Dynamical Love numbers of black holes: Theory and gravitational waveforms

TL;DR

Schwarzschild black holes have vanishing static tidal Love numbers, but the dynamical, frequency-dependent response is nontrivial. The authors deploy the Teukolsky formalism in advanced null coordinates to define a full-solution tidal response ${}_{s}\mathcal{F}_{\ell m_z}$, obtaining a clean small-$\omega$ expansion that reproduces linear dissipation and reveals a nonzero conservative TLN at ${\cal O}(\omega^{2})$ with a logarithmic running. In a post-Newtonian embedding, dynamical TLNs contribute an 8PN phase correction to gravitational waves, with running terms that must be matched to an EFT Wilson coefficient; however, gauge-invariant matching and realistic amplitudes imply these effects are too small to detect in BH binaries with planned detectors. The work clarifies ambiguities in BH tidal responses and provides a framework for connecting BH perturbation theory to worldline EFT descriptions, with potential observational relevance mainly for neutron stars where dynamical tides can be larger. Overall, the results demonstrate a theoretically consistent, perturbative picture in which Schwarzschild BHs may exhibit a nonzero quadratic dynamical TLN, but the imprint on GWs is unlikely to be observable.

Abstract

In General Relativity, the static tidal Love numbers of black holes vanish identically. Whether this remains true for time-dependent tidal fields -- i.e., in the case of dynamical tidal Love numbers -- is an open question, complicated by subtle issues in the definition and computation of the tidal response at finite frequency. In this work, we analyze the dynamical tidal perturbations of a Schwarzschild black hole to quadratic order in the tidal frequency. By employing the Teukolsky formalism in advanced null coordinates, which are regular at the horizon, we obtain a particularly clean perturbative scheme. Furthermore, we introduce a response function based on the full solution of the perturbation equation which does not depend on any arbitrary constant. Our analysis recovers known results for the dissipative response at linear order and the logarithmic running at quadratic order, associated with scale dependence in the effective theory. In addition, we find a finite, nonvanishing conservative correction at second order in frequency, thereby possibly demonstrating a genuine dynamical deformation of the black hole geometry. Although removing any ambiguity in the dynamical tidal response would require a matching with some gauge-invariant coefficient, we assess the impact of these effects on the gravitational-wave phase. These contributions enter at eighth post-Newtonian order, and can be expressed in terms of generic $\mathcal{O}(1)$ coefficients, which have to be matched to the perturbative result. Regardless of the matching ambiguities, we argue that such corrections are too small to be observable even with future-generation gravitational wave detectors. Moreover, the corresponding phase shifts are degenerate with unknown point-particle contributions entering at the same post-Newtonian order.

Figures (1)

• Figure 1: Relative error on the dynamical TLN parameter $\Lambda_{(2)}$, for an equal-mass, nonspinning binary BH system with individual masses $m_\text{\tiny BH}$, observed by ET (left panel) and LISA (right panel) optimistically assuming a luminosity distance of $d_L = 100\,{\rm Mpc}$ and $d_L = 1\,{\rm Gpc}$, respectively. The colored lines identify different choices of values of the dynamical TLN coefficients $\bar{\lambda}_{(2)}, \beta_{(2)}$. A measurement with $100\%$ accuracy would correspond to $\sigma_{\Lambda_{(2)}}/\Lambda_{(2)}=1$ (dashed horizontal line).