Dynamical Love Numbers

• Dynamical Love Numbers are frequency-dependent coefficients that quantify both conservative deformations and dissipative tidal absorption in compact objects.
• They are derived using perturbative methods like the Regge-Wheeler and Teukolsky equations, complemented by effective field theory to capture internal structure and rotation effects.
• Their influence on gravitational-wave phasing offers a unique probe into horizon-scale physics, resonant tidal phenomena, and potential deviations from classical General Relativity.

Dynamical Love Numbers are frequency-dependent tidal response coefficients that parameterize the linear, multipolar gravitational (or broader, electromagnetic or scalar) response of a compact object—such as a planet, neutron star, or black hole—to time-dependent external perturbing fields. While classical, static Love numbers represent the conservative, equilibrium shape response of an object to a time-independent tide, dynamical Love numbers encode both conservative and dissipative aspects of the response when the external field varies in time. The behavior of dynamical Love numbers is determined by the interior structure, rotation, and theoretical framework (e.g., General Relativity (GR), Newtonian gravity, or Effective Field Theory (EFT)) underlying the compact object's description.

1. Definition, Physical Meaning, and General Features

Given a compact object of mass $M$ subjected to an external tidal field oscillating with frequency $\omega$ and decomposed into spherical (or spin-weighted spheroidal) harmonics $(\ell,m)$, the dynamical Love number $k_{\ell m}(\omega)$ is defined via the large-distance (asymptotic) multipole expansion of the relevant perturbing field (metric, Weyl scalar, or potential): $$\psi_s(r\to\infty)\sim A_{\ell m}(\omega) r^{\ell-s} + B_{\ell m}(\omega) r^{-(\ell+1+s)} + \cdots$$ with $s$ the spin weight. The dimensionless dynamical Love number is

$$k_{\ell m}(\omega) \equiv N_{\ell s} \frac{B_{\ell m}(\omega)}{A_{\ell m}(\omega)}$$

with normalization factor $N_{\ell s}$ chosen for the correct Newtonian and static GR limits.

$k_{\ell m}(\omega)$ is in general complex, with

$$k_{\ell m}(\omega) = \Re k_{\ell m}(\omega) + i\, \Im k_{\ell m}(\omega)$$

where the real (conservative) part encodes the in-phase deformation, and the imaginary (dissipative) part encodes horizon absorption, phase-delay, or other energy-dissipating mechanisms. The dissipative response is always tied to horizon absorption for black holes, or, in the Newtonian case, to friction, viscosity, or radiative processes.

2. Dynamical Love Numbers in Black Hole Spacetimes

Schwarzschild Black Holes

For nonrotating black holes in four-dimensional GR, it is well-established that all static ($\omega \to 0$) Love numbers vanish (i.e., no static tidal deformation; universal "rigidity"). For time-dependent perturbations, however, a nonvanishing dynamical Love number arises at second order in frequency, with characteristic logarithmic running: $$k_{\ell}(\omega) = A_\ell (M\omega)^2 \left[ \ln\left(\frac{r}{2M}\right) + C_\ell \right] + \mathcal{O}(\omega^3)$$ where $A_\ell$ and $C_\ell$ are explicit combinatorial coefficients. The leading dissipative response appears at $\mathcal{O}(\omega)$, linked to the absorption of tidal energy by the horizon: $$\Im k_\ell(\omega) = B_\ell\, M\omega + \mathcal{O}(\omega^3)$$ with $B_\ell$ given by a factorial combination of $\ell$ (Combaluzier--Szteinsznaider et al., 4 Nov 2025, Chakraborty et al., 30 Jul 2025).

These results are obtained by solving the Regge-Wheeler or Zerilli equations in the frequency domain, imposing horizon-regular (purely ingoing) boundary conditions and expanding solutions in the small-frequency regime. The appearance of the logarithmic term in $k_{\ell}(\omega)$ manifests the renormalization-group running of the effective finite-size (tidal) operator in the point-particle EFT matched to GR, with the logarithmic coefficient fixed by the structure of classical UV divergences.

