Eikonal quasinormal modes and photon orbits of deformed Schwarzschild black holes

TL;DR

This paper explores how the eikonal correspondence between quasinormal modes and photon orbits extends to less symmetric, deformed Schwarzschild black holes, revealing that an averaged photon orbit radius maintains this relationship under small spacetime deformations.

Contribution

It demonstrates that the eikonal correspondence persists in axisymmetric, deformed Schwarzschild spacetimes through an averaged photon orbit radius, extending previous symmetry-dependent results.

Findings

Eikonal correspondence holds with an averaged photon orbit radius in deformed spacetimes.

The averaged radius aligns with the potential peak in the wave equation.

First-order deformations allow explicit identification of the correspondence.

Abstract

The geometric optics approximation provides an interpretation for eikonal correspondence that, in black-hole-containing spacetimes, connects high-frequency black hole quasinormal modes with closed photon orbits around said black hole. This correspondence has been identified explicitly for Schwarzschild, Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes, the violation of which can be a potential hint toward physics beyond General Relativity. Notably, the aforementioned black hole spacetimes have sufficient symmetries such that both the geodesic equations and the master wave equations are separable. The identification of the correspondence seems to largely rely on these symmetries. One naturally asks how the eikonal correspondence would appear if the spacetime were less symmetric. For a pioneering work in this direction, we consider in this paper a deformed Schwarzschild spacetime retaining only axisymmetry and stationarity. We show that up to the first order of spacetime deformations the eikonal correspondence manifests through the definition of the \textit{averaged} radius of trapped photon orbits along their one period. This averaged radius overlaps the potential peak in the master wave equation, which can be defined up to the first order of spacetime deformations, allowing the explicit identification of the eikonal correspondence.