Dynamical Tidal Love Numbers

• Dynamical Tidal Love Numbers are frequency-dependent coefficients that quantify the linear tidal response of gravitating bodies to time-dependent external forces, incorporating both conservative and dissipative effects.
• They are derived using diverse methods such as Newtonian mode sums, relativistic perturbation theory, and effective field theory, revealing key insights into tidal interactions in rotating stars, planets, and compact objects.
• Empirical studies, including Jupiter’s 4% correction in its quadrupolar Love number and analyses of black-hole responses, illustrate their practical role in probing internal structure and rotational coupling.

Dynamical tidal Love numbers are frequency-dependent coefficients that quantify the linear tidal response of a gravitating body to a time-dependent external field. In the simplest formulation, they generalize the static or hydrostatic Love numbers by retaining the forcing frequency and therefore the coupling to internal degrees of freedom such as normal modes, inertial motions, or higher-derivative worldline operators. Across the contemporary literature, the same underlying idea appears in several technically distinct settings: Newtonian mode sums for rotating stars and giant planets, membrane and viscoelastic descriptions of satellites and solid bodies, Teukolsky- or Regge–Wheeler–based perturbation theory for compact objects and black holes, and point-particle effective field theory descriptions in which dynamical response is encoded by frequency-dependent Wilson coefficients (Lai, 2021, Pnigouras et al., 2022, Dewberry et al., 2021, Combaluzier--Szteinsznaider et al., 4 Nov 2025, Chakraborty et al., 30 Jul 2025, Jarequi et al., 12 Mar 2026).

1. Definition and conceptual scope

A dynamical tidal Love number is a response coefficient at finite forcing frequency. In a rotating-fluid planetary formulation, for a given spherical-harmonic component $(l,m)$, the Love number is defined by

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$

where $U$ is the imposed tidal potential and $\delta\Phi$ is the self-gravitational response at the surface (Lai, 2021). In rapidly rotating planets and stars, the same quantity is written componentwise as

$k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$

with forcing frequency

$\omega_m=m(\Omega_o-\Omega),$

so that the response is explicitly evaluated in the rotating frame (Dewberry et al., 2021).

In relativistic response theory, the induced multipole moments are commonly written as

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$

and the frequency-dependent coefficient is expanded as

$$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$

with $\kappa_{\ell m}$ the conservative part and $\nu_{\ell m}$ the dissipative part (Luca et al., 2024). Closely related EFT treatments instead use retarded response kernels $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$0, whose low-frequency expansions define static, dissipative, and dynamical coefficients order by order in $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$1 (Combaluzier--Szteinsznaider et al., 4 Nov 2025).

The static or hydrostatic Love number is the zero-frequency limit only when that limit is regular. Several recent works stress that the $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$2 limit of a dynamical response need not coincide with the result of solving the exactly static problem. For Kerr-like compact objects with reflectivity, the zero-frequency limit of the frequency-dependent tidal Love numbers is discontinuous, so the strictly static TLNs differ from the static limit of the frequency-dependent TLNs (Chakraborty et al., 2023). This suggests that “dynamical” and “static” are not merely different computational approximations, but can correspond to inequivalent response coefficients.

2. Response formalisms and mode decompositions

In Newtonian fluid theory, the dynamical Love number is naturally expressed as a sum over normal modes. For Jupiter, the mode-amplitude equation in the corotating frame leads to the central formula

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$3

where $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$4 is the tidal overlap, $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$5 a rotationally modified normalization, and $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$6 the detuning from forcing (Lai, 2021). The same logic appears in the Newtonian theory of spinning stars, where the effective Love number is written as a mode sum over the stellar oscillation spectrum, but only after a phase-space expansion is used to respect rotating-star symplectic orthogonality (Pnigouras et al., 2022).

For a nonrotating star, the effective Love number takes the form

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$7

where $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$8 is the mode mass multipole moment and $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$9 the mode normalization (Pnigouras et al., 2022). In the rotating case, the full response contains contributions from both prograde and retrograde mode branches, and the Love-number expression acquires explicit $U$0-mixing because rotation breaks spherical symmetry (Pnigouras et al., 2022).