Kerr Black Holes

For Kerr (rotating) black holes with subextremal spin ($a < M$), the static limit of all Love numbers vanishes, even when $m \neq 0$. However, at nonzero frequency, dynamical Love numbers become nonzero: $$k_{\ell m}(\omega) = A_{\ell m}(a) (M\omega)^{2\ell+1} \left[1 + B_{\ell m}(a) \ln(M\omega) + \mathcal{O}(\omega)\right]$$ with explicit expressions for $A_{\ell m}$ and $B_{\ell m}$ in terms of spin parameter $a$ and $m$. The scaling $(M\omega)^{2\ell+1}$ indicates extreme suppression for astrophysically relevant frequencies ($M\omega\ll1$) (Perry et al., 2023).

Extremal Kerr Black Holes

In the extremal Kerr limit ($a=M$), $m\neq0$ modes acquire a finite real static Love number: $$k_{\ell m}^{\text{static, ext}} = -\frac{(i m)^{2\ell+1} \Gamma(-2\ell-1)\Gamma(\ell+1)}{\Gamma(2\ell+1)\Gamma(-\ell)}$$ while $m=0$ modes still have vanishing static Love numbers. The general dynamical Love number admits the closed low-frequency expansion: $$k_{\ell m}(\omega) = -\frac{(-1)^s i m^{2\ell+1}(\ell-s)!(\ell+s)!}{2(2\ell)!(2\ell+1)!}\left[1 + Q_\ell\,\omega + \mathcal{O}(\omega^2)\right]$$ where $Q_\ell$ contains both real and imaginary parts, so a purely imaginary (dissipative) piece appears at $O(\omega^0)$ and conservative/dissipative structure at $O(\omega^1)$ (Perry et al., 27 Dec 2024).

The calculation is efficiently carried out via the Leaver-MST method, which constructs the full solution as sums over hypergeometric (Coulomb) functions, connecting horizon-regular and infinity-regular expansions via analytically tractable recurrence relations.

3. Extensions: Planets, Neutron Stars, and Exotic Compact Objects

Newtonian Stars and Planets

For rotating fluid planets and non-relativistic stars, the dynamical Love number is naturally expressed as a mode sum over the normal oscillation modes: $$k_{lm}(\omega) = \frac{2\pi G}{(2l+1)R^{2l+1}} \sum_\alpha \frac{Q_{\alpha,lm}^2}{\varepsilon_\alpha(\omega_\alpha - \omega)}$$ where $Q_{\alpha,lm}$ are mode overlaps, $\omega_\alpha$ are mode frequencies, and $\varepsilon_\alpha$ normalization constants (Lai, 2021, Pnigouras et al., 2022). The resonance structure is prominent—when $\omega$ approaches a normal mode, $k_{lm}(\omega)$ exhibits a sharp resonant peak with width set by damping.

Rotation induces corrections at $\mathcal{O}(\Omega)$ in the dynamical Love number (for slow rotation), with prograde and retrograde mode contributions. For giant planets like Jupiter, the observed dynamical $k_2$ is suppressed by $\sim 4\%$ compared to the static, hydrostatic prediction, explained entirely by the dominance of the rotation-modified $f$-mode in $k_2(\omega)$.

Kerr-like Exotic Compact Objects (ECOs)

For horizonless, ultracompact objects with partial reflectivity, the frequency-dependent Love number $K_2(\omega)$ is analytically computable and displays key features:

• Linear-in-frequency rise at low frequencies for arbitrary nonzero reflectivity.
• Resonant oscillations tied to QNM frequencies.
• Suppression of $K_2$ with increasing spin.
• Discontinuity between the strictly static $K_2^{\text{static}}$ and $\lim_{\omega\to 0} K_2(\omega)$ due to the breakdown of independence of horizon-outgoing solutions in the static Teukolsky problem (Chakraborty et al., 2023).

This behavior is model-independent for small horizon deviations and informs parameterized waveform modeling for high-precision GW data analysis.

4. Renormalization, Effective Field Theory, and Scale Dependence

Dynamical Love numbers are best understood within the worldline EFT approach, which integrates out short-distance gravitational, electromagnetic, or matter interactions to produce higher-derivative (finite-size, "multipole") operators in the effective action. The tidal response coefficient (e.g., quadrupolar $\lambda_2(\omega)$) develops a scale dependence (running) with respect to the renormalization point $L$: $\lambda_\ell(L) = \const - \beta_{k_\ell} \ln\left(\frac{L}{r_h}\right)$ with $\beta_{k_\ell}$ given by overlap integrals of the background perturbation and perturbing EFT operator (Barbosa et al., 30 Jan 2025, Mandal et al., 2023). The running is logarithmic for $l\geq 3$ in neutral, $l\geq 2$ in charged black holes, and associated with classical UV divergences.

