Abstract
AbstractWe 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.
Funder
National Science Foundation
Clay Mathematics Institute
Alfred P. Sloan Foundation
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference107 articles.
1. Angelopoulos, Y., Aretakis, S., Gajic, D.: Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes. Adv. Math. 323, 529–621 (2018)
2. Angelopoulos, Y., Aretakis, S., Gajic, D.: A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes. Ann. PDE 4(2), 15 (2018)
3. Angelopoulos, Y., Aretakis, S., Gajic, D.: Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity. J. Hyperbolic Differ. Equ. 16(01), 1–34 (2019)
4. Angelopoulos, Y., Aretakis, S., Gajic, D.: Late-time tails and mode coupling of linear waves on Kerr spacetimes. Preprint. arXiv:2102.11884 (2021)
5. Angelopoulos, Y., Aretakis, S., Gajic, D.: Price’s law and precise late-time asymptotics for subextremal Reissner–Nordström black holes. Preprint. arXiv:2102.11888 (2021)
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