Superradiant instabilities for short-range non-negative potentials on Kerr spacetimes and applications

TL;DR

This paper demonstrates that adding short-range non-negative potentials to wave equations on Kerr spacetimes can induce real and exponentially growing modes, contrasting with the stability observed in stationary asymptotically flat spacetimes.

Contribution

It constructs explicit real and growing mode solutions for wave equations with potentials on Kerr spacetimes, showing instability phenomena absent in similar stationary settings.

Findings

Existence of real mode solutions with potentials on Kerr spacetimes.

Construction of spacetimes with trapped null geodesics admitting growing modes.

Failure of zero-frequency stability criterion on Kerr spacetime.

Abstract

The wave equation $\square_{g_{M,a}}\psi=0$ on subextremal Kerr spacetimes $(\mathcal{M}_{M,a},g_{M,a})$, $0<|a|<M$, does not admit real mode solutions, as was established by Shlapentokh-Rothman. In this paper, we show that the absence of real modes does not persist under the addition of an arbitrary short-range non-negative potential $V$ to the wave equation or under changes of the metric $g_{M,a}$ in the far away region of $\mathcal{M}_{M,a}$ (retaining the causality of $T$ there). In particular, we first establish, for any $0<|a|<M$, the existence of real mode solutions $\psi$ to equation $\square_{g_{M,a}}\psi-V\psi=0$, for a suitably chosen time-independent real potential $V$ with compact support in space, satisfying $V\ge0$. Exponentially growing modes are also obtained after perturbing the potential $V$. Then, as an application of the above result, we construct a family of spacetimes $(\mathcal{M}_{M,a},g_{M,a}^{(def)})$ which are compact in space perturbations of $(\mathcal{M}_{M,a},g_{M,a})$, have the same symmetries as $(\mathcal{M}_{M,a},g_{M,a})$ and moreover admit real and exponentially growing modes. These spacetimes contain stably trapped null geodesics, but we also construct a more complicated family of spacetimes with normally hyperbolic trapped set, admitting real and exponentially growing modes, at the expense of having conic asymptotics. The above results are in contrast with the case of stationary asymptotically flat (or conic) spacetimes $(\mathcal{M},g)$ with a globally timelike Killing field $T$, where real modes for equation $\square_{g}\psi-V\psi=0$ are always absent, giving a useful zero-frequency continuity criterion for showing stability for a smooth family of equations $\square_{g}\psi-V_{\lambda}\psi=0$, with $\lambda\in[0,1]$ and $V_{0}=0$. We show explicitly that this criterion fails on Kerr spacetime.