For rapidly rotating, centrifugally distorted planets and stars, the non-dissipative response is similarly written as

$U$1

and the total Love number is $U$2 (Dewberry et al., 2021). The explicit $U$3-sum is the centrifugal-distortion correction: in a spherical body only $U$4 survives, but in an oblate rotator a sectoral forcing component can drive a tesseral response (Dewberry et al., 2021).

In relativistic EFT language, the same frequency dependence is encoded in retarded kernels rather than explicit fluid modes. For Schwarzschild black holes, the odd- and even-parity kernels are expanded as

$U$5

$U$6

so that $U$7 are static TLNs, $U$8 dissipative coefficients, and $U$9 dynamical Love numbers in the paper’s convention (Combaluzier--Szteinsznaider et al., 4 Nov 2025). A plausible implication is that the mode-sum and EFT viewpoints are complementary low- and high-level descriptions of the same linear response structure.

3. Fluid planets and stars: rotation, resonances, and mode content

The most direct astrophysical application is to giant planets and rotating stars, where the Love numbers are dominated by a small subset of global modes. For Jupiter, the measured quadrupolar Love number

$\delta\Phi$0

is below the hydrostatic value

$\delta\Phi$1

and the finite-frequency response of rotation-modified $\delta\Phi$2-modes was proposed as a simple explanation of the $\delta\Phi$3 anomaly (Lai, 2021). In an $\delta\Phi$4 polytropic model applied to Jupiter–Io, the calculation gives

$\delta\Phi$5

so the dynamical correction is a 4% reduction relative to the hydrostatic value (Lai, 2021).

The same paper argues that the Love number is usually dominated by the response of the rotation-modified $\delta\Phi$6-modes, while inertial-mode contributions are negligible because their overlaps are suppressed as $\delta\Phi$7 (Lai, 2021). It also shows that sufficiently strong stratification in a large region of the interior can produce $\delta\Phi$8-mode resonances that significantly alter $\delta\Phi$9, turning dynamical Love numbers into probes of internal stratification (Lai, 2021).

For spinning Newtonian stars, the conceptual result is that rotation affects the dynamical and static problems differently. In a slow-rotation expansion,

$k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$0

so the dynamical tide is corrected already at first order in spin, whereas the static tide changes only at second order (Pnigouras et al., 2022). This first-order sensitivity arises because finite-frequency forcing distinguishes prograde and retrograde propagation; in the static limit those terms cancel (Pnigouras et al., 2022).

Rapid rotation amplifies these effects. In centrifugally distorted $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$1 polytropes rotating up to $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$2 of breakup, $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$3-modes remain dominant at nearly all rotation rates, but retrograde inertial modes can produce strong tesseral resonances, and at $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$4 of breakup prograde $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$5- and $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$6-modes mix strongly (Dewberry et al., 2021). For $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$7, the hydrostatic values listed for an $k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$8 polytrope are

$k_{\ell m}=\frac{\delta\Phi_{\ell m}}{U_{\ell m}},$9

whereas at the orbital frequencies of the Galilean satellites the tesseral Love numbers become much larger, for example

$\omega_m=m(\Omega_o-\Omega),$0

(Dewberry et al., 2021). This suggests that, in rapid rotators, tesseral Love numbers are especially sensitive to rotational coupling and inertial resonances.

A distinct but related result concerns Jupiter’s high-degree tesseral response. An analytic perturbative treatment of $\omega_m=m(\Omega_o-\Omega),$1 shows that the hydrostatic $\omega_m=m(\Omega_o-\Omega),$2 is dominated by the tidal response at $\omega_m=m(\Omega_o-\Omega),$3 coupled into the spherical harmonic $\omega_m=m(\Omega_o-\Omega),$4 by the planet’s oblate figure, and that high-degree tesseral Love numbers are dominated by lower-degree Love numbers when they are primarily hydrostatic (Idini et al., 2021). The same abstract states that, after including the coupling from the well-understood $\omega_m=m(\Omega_o-\Omega),$5 dynamical tides $\omega_m=m(\Omega_o-\Omega),$6, Jupiter’s hydrostatic $\omega_m=m(\Omega_o-\Omega),$7 requires an unknown dynamical effect to produce a fractional correction $\omega_m=m(\Omega_o-\Omega),$8 in order to fit Juno’s observation within $\omega_m=m(\Omega_o-\Omega),$9 (Idini et al., 2021).

