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

Amos Ori
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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 $Ω$) to the MP.

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Amos Ori. 2003-12-02. Harmonic-gauge dipole metric perturbations for weak-field circular orbits in Schwarzschild spacetime. https://doi.org/10.1103/physrevd.70.124027

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