Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes

Abstract

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordström families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law τ −3 as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

Strong cosmic censorship for a scalar field in a Born-Infeld–de Sitter black hole

“…When the perturbation decays slowly enough, SCC could be valid. In fact, a perturbation in an asymptotically flat black hole satisfies an inverse power law decay [13][14][15], which ensures the mass-inflation mechanism is strong enough to render the Cauchy horizon unstable upon perturbation [8,16]. On the other hand, it was observed that a remnant perturbation can exponentially decay in a black hole in asymptotically dS space-time [17][18][19][20][21][22][23][24], which implies that the perturbation might have chance to decay fast enough to violate SCC.…”

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]:…”

Strong cosmic censorship conjecture in higher curvature gravity

“…On the other hand, for asymptotically de Sitter spacetimes, the perturbations at late times also decay exponentially, which has the possibility of being cancelled by the exponential growth at the Cauchy horizon, leading to extension of the spacetime beyond the Cauchy horizon [9,10]. More precisely, for the case of asymptotically de Sitter spacetimes, e.g., Reissner-Norsdröm-de Sitter black hole, the perturbation attains an exponentially decaying late-time tail Φ ∼ e −αu , where α = −Im(ω) is the spectral gap related to the lowest-lying quasi-normal frequency.…”