分歧球邻域标量扰动的渐近性
Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere
摘要：
在我们以前的工作[Y。Angelopoulos，S. Aretakis和D. Gajic，Late-time assistantics for the wave equation on spherically symmetric stationary backgrounds，in Advances in Mathematics 323（2018），529-621]我们证明了在史瓦西背景上具有光滑、紧支持的初始数据的波动方程的解的精确首阶后期渐近中的系数与时间反演的纽曼-彭罗斯常数（TINP）成比例，也就是相关时间积分的纽曼-彭罗斯常数。时间积分（以及TINP常数）被正则地定义在任何柯西超曲面的依赖域中，沿着该柯西超曲面，固定的Killing场是非零的。结果表明，在与未来事件视界相交的Cauchy超曲面上，可以得到关于初始数据的后期多项式尾的显式表达式.在本文中，我们通过一个新的几何解释的TINP常数在柯西超曲面上的修改的梯度通量，将上述结果推广到柯西超曲面相交的分歧球。我们表明，没有呼吁的时间积分建设，一般的守恒定律，这些梯度通量。这使得我们可以用柯西超曲面上的初始数据来表示TINP常数，因为柯西超曲面的时间积分结构是不存在的。
英文摘要：
In our previous work [Y. Angelopoulos, S. Aretakis, and D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric stationary backgrounds, in Advances in Mathematics 323 (2018), 529-621] we showed that the coefficient in the precise leading-order late-time asymptotics for solutions to the wave equation with smooth, compactly supported initial data on Schwarzschild backgrounds is proportional to the time-inverted Newman-Penrose constant (TINP), that is the Newman-Penrose constant of the associated time integral. The time integral (and hence the TINP constant) is canonically defined in the domain of dependence of any Cauchy hypersurface along which the stationary Killing field is non-vanishing. As a result, an explicit expression of the late-time polynomial tails was obtained in terms of initial data on Cauchy hypersurfaces intersecting the future event horizon to the future of the bifurcation sphere. In this paper, we extend the above result to Cauchy hypersurfaces intersecting the bifurcation sphere via a novel geometric interpretation of the TINP constant in terms of a modified gradient flux on Cauchy hypersurfaces. We show, without appealing to the time integral construction, that a general conservation law holds for these gradient fluxes. This allows us to express the TINP constant in terms of initial data on Cauchy hypersurfaces for which the time integral construction breaks down.
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