Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole

Understanding the behaviour of linear waves on black hole backgrounds is a central problem in general relativity, intimately connected with the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein equations--a major open question in the subject. Nonetheless, it is only very recently that even the most basic boundedness and quantitative decay properties of linear waves have been proven in a suitably general class of black hole exterior space...

We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics,...

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determ...

We consider the Cauchy problem for the (non-linear) Maxwell-Charged-Scalar-Field equations with spherically symmetric initial data, on a sub-extremal Reissner--Nordstr\"{o}m or Schwarzschild exterior space-time. We prove that the solutions are bounded and decay at an inverse polynomial rate towards time-like infinity and along the black hole event horizon, provided the charge of the Maxwell equation is sufficiently small. This condition is in particular satisfied for small da...

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as $t^{-2\ell-3}$ for $t \to \infty$ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be $t^{-2\ell-4}$. We give a proof of $t^{-2\ell-2}$ decay for general data in the form of weighted $L^1$ to $L^\infty$ bounds for solutions of the Regge--Wheeler equ...

We study the behaviour of smooth solutions to the wave equation, $\square_g\psi=0$, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards $r=0$ and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entir...

In this paper, we derive the early-time asymptotics for fixed-frequency solutions $\phi_\ell$ to the wave equation $\Box_g \phi_\ell=0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on $\mathcal I^-$ and polynomially decaying data, $r\phi_\ell\sim t^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $\partial...

We present short proofs of integrated local energy decay estimates on Schwarzschild, extremal Reissner-Nordstr\"om, and Schwarzschild-de Sitter spacetimes. The proofs employ novel global physical space multipliers, which, besides their remarkable simplicity, (a) are directly derivable from the geodesic flow, (b) do not require decomposition into spherical harmonics, and (c) whose boundary terms can be controlled by the conserved $T$-energy alone. We also elaborate on the inti...

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is ...

We prove that sufficiently regular solutions to the wave equation $\Box_{g_K}\Phi=0$ on the exterior of a sufficiently slowly rotating Kerr black hole obey the estimates $|\Phi|\leq C (t^*)^{-3/2+\eta}$ on a compact region of $r$. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski and Schwarzschild spacetime. This result improves the known robust decay rates that are proved using the vector field method in ...