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Static response and Love numbers of Schwarzschild black holes

Lam Hui, Austin Joyce, Riccardo Penco, Luca Santoni, Adam R. Solomon

TL;DR

<3-5 sentence high-level summary>This work provides a comprehensive, action-based framework to compute the static linear response of Schwarzschild black holes to external massless perturbations of spins $0$, $1$, and $2$ in arbitrary spacetime dimension $D$, unifying the treatment across spins and backgrounds. By decomposing perturbations into spherical harmonics and deriving gauge-fixed quadratic actions, the authors obtain Schrödinger-type radial equations and exact Love-number expressions via hypergeometric (and Heun in special cases) analysis, then connect these results to a worldline EFT that encodes the BH's static responses as Wilson coefficients. A central result is the universal vanishing of all Love numbers in four dimensions, with new higher-$D$ insights showing nonzero vector-type gravitation Love numbers and dimension-dependent electromagnetic and scalar responses. The EFT matching clarifies the operator structure responsible for these responses and provides a framework for testing gravity through precise measurements of BH tidal responses while pointing to potential hidden symmetries in $D=4$ as a subject for further study.

Abstract

We derive the quadratic action for the physical degrees of freedom of massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild--(A)dS background in arbitrary dimensions. We then use these results to compute the static response of asymptotically flat Schwarzschild black holes to external fields. Our analysis reproduces known facts about black hole Love numbers, in particular that they vanish for all types of perturbation in four spacetime dimensions, but also leads to new results. For instance, we find that neutral Schwarzschild black holes polarize in the presence of an electromagnetic background in any number of spacetime dimensions except four. Moreover, we calculate for the first time black hole Love numbers for vector-type gravitational perturbations in higher dimensions and find that they generically do not vanish. Along the way, we shed some light on an apparent discrepancy between previous results in the literature, and clarify some aspects of the matching between perturbative calculations of static response on a Schwarzschild background and the point-particle effective theory

Static response and Love numbers of Schwarzschild black holes

TL;DR

<3-5 sentence high-level summary>This work provides a comprehensive, action-based framework to compute the static linear response of Schwarzschild black holes to external massless perturbations of spins , , and in arbitrary spacetime dimension , unifying the treatment across spins and backgrounds. By decomposing perturbations into spherical harmonics and deriving gauge-fixed quadratic actions, the authors obtain Schrödinger-type radial equations and exact Love-number expressions via hypergeometric (and Heun in special cases) analysis, then connect these results to a worldline EFT that encodes the BH's static responses as Wilson coefficients. A central result is the universal vanishing of all Love numbers in four dimensions, with new higher- insights showing nonzero vector-type gravitation Love numbers and dimension-dependent electromagnetic and scalar responses. The EFT matching clarifies the operator structure responsible for these responses and provides a framework for testing gravity through precise measurements of BH tidal responses while pointing to potential hidden symmetries in as a subject for further study.

Abstract

We derive the quadratic action for the physical degrees of freedom of massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild--(A)dS background in arbitrary dimensions. We then use these results to compute the static response of asymptotically flat Schwarzschild black holes to external fields. Our analysis reproduces known facts about black hole Love numbers, in particular that they vanish for all types of perturbation in four spacetime dimensions, but also leads to new results. For instance, we find that neutral Schwarzschild black holes polarize in the presence of an electromagnetic background in any number of spacetime dimensions except four. Moreover, we calculate for the first time black hole Love numbers for vector-type gravitational perturbations in higher dimensions and find that they generically do not vanish. Along the way, we shed some light on an apparent discrepancy between previous results in the literature, and clarify some aspects of the matching between perturbative calculations of static response on a Schwarzschild background and the point-particle effective theory

Paper Structure

This paper contains 61 sections, 254 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Outline:
  3. Conventions:
  4. The Schwarzschild--(anti-)de Sitter geometry
  5. Linearized fields in the black hole background
  6. Spin-0 field
  7. Massless spin-1 field
  8. Decomposition of perturbations
  9. Spin-1 vector mode
  10. Spin-1 scalar mode
  11. Electric-magnetic duality in $D=4$
  12. Massless spin-2 field
  13. Decomposition of perturbations
  14. Spin-2 tensor sector
  15. Spin-2 vector sector (odd parity)
  16. ...and 46 more sections

Figures (1)

  • Figure 1: Diagrammatic representation of static response in the EFT. The thick grey line is the worldline of the point particle. One of the external fields (wavy lines) behaves as a background---denoted by the red $\times$---which induces a response that can be read off at infinity. The computation of the static response can be carried out diagrammatically using this Feynman diagram.