Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere

Abstract

In our previous work [Angelopoulos, Y., Aretakis, S. and Gajic, D. Latetime asymptotics for the wave equation on spherically symmetric stationary backgrounds. 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. ON, M1C 1A4, Canada,

Late-time asymptotics for the wave equation on extremal Reissner–Nordström backgrounds

“…On the other hand, the heuristic work of Leaver [42] related the late-time power law to the branch cut at ω = 0 in the Laplace transform of Green's function for each fixed angular frequency. This is consistent with the results of [24,25], in view of the fact that the geometric origin of the constant I (1) 0 [ψ] is related to an obstruction to the invertibility of the time operator T = ∂ t in a suitable function space (and hence is related to the ω = 0 frequency in the Fourier space). Indeed, restricting (strictly) to the future of the bifurcation sphere where T = 0, we have that an obstruction to the invertibility of the operator T is the existence of a conservation law along the null infinity I + : For solutions ψ to the wave equation (1.1) on Reissner-Nordström spacetimes, the limits…”

“…Here τ denotes a global time parameter and ω ∈ S 2 . 2 The following global quantitative estimates which establish rigorously the above asymptotics were obtained for general sub-extremal Reissner-Nordström spacetimes in [24,25]:…”

“…It is important to emphasize that the approach of [24,25] is based on purely physical space techniques. On the other hand, the heuristic work of Leaver [42] related the late-time power law to the branch cut at ω = 0 in the Laplace transform of Green's function for each fixed angular frequency.…”

“…A definitive proof of the precise late-time asymptotics of solutions to the wave equation on the general sub-extremal Reissner-Nordström backgrounds, including the celebrated Schwarzschild family of black holes, was obtained in the recent series of papers [23][24][25] confirming, in particular, Price's heuristics (for more details and references see Section 1.2).…”

Horizon Hair of Extremal Black Holes and Measurements at Null Infinity

“…If follows that the unique obstruction to inverting the operator T 2 is the non-vanishing of I (1) [ψ]. The relevance of I (1) [ψ] became apparent in [55] where the precise latetime asymptotics were obtained for compactly supported initial data: The following expression of I (1) [ψ] was obtained in terms of compactly supported initial data on Σ 0 in [56]:…”