On the gravitational polarizability of black holes

TL;DR

This work defines and computes the gravitational shape Love numbers $h_l$ for nonrotating black holes, quantifying horizon deformation under external tidal fields and comparing them to electromagnetic analogs. Using Weyl metrics, the authors derive a horizon-curvature expansion, showing that in the linear, far-away limit the multipole coefficients scale as $c_l\approx n_l T_l M^l$ and yield $h_l=\frac{l+1}{l-1}\frac{(l!)^2}{2(2l)!}$, with $h_2=1/4$, $h_3=1/20$, $h_4=1/84$, and rapid decay for large $l$. The electromagnetic case yields $h_l^{EM}=\frac{l!(l+1)!}{(2l)!}$ and the relation $h_l^{EM}=2(l-1)h_l$, with near-horizon limits $h_l^{EM}(2M)=2l+1$ indicating a conducting-sphere-like horizon response as a charge approaches the horizon. In the near-horizon gravitational limit, a test mass $m$ yields a dressed response with $c_l^{lin}(m)|_{b\to M}=(2l+1)\frac{m}{\epsilon}$, leading to a localized horizon bulge described by an explicit $\delta R(\mu)/R_0$ form and a logarithmic spike at the pole, underscoring the dramatic horizon-level bending of spacetime in this regime. Overall, the paper clarifies the gravitational and electromagnetic polarizabilities of BH horizons and highlights the contrast with Newtonian and flat-space conductor analogies, including connections to neutron-star results via compactness.

Abstract

The gravitational polarizability properties of black holes are compared and contrasted with their electromagnetic polarizability properties. The "shape" or "height" multipolar Love numbers h_l of a black hole are defined and computed. They are then compared to their electromagnetic analogs h_l^{EM}. The Love numbers h_l give the height of the l-th multipolar "tidal bulge" raised on the horizon of a black hole by faraway masses. We also discuss the shape of the tidal bulge raised by a test mass m, in the limit where m gets very close to the horizon.

Figures (1)

• Figure 1: The shape of a meridian section of the "tidal bulge" raised by a test-mass $m$ located at the north pole of the horizon of a BH.