The calculation of dynamical Love numbers then involves matching classical GR (or alternative theory) solutions to the EFT response, with finite, scheme-dependent contributions remaining after logarithmic subtraction. These can, in principle, encode additional physical information—such as internal structure, heavy-field loops, or new physics at the horizon.

5. Observational Implications and Gravitational-Wave Signatures

Dynamical Love numbers yield corrections to GW phasing in binary inspirals, entering at high post-Newtonian (PN) order:

• For black holes, the conservative tidal term from $k_{2}(\omega)$ enters at 8PN in the GW phase, while the dissipative term enters at 4PN ($\nu_{2}(\omega)\sim M\omega$) in Schwarzschild and 2.5PN in Kerr (Chakraborty et al., 30 Jul 2025).
• The scale and frequency dependence induces logarithmic corrections, but the magnitude of these dynamical tidal contributions for black holes is orders of magnitude below current and next-generation GW detector sensitivity.
• However, for resonance excitation in objects with internal structure, the phase shift associated with the dynamical Love number can be large and accessible to LISA-like missions, scaling as $\Delta \Phi_{\mathrm{Res}} \propto k_2/q$ for mass ratio $q$ in EMRIs (Avitan et al., 2023).

In the context of new physics, such as dark-sector charges or Planck-scale horizon modifications, running Love numbers (from EFT corrections) and frequency-dependent $K_2(\omega)$ for Kerr-like ECOs may generate $\mathcal{O}(1)$ effects, especially near extremality or in charged scenarios. Consequently, GWs provide a direct probe of horizon-scale microphysics through precise measurement, or bounds, on dynamical Love numbers.

6. Mathematical and Computational Methodologies

The computation of dynamical Love numbers proceeds via:

• Solution of the radial Teukolsky equation (or Regge-Wheeler/Zerilli in Schwarzschild) with appropriate horizon-regular and far-field boundary conditions.
• Series expansion techniques (hypergeometric, Leaver-MST recurrence relations) for low-frequency (near-horizon) analysis; determination of connection coefficients translating horizon-ingoing to infinity-regular solutions.
• Matching in Effective Field Theory: Born series expansion of Green’s functions, dimensional regularization and renormalization to extract universal and scheme-dependent coefficients (Combaluzier--Szteinsznaider et al., 4 Nov 2025).
• Mode-sum and phase-space methods in Newtonian stars, extending to slow-rotation via symplectic orthogonality and multi-mode coupling analysis (Pnigouras et al., 2022).
• Decomposition—case-dependent—into real (conservative) and imaginary (dissipative) parts, or via expansion in powers of frequency to isolate order-by-order physical content (Bhatt et al., 2023).

These formalisms are robust to generalization for various internal structures, equation of state effects, and higher-derivative corrections.

7. Controversies and Current Open Questions

• The strict vanishing of static Love numbers in GR is a notable special case; generic modifications (EFT, charge, non-GR theories, and horizonless objects) render dynamical Love numbers nonzero, often at leading order in frequency.
• There exist two inequivalent definitions of tidal Love numbers for rotating black holes in the relativistic regime: splitting by real/imaginary parts or by separation into $\omega$-independent and $\omega$-dependent parts, leading to different interpretations—especially for dissipative effects (Bhatt et al., 2023).
• For charged scalar fields in Kerr-Newman backgrounds, the real part of the dynamical Love number is discontinuous as scalar charge $q\to0$; $k_{\ell m}(q\to 0) \propto 1/q$, yet vanishes identically at $q=0$ (Ma et al., 19 Aug 2024).
• The discontinuity between static Love numbers and the $\omega\to0$ limit of dynamical Love numbers in ECOs and certain black hole spacetimes reflects physical differences in boundary condition structure and the noncommutativity of frequency limits.

Debate continues about the physical measurability and interpretive uniqueness of these effects, especially in light of potential degeneracies with higher-PN point-particle corrections and uncertainties in GW phase modeling.

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In summary, dynamical Love numbers provide a rigorous, theoretically grounded, and observationally motivated framework for quantifying the time-dependent tidal response of compact objects, including black holes, neutron stars, and planets. They unify the mathematical formalism of perturbation theory, effective field theory, and stellar oscillations, while revealing novel connections between classical GR, quantum corrections, and possible new physics at the horizon scale.