maxwell equations and [3] for a discussion of the extremal case |a|=M . Concerning the linearised Einstein equations themselves, work on the wave equation easily generalises to establish physical space decay on certain quantities, for instance those gauge-invariant quantities satisfying the Regge-Wheeler equation on Schwarzschild [32], [7], [29]. For the full system of linearised gravity however, both problems (2a) and ( 2b) have remained open until today. We note explicitly that even the question of uniform boundedness, let alone decay, for the gauge-invariant quantities satisfying the Teukolsky equation on Schwarzschild has remained open.(3) The full non-linear stability of Schwarzschild as a solution to the Cauchy problem for the non-linear Einstein vacuum equations (2). This is the definitive formulation of the fundamental question. See our previous [18] for a precise statement of the conjecture in the language of the Cauchy problem for (2). In analogy with 2. above, one could distinguish between questions of (3a) orbital stability and (3b) asymptotic stability. Experience from non-linear problems, however, in particular the proof of the non-linear stability of Minkowski space [14] referred to earlier (see also [47], [5]), indicates that (3a) and (3b) are naturally coupled.( 2 ) Since non-linear stability is thus necessarily a question of asymptotic stability, the "Schwarzschild" problem is more correctly re-phrased as the non-linear asymptotic stability of the Kerr family in a neighbourhood of Schwarzschild. For even if one restricts to small perturbations of Schwarzschild, it is expected that generically, spacetime dynamically asymptotes to a very slowly rotating Kerr solution with a =0. Since in the context of a non-linear stability proof, one effectively must "linearise" around the solution one expects to approach, this suggests that to resolve the full ( 1 ) Thus, the mode analysis can be an effective tool to show instability, but never, on its own, stability. For instability results for related problems proven via the existence of unstable modes, see [70], [28] and references therein. See also discussion in [78].( 2 ) This coupling arises from the super-criticality of the Einstein vacuum equations (2). Note that under spherical symmetry (where the vacuum equations must be replaced, however, by a suitable Einstein-matter system to restore a dynamical degree of freedom) this super-criticality is broken in the presence o...

Teukolsky formalism for nonlinear Kerr perturbations

“…ab are determined by transport equations (ODEs) from the nonlinear terms T (n) ab in the n-th order Einstein tensor (involving h (1) ab , . .…”

“…ab is the background Kerr metric. The equation for h (1) ab is of course just the source free linearized Einstein equation in the absence of matter sources, whereas the higher order corrections for n > 1 satisfy an equation of the form…”

“…etc. As described in detail in section III, at zeroth order, we can apply the decomposition h (1) ab = g (1) ab + (L ξ (1) g (0) ) ab + Re(S † Φ (1) ) ab , where Φ (1) is a Hertz potential, i.e., a (−4, 0) solution to O † Φ (1) = 0, and where in this section,…”

Theoretical cosmology

“…In fact, numerical computations are now commonly used to address fundamental issues within full GR cosmology [73,74,75,76,77,214]. 4 A full proof of the linear stability of Schwarzschild spacetime has recently been established [40]. The non-linear stability of the Schwarzschild spacetime is still elusive [41] (however, see [121]).…”

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

“…Theorem 6.4. (Full linear stability of Schwarzschild, Dafermos et al, [26]). Solutions for the linearisation of the Einstein equations around Schwarzschild arising from regular admissible 14 data remain bounded in the exterior and decay (with respect to a hyperboloidal foliation) to a linearised Kerr solution.…”