On the instability of ultracompact horizonless spacetimes

TL;DR

This paper investigates the linear stability of horizonless ultracompact objects with light rings, revealing that such objects are unstable when they contain ergoregions but can be stable outside ergoregions, depending on their compactness.

Contribution

It demonstrates the conditions under which horizonless, ultracompact spacetimes are linearly unstable or stable, highlighting the role of ergoregions and the surface location.

Findings

Objects with ergoregions are linearly unstable.

Objects outside the ergoregion are stable on long timescales.

Zero-modes are associated with the ergoregion instability.

Abstract

Motivated by recent results reporting the instability of horizonless objects with stable light rings, we revisit the linearized stability of such structures. In particular, we consider an exterior Kerr spacetime truncated at a surface where Dirichlet conditions on a massless scalar are imposed.This spacetime has ergoregions and light rings when the surface is placed sufficiently deep in the gravitational potential. We establish that the spacetime is linearly, mode-unstable when it is sufficiently compact, in a mechanism associated with the ergoregion. In particular, such instability has associated zero-modes. At large multipole number the critical surface location for zero modes to exist is precisely the location of the ergosurface along the equator. We show that such modes don't exist when the surface is outside the ergoregion, and that any putative linear instability mechanism acts on timescales $\tau \gtrsim 10^5 M$, where $M$ is the black hole mass. Our results indicate therefore that at least certain classes of objects are linearly stable in the absence of ergoregions, even if rotation and light rings are present.