Static Tidal Love Tensor

• The static tidal Love tensor is a linear response coefficient that maps time-independent tidal fields to induced multipole moments in gravitating objects.
• It generalizes scalar Love numbers to tensor forms, capturing complex dependencies in rotating, charged, higher-dimensional, and modified gravity contexts.
• Extraction techniques include asymptotic matching, effective field theory formulations, and symmetry-based methods that distinguish conservative from dissipative responses.

A static tidal Love tensor is the tensorial linear-response coefficient that relates an externally applied, time-independent tidal field to the induced multipole structure of a gravitating object. In the simplest nonspinning setting it reduces to scalar Love numbers, but with spin, extra fields, higher dimensions, or coupled gravito-electromagnetic sectors, the response generally becomes a tensor or matrix acting on STF multipoles or harmonic coefficients. In asymptotic language, it is the coefficient of the decaying response branch relative to the growing source branch; in worldline EFT it is a Wilson coefficient multiplying static tidal operators; and in response theory it is the static limit of the retarded multipole kernel (Luca et al., 2024, Tiec et al., 2020, Parra-Martinez et al., 23 Oct 2025).

1. Definition and kinematic structure

In Newtonian and relativistic multipolar language, the applied tidal field is encoded by STF tensors such as $\mathcal{E}_L$ and $\mathcal{B}_L$, while the induced response is encoded by multipole moments such as $I_L$, $\delta M_L$, or $\delta S_L$. For a nonspinning body the response is scalar-like,

$I_L=\lambda_\ell\,\mathcal{E}_L,$

or, for the quadrupole,

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$

The asymptotic field then splits into a growing source term and a decaying induced term. In one standard form,

$$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$

so the Love coefficient is the ratio of the $r^{-\ell-1}$ response to the $r^\ell$ source (Luca et al., 2024, Das et al., 2020).

For rotating bodies this scalar description is insufficient. In the Newtonian rotating-body discussion and in its Kerr generalization, the induced multipole can depend on the full tensor structure of the applied field,

$\mathcal{B}_L$0

and at quadrupolar order one may write

$\mathcal{B}_L$1

Here the “static tidal Love tensor” is the linear map $\mathcal{B}_L$2 or, more generally, $\mathcal{B}_L$3, rather than a single scalar coefficient (Tiec et al., 2020, Tiec et al., 2020).

A related frequency-space decomposition writes the complex tidal coefficient as

$\mathcal{B}_L$4

where $\mathcal{B}_L$5 is the conservative deformational Love number and $\mathcal{B}_L$6 the dissipative response coefficient. In strictly static problems, the focus is usually on the conservative part, but rotating black holes provide examples where a nontrivial imaginary response can survive at $\mathcal{B}_L$7 in the relevant mode decomposition (Perry et al., 2023).

2. Effective-field-theory formulation and extraction procedures

In worldline EFT, static tidal response is encoded by local operators on the compact-object worldline. For the standard gravitational sector this takes the form

$\mathcal{B}_L$8

or, more generally,

$\mathcal{B}_L$9

At nonlinear order the response is packaged into higher-rank Wilson coefficients,

$I_L$0

which are the nonlinear static Love tensors in the EFT sense (Parra-Martinez et al., 23 Oct 2025, Ivanov et al., 2022).

The same logic extends to theories with extra fields. In Horndeski gravity the point-particle action contains

$I_L$1

so there are tensor/tensor, scalar/scalar, and mixed Love numbers (Diedrichs et al., 14 Jan 2025).

Response theory packages the same information in the retarded kernel,

$I_L$2

with the static limit becoming local in time,

$I_L$3

This identifies the Love tensor with the zero-frequency part of the multipole Green’s function (Luca et al., 2024).

A central technical issue is extraction. In GR for ordinary static stellar perturbations, matching the $I_L$4 and $I_L$5 terms is straightforward. In scalar-tensor theories this can fail. In Horndeski neutron stars the $I_L$6 coefficient contains both a genuine tidal response and an additional long-range contribution from the mass and scalar charge, so the correct Love numbers require subtracting analytically derived contamination terms rather than naively equating the full $I_L$7 coefficient to the tidal response (Diedrichs et al., 14 Jan 2025). A different ambiguity appears in odd-parity EFTs with a timelike scalar profile, where integer $I_L$8 causes overlap between the growing and decaying asymptotic series; there the TLNs are extracted uniquely by analytic continuation of the multipole index $I_L$9 to non-integer values (Barura et al., 2024).

3. Vanishing theorems in four-dimensional general relativity

For asymptotically flat, four-dimensional black holes in GR, the dominant result in the conservative static sector is vanishing. For Schwarzschild black holes, the static linear Love numbers vanish, and this persists beyond linear order. Solving the static Einstein equations to second order in the external tidal field and matching to the point-particle EFT gives

$\delta M_L$0

so the quadratic polar/even-parity response is fully reproduced by Einstein–Hilbert plus point-particle dynamics with no tidal operators turned on (Riva et al., 2023).

