论文标题
Price的法律的尖锐版本，用于渐近平面上的波浪衰减
A sharp version of Price's law for wave decay on asymptotically flat spacetimes
论文作者
论文摘要
我们在一类固定的渐近平面$（3+1）$ - 尺寸空间（包括亚超级kerr黑洞）上的标量波方程的解决方案$ ϕ（t，x）$的明确指定订单$ - 标量波方程的明确定律。我们在完整的正向因果锥中的精确渐近学尤其暗示$ ϕ（t，x）= c t^{ - 3}+\ mathcal o（t^{ - 4+}）$ | | x | $，其中$ c \ in \ mathbb c $是一个明确的常数。这种衰减也沿着kerr的空间上的事件范围保持，因此卢克·史比尔斯基（Luk-Sbierski）在库奇（Cauchy Horizon）无条件的线性标量不稳定性上产生了结果。此外，我们证明了辐射场的逆二次衰减，并具有明确的领先顺序项。我们建立了与立方空间衰减相反的固定电势散射的类似结果。在Schwarzschild时空中，我们证明了$ t^{ - 2 L-3} $衰减，具有角度频率至少$ l $的波和$ t^{ - 2 L-4} $衰减的波浪，而最初是静态的。这最终使Price的定律完全普遍为线性标量波。 证明的核心是对低能量的分解分析。与其明确地构建其Schwartz内核，不如使用Melrose率先使用的限制吸收原理进行更直接地进行，并最近通过Vasy扩展到零能量限制。
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 $ϕ(t,x)=c t^{-3}+\mathcal O(t^{-4+})$ for bounded $|x|$, where $c\in\mathbb C$ 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.
