Near-horizon modes and self-adjoint extensions of the Schroedinger operator

TL;DR

This paper studies scalar field dynamics near black hole horizons, revealing that low-energy modes are long-lived and governed by a Schroedinger operator with multiple self-adjoint extensions, which are physically equivalent for statistical analyses.

Contribution

It demonstrates the role of self-adjoint extensions in modeling near-horizon scalar modes and proposes a physical criterion for selecting relevant extensions.

Findings

Low-energy modes are long-lived near the horizon.

Self-adjoint extensions are parameterized by U(1) and are physically equivalent.

Results apply broadly to non-extremal, spherically symmetric black holes.

Abstract

We investigate the dynamics of scalar fields in the near-horizon exterior region of a Schwarzschild black hole. We show that low-energy modes are typically long-living and might be considered as being confined near the black hole horizon. Such dynamics are effectively governed by a Schroedinger operator with infinitely many self-adjoint extensions parameterized by $U(1)$, a situation closely resembling the case of an ordinary free particle moving on a semiaxis. Even though these different self-adjoint extensions lead to equivalent scattering and thermal processes, a comparison with a simplified model suggests a physical prescription to chose the pertinent self-adjoint extensions. However, since all extensions are in principle physically equivalent, they might be considered in equal footing for statistical analyses of near-horizon modes around black holes. Analogous results hold for any non-extremal, spherically symmetric, asymptotically flat black hole.