This vanishing has also been elevated to an all-orders symmetry statement. In the static sector of four-dimensional GR, after Kaluza–Klein decomposition and dualization of the graviphoton, the Einstein–Hilbert action becomes a nonlinear sigma model on the hyperbolic plane with an accidental $\delta M_L$1 symmetry. The crucial discrete element acts as

$\delta M_L$2

and, together with the mass spurion transformation $\delta M_L$3, forbids the static tidal coefficients that would otherwise appear in the tail expansion. In this formulation, static Love numbers vanish and do not run, and the same symmetry excludes all nonlinear static tides for four-dimensional Schwarzschild black holes (Parra-Martinez et al., 23 Oct 2025).

A complementary explanation uses hidden near-zone symmetry. In the properly defined near-zone approximation, massless-field perturbations on Kerr organize into representations of a hidden

$\delta M_L$4

algebra. For static perturbations the regular solution becomes a polynomial of degree $\delta M_L$5, which eliminates the $\delta M_L$6 tail and therefore the Love number. In four-dimensional Schwarzschild, the static perturbations belong to finite-dimensional representations of dimension

$\delta M_L$7

This provides an algebraic explanation of vanishing static Love numbers (Charalambous et al., 2021).

A further gauge-invariant argument comes from on-shell amplitudes. Comparing the near-field scattering amplitude in GR with the worldline EFT amplitude shows that, for four-dimensional Kerr black holes, the static Love numbers vanish for spin-0, spin-1, and spin-2 perturbations. The decisive quantity is the elastic near-field phase shift: $\delta M_L$8 In that framework, the static finite-size Wilson coefficients vanish identically (Ivanov et al., 2022).

4. Rotation, dissipation, and charged black holes

The most visible conceptual tension in the literature concerns Kerr. One line of work defines a genuine tensorial static response for spinning black holes under nonaxisymmetric perturbations. In the static Teukolsky analysis of a Kerr black hole embedded in a weak, slowly varying tidal environment, the response vanishes for Schwarzschild and for axisymmetric Kerr perturbations, but becomes nonzero for spinning, nonaxisymmetric perturbations. At quadrupolar order and linear order in spin,

$\delta M_L$9

and the tensorial coupling is

$\delta S_L$0

In this definition, the static tidal Love tensor measures a spin-induced, anisotropic coupling between the tidal field, the spin vector, and the induced quadrupole (Tiec et al., 2020, Tiec et al., 2020).

Later work, however, isolates the conservative static response differently. In “Dynamical Love Numbers for Kerr Black Holes,” the static Love numbers are stated to vanish,

$\delta S_L$1

even though the dissipative coefficient

$\delta S_L$2

can remain nonzero because the black hole rotates. The same paper then finds nonzero dynamical tidal coefficients for $\delta S_L$3, with logarithmic behavior and generally nonzero real and imaginary parts (Perry et al., 2023). Combined with the on-shell scattering analysis and the hidden-symmetry argument, this suggests that different papers are isolating different pieces of the stationary Kerr response: conservative static deformability, dissipative/frame-dragging response, or full mode-dependent multipolar response (Ivanov et al., 2022, Charalambous et al., 2021).

Charged black holes show the same sensitivity to setup. For an isolated extremal Reissner–Nordström black hole the static Love number vanishes,

$\delta S_L$4

because the physical isolated limit corresponds to $\delta S_L$5 and $\delta S_L$6. But in an exact axially symmetric electrovacuum configuration describing a central ERN black hole embedded in the field of distant companions, both coefficients are present,

$\delta S_L$7

and the nonlinear static Love number is finite and nonzero to all orders in the external field, controlled by the ratio $\delta S_L$8 (Gounis et al., 2 Dec 2025).

Higher-derivative corrections change the static sector more radically. For extremal charged black holes in Einstein–Maxwell EFT, the parity-odd tensor sector with $\delta S_L$9 acquires nonzero static Love numbers with logarithmic running. The running appears through asymptotic terms such as $I_L=\lambda_\ell\,\mathcal{E}_L,$0 exactly at response order, and the cross gravito-electromagnetic runnings are equal because the worldline EFT contains a unique mixed operator,

$I_L=\lambda_\ell\,\mathcal{E}_L,$1

In this setting, vanishing is a property of pure Einstein–Maxwell theory with $I_L=\lambda_\ell\,\mathcal{E}_L,$2, while four-derivative corrections generate nonzero static tidal response (Noumi et al., 28 Jan 2026).

