Love at First Loop: Quantum Deformability of Near-Extremal Black Holes

The static Love numbers of black holes vanish in classical, four-dimensional, asymptotically flat General Relativity. We study how this changes for the response of near-extremal Reissner–Nordström black holes to a minimally coupled massless scalar, at energies where the Schwarzian mode of the near-horizon throat fluctuates strongly. The black hole then responds as a quantum system with a spectrum of states. Its static response is a polarizability, given by virtual transitions between them. For $\mathrm{\ell }=0$ the static Love number vanishes, by the shift symmetry of the scalar, but for multipoles $\mathrm{\ell }\ge 1$ the static Love numbers are nonzero: the finite rotational energy of the spin-$\mathrm{\ell }$ intermediate states spoils the cancellation responsible for the classical result. Since the throat no longer has a classical geometry from which to read off a tidal response, we define it through the retarded Green’s function, and compute its full frequency dependence at leading order in $r_{+}\omega$. Its conservative part requires renormalization: the near-horizon theory fixes the coefficient of its logarithm, and with it a dependence on the energy that no local term can remove, while its finite local terms are matching data of the four-dimensional theory. The dissipative part is finite and scheme independent. For $\mathrm{\ell }\ge 1$, it vanishes at low frequencies when the energy is below the rotational gap, while at the gap threshold it becomes non-analytic in frequency and produces a late-time $t^{-3\mathrm{/}2}$ tail.