Black hole stereotyping: Induced gravito-static polarization

TL;DR

This work analyzes the static finite-size response of black holes by formulating the static sector of the black hole effective action and computing the induced gravito-static polarization constants (Love numbers) for Schwarzschild BHs in arbitrary dimensions. Using a combination of microscopic perturbation theory and effective field theory matching, the authors derive explicit expressions for the Love numbers in terms of the dimensionless ratio l/(d-3) and horizon scales, demonstrating that 4d Love numbers vanish while higher dimensions yield nonzero values. They identify a classical renormalization-group flow for half-integer hat l and discuss negative values as potential non-spherical instabilities, linking hypergeometric-function solutions to EFT counterterms. The results challenge the notion that black holes have no non-minimal world-line couplings in d>4 and provide a detailed map of BH finite-size effects across dimensions, with implications for higher-dimensional BH dynamics and stability.

Abstract

We discuss the black hole effective action and define its static subsector. We determine the induced gravito-static polarization constants (electric Love numbers) of static black holes (Schwarzschild) in an arbitrary dimension, namely the induced mass multipole as a result of an external gravitational field. We demonstrate that in 4d these constants vanish thereby settling a disagreement in the literature. Yet in higher dimensions these constants are non-vanishing, thereby disproving (at least in d>4) speculations that black holes have no effective couplings beyond the point particle action. In particular, when l/(d-3) is half integral these constants demonstrate a (classical) renormalization flow consistent with the divergences of the effective field theory. In some other cases the constants are negative indicating a novel non-spherical instability. The theory of hypergeometric functions plays a central role.

Figures (7)

• Figure 1:
• Figure 2: Experimental set-up for measuring the induced gravito-static polarization constants. The black hole (gray) is positioned at an equilibrium point between two fixed stars (black). Its mass multipole moments are measured on a far away envelope (dashed sphere).
• Figure 3: The idea behind the black hole effective action: the full black hole space-time geometry is replaced by a point particle together with some effective interactions with its slowly varying background.
• Figure 4: A graph of the raw gravito-static Love numbers ${\hat{\lambda}}$ of a black hole as a function of ${\hat{l}}\equiv l/(d-3)$ where $l$ is the spherical harmonic index, and $d$ is the total space-time dimension. Units are such that $\rho_0=1$ where $\rho_0$ is the location of the horizon in isotropic coordinates (\ref{['def:rho0']})
• Figure 5: All necessary diagrams to match $\lambda_2$ in isotropic coordinates. Solid, dashed and wavy lines represent propagators of $\phi,\psi$ and $\hat{\sigma}$ respectively. Cross indicates insertion of $\bar{\psi}$. Wave numbers flow into the vertex.