4. Ocean worlds and viscoelastic bodies

In icy satellites with subsurface oceans, the dynamical correction is carried primarily by the ocean rather than the solid shell. In membrane theory for Europa- and Titan-like bodies, the Love numbers are written as

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$0

and the dynamical modification enters through

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$1

with the key inertial term

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$2

(Beuthe, 2015). For a homogeneous incompressible ocean over a rigid mantle,

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$3

and in the shallow-ocean limit

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$4

so the correction scales like $$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$5 and becomes large when the ocean is shallow (Beuthe, 2015). The same work identifies the resonance as the ocean’s surface gravity mode, and for Europa with a 10 km crust gives a representative resonant thickness

$$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$6

(Beuthe, 2015).

In solid viscoelastic bodies, the relevant dynamical quantity is usually the dissipative combination $$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$7, identified with the imaginary part of the complex Love number: $$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$8 A damped mass-spring simulation of a Kelvin–Voigt self-gravitating sphere reproduces the expected “kink” shape of this function as a function of frequency: it is proportional to frequency at small $$I_{\ell m}\left(\omega\right) =-\frac{\left(\ell-2\right)!}{(2\ell-1)!!}k_{\ell m}(\omega) r_+^{2\ell+1} \mathcal{E}_{\ell m}\left(\omega\right),$$9, peaks near $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$0, and declines at higher $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$1 (Frouard et al., 2016). The inversion formula used in the simulations is

$$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$2

so the dynamical Love response is inferred directly from tidal drift (Frouard et al., 2016). The paper states that the measured quality function is about 30% larger than the analytic prediction, with $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$3, showing that the frequency dependence is robust but the absolute normalization is only approximate (Frouard et al., 2016).

A common misconception is that any paper relating Love numbers to an instability criterion is therefore computing dynamical Love numbers. The AdS-bubble study explicitly does not compute genuinely frequency-dependent Love numbers $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$4; it works with static, time-independent perturbations and uses the sign of the resulting static TLNs to compare with a separate dynamical instability criterion (Chen et al., 11 Jun 2026). This distinction is terminologically important because “dynamical” may refer either to the response coefficient itself or to an instability of the background system.

5. Compact objects and black holes

For compact objects, the dynamical problem is most naturally formulated in terms of asymptotic coefficients of wave-equation solutions. In Kerr-like compact objects with a reflective surface, the response is packaged into a frequency-domain function

$$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$5

and is extracted from the decaying term in the asymptotic expansion of $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$6 (Chakraborty et al., 2023). For non-rotating objects, the quadrupolar response can be written as

$$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$7

where $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$8 is the reflectivity and $$k_{\ell m} \simeq \kappa_{\ell m} + {\rm i}\nu_{\ell m}\left(\omega - m \Omega\right) + \dots,$$9 as $\kappa_{\ell m}$0 (Chakraborty et al., 2023). For generic $\kappa_{\ell m}$1, the zero-frequency dynamical TLN vanishes; only when

$\kappa_{\ell m}$2

does the limit remain finite (Chakraborty et al., 2023). The same paper shows explicitly that the strictly static TLNs,

$\kappa_{\ell m}$3

differ from the $\kappa_{\ell m}$4 limit of the dynamical response (Chakraborty et al., 2023).

For Kerr black holes, the static Love numbers vanish identically, but several finite-frequency analyses find nontrivial behavior. A low-frequency hidden-conformal treatment gives a response coefficient $\kappa_{\ell m}$5 proportional to a gamma-function ratio times

$\kappa_{\ell m}$6

and argues that the corresponding dynamical tidal coefficients are generically non-zero and exhibit logarithmic behavior (Perry et al., 2023). The Schwarzschild limit of that formula yields an $\kappa_{\ell m}$7 response structure proportional to $\kappa_{\ell m}$8 (Perry et al., 2023).

A more directly perturbative Schwarzschild calculation in advanced null coordinates finds

$\kappa_{\ell m}$9

so that the linear term is purely dissipative and the quadratic term contains both the logarithmic running and a finite conservative piece (Chakraborty et al., 30 Jul 2025). The corresponding Love number inferred from the real part is

$\nu_{\ell m}$0

although the same work cautions that proper EFT matching is still required before this can be treated as a definitive observable prediction (Chakraborty et al., 30 Jul 2025).

