A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

Abstract

We prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes away from extremality

“…(For a direct positive commutator proof of these trapping estimates, albeit not in a semiclassical setting, see [Hin21b,§3].) Near the event horizon r = re on the other hand, we can follow [Vas13, § §4.6 and 6.4], which applies in the present subextremal Kerr context (see also [Hin21a,Lemma 4.3] and [HHV21, Theorem 4.3]).…”

“…Suppose r is a variable order function that is monotone along the Hamiltonian flow of the principal symbol of tf (1), and which satisfies r > 1 2 , resp. r < − 1 2 29 This is the conjugation of the model operator in [Hin21a,Definition 2.20] by e ir . We remark that in [Hin21a], which is based on [Vas21c], the analytic setup focuses on precise second microlocal/module regularity at the outgoing radial set, whereas in the present paper variable order estimates are sufficient.…”

“…8 The quantitative main result of [SR15] was a key input in the proof of decay of solutions of the wave equation on subextremal Kerr spacetimes by Dafermos-Rodnianski-Shlapentokh-Rothman [DRSR16]. The merely qualitative mode stability result is sufficient for this purpose as well if one uses it, in conjunction with strong (Fredholm and high energy) estimates for the spectral family, to exclude the presence of a nontrivial nullspace of the spectral family for Im σ ≥ 0; see [Hin21a] and also Propositions 3.17, 3.18, and 3.21.…”

On pointwise decay of waves

“…Note how the accelerated decay for higher values of ℓ mirrors what we saw for the surfaces of revolution in Theorem 3.1. Hintz [42] recently closed the gap of the missing power of t and thus finished the proof of Price's law. Theorem 3.9.…”

“…In general relativity, unstable closed geodesics arise naturally in the study of the linear wave evolution on the background of both Schwarzschild and Kerr black holes. A substantial amount of work has accumulated around this topic, see for example the early works by Blue and Soffer, see [7] as well the very recent study of Price's law by Hintz [42]. The latter paper was preceded by the work of Tataru [76], as well as the results by Donninger, Soffer and the author [28] on the spatially local, but temporally global, decay of linear waves on Schwarzschild.…”