Harmonic-gauge dipole metric perturbations for weak-field circular orbits in Schwarzschild spacetime

TL;DR

This paper computes the harmonic-gauge dipole metric perturbations caused by a particle in a weak-field circular orbit around a Schwarzschild black hole, revealing anomalous behavior in the Newtonian limit and deriving higher-order corrections.

Contribution

It provides explicit expressions for the metric perturbations in the Newtonian limit and uncovers their anomalous growth as the black hole mass approaches zero, highlighting pathologies.

Findings

Metric perturbations grow as black hole mass decreases.

Perturbations do not approach flat-space Coulomb values in the Newtonian limit.

Next-order correction in orbital frequency is derived.

Abstract

We calculate the harmonic-gauge even l=1 mode of the linear metric perturbation (MP) produced by a particle in a weak-field circular orbit around a Schwarzschild black hole (BH). We focus on the Newtonian limit, i.e. the limit in which the mass M of the central BH approaches zero (while fixing the orbital radius and the small-object mass), and obtain explicit expressions for the MP in this limit. We find that the MP are anomalous in this limit, namely, they do not approach their standard, Coulomb-like, flat-space values. Instead, the MP grows on approaching the BH, and this growth becomes worse as M decreases. This anomalous behavior leads to some pathologies which we briefly discuss. We also derive here the next-order correction (in the orbital frequency $\Omega $) to the MP.