Kerr metric perturbations for second-order self-force theory
Abstract
We present a calculation of the first-order metric perturbation arising from a point mass moving on a circular, equatorial orbit in Kerr spacetime. The metric perturbation is specifically aimed at meeting the requirements of second-order self-force theory: it is highly accurate; incorporates all contributions, including those associated with perturbations to the black hole's mass and angular momentum; is in Lorenz gauge near the worldline and in (near) Bondi--Sachs gauge at large radius; and can be efficiently evaluated across the entire exterior spacetime. The Lorenz gauge gives a well-understood singular structure at the particle, while the (near) Bondi--Sachs gauge ensures the asymptotic fall-off needed for an asymptotically flat second-order source. Our formalism reconstructs the metric from separable Teukolsky-type equations, decomposed into spin-weighted spheroidal harmonic modes and solved spectrally on hyperboloidal slices. The resulting (ℓ,m)-mode sums converge rapidly across most of the domain, but slowly near the point mass; we overcome this with a novel use of a puncture to extract accurate m-mode data, yielding output accurate to ∼ 10−10 per m-mode and suitable for use at second order. Our metric-perturbation results cover a range of Kerr spins −0.9 ≤ a/M ≤ 0.9 and orbital radii out to 50M. In addition to its immediate application in second-order self-force calculations, our results can also be used as input to other calculations involving Kerr metric perturbations, including studies of environmental effects and quasinormal modes.
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BibTeX
@misc{leather2026kerr,
title = {{Kerr metric perturbations for second-order self-force theory}},
author = {Benjamin Leather and Samuel D. Upton and Patrick Bourg and Kevin O. Cunningham and Sam R. Dolan and Conor Dyson and Zachary Nasipak and Nami Nishimura and Andrew Spiers and Maarten van de Meent and Barry Wardell},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03888},
eprint = {2611.03888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03888},
}
RIS
TY - UNPB
TI - Kerr metric perturbations for second-order self-force theory
AU - Benjamin Leather
AU - Samuel D. Upton
AU - Patrick Bourg
AU - Kevin O. Cunningham
AU - Sam R. Dolan
AU - Conor Dyson
AU - Zachary Nasipak
AU - Nami Nishimura
AU - Andrew Spiers
AU - Maarten van de Meent
AU - Barry Wardell
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03888
UR - https://arxiv.org/abs/2611.03888
AB - We present a calculation of the first-order metric perturbation arising from a point mass moving on a circular, equatorial orbit in Kerr spacetime. The metric perturbation is specifically aimed at meeting the requirements of second-order self-force theory: it is highly accurate; incorporates all contributions, including those associated with perturbations to the black hole's mass and angular momentum; is in Lorenz gauge near the worldline and in (near) Bondi--Sachs gauge at large radius; and can be efficiently evaluated across the entire exterior spacetime. The Lorenz gauge gives a well-understood singular structure at the particle, while the (near) Bondi--Sachs gauge ensures the asymptotic fall-off needed for an asymptotically flat second-order source. Our formalism reconstructs the metric from separable Teukolsky-type equations, decomposed into spin-weighted spheroidal harmonic modes and solved spectrally on hyperboloidal slices. The resulting (ℓ,m)-mode sums converge rapidly across most of the domain, but slowly near the point mass; we overcome this with a novel use of a puncture to extract accurate m-mode data, yielding output accurate to ∼ 10^−10 per m-mode and suitable for use at second order. Our metric-perturbation results cover a range of Kerr spins −0.9 ≤ a/M ≤ 0.9 and orbital radii out to 50M. In addition to its immediate application in second-order self-force calculations, our results can also be used as input to other calculations involving Kerr metric perturbations, including studies of environmental effects and quasinormal modes.
ER -