Abstract
We prove Price’s law with an explicit leading order term for solutions phi (t,x) of the scalar wave equation on a class of stationary asymptotically flat (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+}) for bounded |x|, where cin {mathbb {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} decay for waves with angular frequency at least l, and 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.
Highlights
The Schwarzschild spacetime [Sch16] with mass m > 0 is a spherically symmetric solution of the Einstein vacuum equation given by g=− 1 − 2m dt2 + −1 dr + r 2g/ (1.1)r r on Rt × (2m, ∞)r × S2, where g/ is the standard metric on S2
We prove Price’s law with an explicit leading order term for solutions φ(t, x) of the scalar wave equation on a class of stationary asymptotically flat (3+1)-dimensional spacetimes including subextremal Kerr black holes
Our precise asymptotics in the full forward causal cone imply in particular that φ(t, x) = ct−3 + O(t−4+) for bounded |x|, where c ∈ C is an explicit constant. This decay holds along the event horizon on Kerr spacetimes and renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional
Summary
The Schwarzschild spacetime [Sch16] with mass m > 0 is a spherically symmetric solution of the Einstein vacuum equation given by g=−. C is equal to the L2 inner product of a linear combination of the initial data with an ‘extended dual bound state’ u∗(0) which solves ( g + V )∗u∗(0) = 0 We illustrate this on Minkowski space Rt × R3x with metric −dt2 + dx[2]; even in this setting, the result appears to be new: Theorem 1.9 (Sharp asymptotics for wave equations with stationary potentials in a simple special case). Price’s law angular momentum affects the full asymptotic expansion, in particular, whether there are extra logarithmic terms which are not present for a = 0;10 (4) sharp asymptotics for equations with zero energy resonances or bound states This requires significantly more work, as the resolvent has strong singularities at σ = 0; see [HHV21].
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