Stability of Minkowski space and polyhomogeneity of the metric

Abstract

We first give a new proof of the non-linear stability of the (3 + 1)-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. We then show that the metric admits a full asymptotic expansion at infinity, more precisely at the boundary hypersurfaces (corresponding to spacelike, null, and timelike infinity) of a suitable compactification of R 4 adapted to the bending of outgoing light cones. We work in a wave map/DeTurck gauge closely related to the standard wave coordinate gauge. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the non-linear equation using an iteration scheme in which we solve a linearized equation globally at each step; to fix the geometry of null infinity throughout the iteration scheme, we devise a hyperbolic formulation of Einstein's equation which ensures constraint damping. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Furthermore, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

Linear stability of slowly rotating Kerr black holes

“…Remark 14.3. The asymptotic behavior in t * ≤ 0 can be described in great detail, see [HV17] for results in the nonlinear setting; the results for linear waves here are straightforward to obtain using energy methods (see also Lemma 14.5 below), and the statement (2) above is merely the simplest pointwise bound one can prove. As in [HV17], one can moreover show that h in fact has an r −1 leading order term at null infinity I + , viewed as a boundary hypersurface of a suitable compactification of M • (which in r/t χ 0 > > 0 is given by m M in the notation of the reference); we recall the argument in the proof of Lemma 14.5 below when restricting to the region t * ≤ C for any fixed C ∈ R.…”

“…Remark 10.2. On the other hand, in [HV17], we used small CD which however is asymptotically (at I + ) non-trivial (roughly, in the reference we took c = r −1 dt near I + ). Such CD modifications affect (albeit only mildly so for small γ 1 , γ 2 ) the asymptotic behavior of (mode) solutions of the modified linearized gauge-fixed operator.…”

“…Such CD modifications affect (albeit only mildly so for small γ 1 , γ 2 ) the asymptotic behavior of (mode) solutions of the modified linearized gauge-fixed operator. Recall however that CD at I + in [HV17] was implemented only to ensure better decay properties of certain metric components at I + in a nonlinear iteration scheme; hence, for the present linear stability problem, there is no need for such asymptotically non-trivial CD. With an eye towards a possible proof of the nonlinear stability of the Kerr family, we do remark however that this type of CD can be implemented in this paper as well (as an additional small perturbation on top of an already working compactly supported CD); the changes in the behavior of the resolvent are minor, as discussed in a general setting in [Vas19b].…”

“…Beyond our straightforward choice of gauge condition, our construction of L g b with these properties involves the implementation of constraint damping (CD), which was first discussed in [GCHMG05] and played a central role both in numerical work [Pre05] and the nonlinear stability proofs [HV18b,Hin18a,HV17]. In fact, CD is a crucial input already in the proof of Theorem 1.1, as it makes the low energy behavior of the spectral family of L g b non-degenerate and thus perturbation-stable, allowing us to deduce Theorem 1.1 perturbatively from the statement for Schwarzschild parameters b = b 0 ; we discuss this in § §1.1.2-1.1.3.…”

“…Explicitly, L b = 1 2 g b + lower order terms; this is thus a linear wave operator acting on symmetric 2-tensors. Simple linear theory (using the framework of [HV17]) allows us to solve the initial value problem for L b h = 0 up to a hypersurface which is transversal to future null infinity I + and the future event horizon H + , see Figure 1.1.…”