The Love for Three Logarithms: Universal Running of Dynamical Love Numbers
Abstract
The static tidal Love numbers of black holes vanish in general relativity (GR), but are generically non-zero, and in fact 'run' logarithmically with scale, in more general theories or beyond the static limit. We expose a model-agnostic framework for the running of Love numbers at the dynamical level, valid for any tidal perturbation on any static, spherically symmetric, asymptotically flat background, based on the near-zone expansion in powers of the frequency ω and on the monodromy at spatial infinity. A master recurrence formula determines the monodromy generator to all orders in ω2, whose three independent entries fully encode the running of the Love numbers. This structure explains the quadratic nature of the renormalization group equation, here proved to all orders and for any type of Love number. We discuss the scheme dependence of the logarithmic coefficients, which is made transparent by the monodromy form. In particular, this clarifies that the anomalous exponents of the tidal field, the so-called 'renormalized angular momentum' of the Mano-Suzuki-Takasugi formalism, are scheme-invariant and directly computable on any spacetime, without need of explicit solutions. We discuss several applications to black holes, including the derivation of a closed-form formula for the leading dynamical running of axial Love numbers of Reissner-Nordström black holes, proving in particular that it vanishes in the extremal limit. For generic black holes beyond GR, we prove that dynamical running probes deformations of the metric at lower multipoles than the static one, and we illustrate the power of the method by deriving some exact results for the Hayward and Bardeen regular black holes.
Related
- Universal Closed Form for Dynamical Love Numbers of Black Holes
- Universal Dynamical Response to Slow Driving in Chaotic Systems
- Universal Relations Meet Causality: Bounding the Dynamical Love Number of Neutron Stars
- Dynamical Decoupling using Universal Optimal Tracking
- Universal Construction of Generalized Lyapunov Functions for Nonlinear Dynamical Systems Using Physics-Informed Neural Networks
BibTeX
@misc{garciasaenz2026love,
title = {{The Love for Three Logarithms: Universal Running of Dynamical Love Numbers}},
author = {Sebastian Garcia-Saenz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01152},
eprint = {2610.01152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01152},
}
RIS
TY - UNPB
TI - The Love for Three Logarithms: Universal Running of Dynamical Love Numbers
AU - Sebastian Garcia-Saenz
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01152
UR - https://arxiv.org/abs/2610.01152
AB - The static tidal Love numbers of black holes vanish in general relativity (GR), but are generically non-zero, and in fact 'run' logarithmically with scale, in more general theories or beyond the static limit. We expose a model-agnostic framework for the running of Love numbers at the dynamical level, valid for any tidal perturbation on any static, spherically symmetric, asymptotically flat background, based on the near-zone expansion in powers of the frequency ω and on the monodromy at spatial infinity. A master recurrence formula determines the monodromy generator to all orders in ω^2, whose three independent entries fully encode the running of the Love numbers. This structure explains the quadratic nature of the renormalization group equation, here proved to all orders and for any type of Love number. We discuss the scheme dependence of the logarithmic coefficients, which is made transparent by the monodromy form. In particular, this clarifies that the anomalous exponents of the tidal field, the so-called 'renormalized angular momentum' of the Mano-Suzuki-Takasugi formalism, are scheme-invariant and directly computable on any spacetime, without need of explicit solutions. We discuss several applications to black holes, including the derivation of a closed-form formula for the leading dynamical running of axial Love numbers of Reissner-Nordström black holes, proving in particular that it vanishes in the extremal limit. For generic black holes beyond GR, we prove that dynamical running probes deformations of the metric at lower multipoles than the static one, and we illustrate the power of the method by deriving some exact results for the Hayward and Bardeen regular black holes.
ER -