Universal Closed Form for Dynamical Love Numbers of Black Holes
Abstract
Black hole static Love numbers vanish, but their dynamical counterparts do not. We present the entire scheme-independent dynamical response F̅ℓ,s of a Schwarzschild black hole in closed form, to all orders in G, and for every spin s and multipole ℓ. The result is F̅ℓ,s/4πRS2ℓ+1=Φℓ,s(y̅)−1/2η Φℓ,s'(y̅) with y̅=−1/2η2τ and η=iωRS. Here Φℓ,s is simply the leading-log solution to the renormalization group equation, but lifting the running logarithm to τ=log(RS/R)−2∑k≥2ζk ηk−1 resums it to all orders. This tower of Riemann zeta values is the Newtonian phase in disguise: it originates from the long-range 1/r potential in the far zone, and is universal across multipole and spin. Our result exhibits a factorization pinned to three ingredients: the strength of horizon absorption, the running rate in the near zone, and the dressed logarithm in the far zone. Using shell effective field theory, we independently verify our formula for scalar, electromagnetic, and gravitational perturbations, reaching 𝒪(G15).
Versions
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BibTeX
@misc{solon2026universal,
title = {{Universal Closed Form for Dynamical Love Numbers of Black Holes}},
author = {Mikhail P. Solon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.19281},
eprint = {2606.19281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal Closed Form for Dynamical Love Numbers of Black Holes
AU - Mikhail P. Solon
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2606.19281
UR - https://arxiv.org/abs/2606.19281
AB - Black hole static Love numbers vanish, but their dynamical counterparts do not. We present the entire scheme-independent dynamical response F̅_ℓ,s of a Schwarzschild black hole in closed form, to all orders in G, and for every spin s and multipole ℓ. The result is F̅_ℓ,s/4πR_S^2ℓ+1=Φ_ℓ,s(y̅)−1/2η Φ_ℓ,s'(y̅) with y̅=−1/2η^2τ and η=iωR_S. Here Φ_ℓ,s is simply the leading-log solution to the renormalization group equation, but lifting the running logarithm to τ=log(R_S/R)−2∑_k≥2ζ_k η^k−1 resums it to all orders. This tower of Riemann zeta values is the Newtonian phase in disguise: it originates from the long-range 1/r potential in the far zone, and is universal across multipole and spin. Our result exhibits a factorization pinned to three ingredients: the strength of horizon absorption, the running rate in the near zone, and the dressed logarithm in the far zone. Using shell effective field theory, we independently verify our formula for scalar, electromagnetic, and gravitational perturbations, reaching 𝒪(G^15).
ER -