Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere

TL;DR

This paper extends previous results on late-time wave decay on Schwarzschild backgrounds by providing a new geometric interpretation of the TINP constant, enabling analysis on more general Cauchy hypersurfaces near the bifurcation sphere.

Contribution

It introduces a novel geometric approach to express the TINP constant via gradient fluxes, broadening the applicability of late-time asymptotics analysis.

Findings

Established a conservation law for gradient fluxes on Cauchy hypersurfaces.

Expressed the TINP constant in terms of initial data without time integral construction.

Extended late-time asymptotics to hypersurfaces intersecting the bifurcation sphere.

Abstract

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.