Improved decay for quasilinear wave equations close to asymptotically flat spacetimes including black hole spacetimes

TL;DR

This paper establishes improved decay rates for solutions to quasilinear wave equations near asymptotically flat spacetimes, including black hole backgrounds, under weak assumptions on the metric's coefficients.

Contribution

It proves new pointwise decay estimates for quasilinear wave equations on black hole spacetimes with minimal assumptions, extending previous results to more general asymptotically flat metrics.

Findings

Solutions decay at rate $t^{-1- ext{min}( ext{delta},1)}$ near the light cone.

Decay estimates hold for equations with general asymptotically flat metrics and lower order terms.

Results apply to metrics approaching Schwarzschild or Kerr with small angular momentum.

Abstract

We study the quasilinear wave equation $\Box_{g}\phi=0$ where the metric $g = g(\phi,t,x)$ is close to and asymptotically approaches $g(0,t,x)$, which equals the Schwarzschild metric or a Kerr metric with small angular momentum, as time tends to infinity. Under only weak assumptions on the metric coefficients, we prove an improved pointwise decay rate for the solution $\phi$. One consequence of this rate is that for bounded $|x|$, we have the integrable decay rate $|\phi(t,x)| \le Ct^{-1-\min(\delta,1)}$ where $\delta>0$ is a parameter governing the decay, near the light cone, of the coefficient of the slowest-decaying term in the quasilinearity. We also obtain the same aforementioned pointwise decay rates for the quasilinear wave equation $(\Box_{\tilde g} + B^{\alpha}(t,x)\partial_\alpha + V(t,x))\phi=0$ with a more general asymptotically flat metric $\tilde g = \tilde g(\phi,t,x)$ and with other time-dependent asymptotically flat lower order terms.