The EFT-matched Schwarzschild analysis goes further in organizing the response. It defines the retarded kernels $\nu_{\ell m}$1, shows that the static black-hole coefficients vanish, and computes the leading nontrivial small-frequency terms. For example,

$\nu_{\ell m}$2

$\nu_{\ell m}$3

so the $\nu_{\ell m}$4 term is dissipative while the $\nu_{\ell m}$5 term is the universal logarithmic running of the conservative dynamical response (Combaluzier--Szteinsznaider et al., 4 Nov 2025). This paper emphasizes that the finite constants are scheme dependent, whereas the logarithmic running is universal (Combaluzier--Szteinsznaider et al., 4 Nov 2025).

Extremal Kerr behaves differently again. In a Teukolsky–MST treatment, the low-frequency Love number is

$\nu_{\ell m}$6

so the leading-order response is purely imaginary and therefore purely dissipative, while a conservative real part appears at order $\nu_{\ell m}$7 (Perry et al., 2024). A plausible implication is that finite-frequency black-hole response is best viewed as a hierarchy of conservative and dissipative coefficients rather than as a single yes-or-no statement about whether black holes “have Love numbers.”

6. Effective field theory, renormalization, and definitional issues

The EFT viewpoint turns dynamical Love numbers into Wilson coefficients. For compact stars, the point-particle action can be organized as

$\nu_{\ell m}$8

so that $\nu_{\ell m}$9 is the static TLN, odd-$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$00 coefficients are dissipative, and even-$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$01 coefficients are conservative dynamical TLNs (Jarequi et al., 12 Mar 2026). In the electric quadrupole sector, the response function is written as

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$02

which makes the distinction between static, dissipative, and dynamical pieces explicit (Jarequi et al., 12 Mar 2026). The same paper states that for non-viscous neutron stars the dissipative term vanishes, so the main new result is the NNLO conservative dynamical TLN and its renormalization-group equation (Jarequi et al., 12 Mar 2026).

A closely related PN/EFT treatment of gravitoelectric quadrupolar dynamical tides introduces explicit oscillator degrees of freedom $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$03 and finds that the first post-adiabatic tidal Wilson coefficient $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$04 must be renormalized at 3PN. The renormalization-group equation is

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$05

with solution

$k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$06

(Mandal et al., 2023). This paper is careful that $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$07 is the post-adiabatic Love number, not the ordinary adiabatic Love number $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$08, so the renormalized coefficient is a higher-order dynamical response parameter rather than the static deformability itself (Mandal et al., 2023).

A recurring conceptual issue is the relation between Love numbers and Green’s functions. One response-theory analysis stresses that the retarded Green’s function contains not only the instantaneous tidal response but also radiation-reaction effects, especially non-analytic tail terms such as $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$09, so dynamical Love numbers should not be naively identified with the full retarded Green’s function in asymptotically flat black-hole spacetimes (Luca et al., 2024). By contrast, in BTZ black holes, where the absence of radiative modes eliminates tails, the Green’s function can be linked directly to the Love coefficient (Luca et al., 2024). This suggests that, in four-dimensional asymptotically flat gravity, part of the current controversy is definitional rather than purely computational.

Another terminological point concerns low-frequency fluid stars in general relativity. The gravitomagnetic response of an irrotational fluid star is not a computation of modern frequency-dependent $k_{lm}=\frac{\delta\Phi}{U}\Big|_{r=R},$10, but it shows that even in an adiabatic slow-tide limit, internal motions driven by relativistic circulation conservation can qualitatively change the Love numbers. In particular, the gravitomagnetic Love numbers are positive in strict hydrostatic equilibrium and negative in the irrotational state (Landry et al., 2015). This is not itself a dynamical Love-number calculation in the modern sense, but it identifies a low-frequency internal-motion effect that any truly dynamical theory should reproduce in the appropriate limit (Landry et al., 2015).

Taken together, these developments indicate that “dynamical tidal Love number” now denotes a family of related but not identical objects: mode-sum susceptibilities in fluid bodies, frequency-expanded Wilson coefficients in EFT, asymptotic response coefficients in relativistic perturbation theory, and sometimes low-frequency conservative pieces after radiation-reaction subtraction. The common content is finite-frequency tidal response; the precise observable depends on the formulation and on how source, response, dissipation, and running are separated (Luca et al., 2024, Combaluzier--Szteinsznaider et al., 4 Nov 2025, Jarequi et al., 12 Mar 2026).