5. Modified gravity, extra fields, and matter-supported compact objects

Outside vacuum GR, nonzero static tidal response is common. In four-dimensional quadratic gravity,

$I_L=\lambda_\ell\,\mathcal{E}_L,$3

worldline EFT and one-point-function reconstruction show that higher-curvature terms induce nonzero but scale-independent static tidal responses. The Love numbers are generally nonzero, but the Wilson coefficients do not run,

$I_L=\lambda_\ell\,\mathcal{E}_L,$4

because the response appears already at tree level and no logarithms are generated in the relevant one-point functions. In this theory, Yukawa tails generated by the massive spin-2 and spin-0 modes produce nonzero scalar and tensor tidal coefficients such as

$I_L=\lambda_\ell\,\mathcal{E}_L,$5

and, for scalar dipole perturbations, a generically nonzero $I_L=\lambda_\ell\,\mathcal{E}_L,$6 that reduces to $I_L=\lambda_\ell\,\mathcal{E}_L,$7 in the GR limit (Bhattacharyya et al., 4 Aug 2025).

In scalar-tensor/Horndeski gravity, the static Love tensor enlarges to include tensor, scalar, and mixed sectors. The asymptotic coefficients $I_L=\lambda_\ell\,\mathcal{E}_L,$8 and $I_L=\lambda_\ell\,\mathcal{E}_L,$9 each split into a tidal piece and a non-Love-number contamination piece. For a minimally coupled scalar field the contamination terms are

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$0

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$1

In the DEF model, neglecting these extra contributions can shift inferred Love numbers by up to about $$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$2 for $$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$3 at $$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$4, and typically by $$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$5 (Diedrichs et al., 14 Jan 2025).

Odd-parity black-hole TLNs in scalar-tensor EFTs with a timelike scalar profile provide another split between stealth and non-stealth backgrounds. For stealth Schwarzschild,

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$6

because the odd-mode generalized Regge–Wheeler equation is equivalent to the GR one after a simple frequency rescaling. For the Hayward background, the TLNs are generically nonzero and can develop logarithmic running; for example,

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$7

while

$$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$8

exhibits running at $$\mathcal{Q}_{ij}=-\Lambda \mathcal{E}_{ij},\qquad k_2=\frac{3}{2}\Lambda R^{-5}.$$9 (Barura et al., 2024).

For matter-supported compact stars, the static response is nonzero already in GR. In anisotropic compact stars one uses the standard relation

$$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$0

with $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$1 determined by compactness $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$2 and the surface variable

$$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$3

In the anisotropic Korkina–Orlyanskii-type model, anisotropy enhances $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$4 for fixed compactness, while $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$5 as $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$6, consistent with the black-hole limit (Das et al., 2020).

6. Higher dimensions, AdS, and Green-function formulations

In higher dimensions and non-asymptotically flat settings, the static tidal Love tensor often becomes explicitly matrix-valued. For five-dimensional Myers–Perry black holes, the electromagnetic response is expanded in modified spherical harmonics on $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$7, and rotation mixes multipoles. The resulting static Love tensor is

$$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$8

so it is diagonal in $$U_{\text{\tiny tot}}= -\frac{M}{r} -\sum_{\ell, m} Y_{\ell m} \left[\frac{(\ell-2)!}{\ell !} \mathcal{E}_{\ell m} r^\ell - \frac{(2\ell-1)!!}{\ell !} \frac{I_{\ell m}}{r^{\ell+1}}\right],$$9 but not in $r^{-\ell-1}$0. A single source mode can induce higher-$r^{-\ell-1}$1 response modes, making the response lower triangular in harmonic order. In the electric polarization the problem reduces to the scalar one, while the magnetic polarization and vector-type gravitational perturbations reduce to special Heun equations admitting exact hypergeometric solutions. Vanishing occurs only under special conditions such as $r^{-\ell-1}$2, $r^{-\ell-1}$3, or the corresponding integer selection rules (Yu, 16 May 2026).

Higher-dimensional asymptotically flat black holes also exhibit a structured pattern of vanishing and running. In the scalar sector, the on-shell amplitude analysis gives Love numbers that vanish when

$r^{-\ell-1}$4

is an integer, run logarithmically when $r^{-\ell-1}$5 is half-integer, and are otherwise constant numbers of order $r^{-\ell-1}$6 (Ivanov et al., 2022). The all-orders symmetry analysis in higher-dimensional gravity reproduces this pattern in electric and tensor sectors and extends it to nonlinear static tides, with the allowed vanishing or running determined by the parity of $r^{-\ell-1}$7 (Parra-Martinez et al., 23 Oct 2025).

In anti-de Sitter space the interpretation changes. For Schwarzschild-AdS$r^{-\ell-1}$8, static Love numbers are ratios of normalizable to non-normalizable perturbation amplitudes and thus measure geometric polarization of the dual CFT plasma on $r^{-\ell-1}$9. In Regge–Wheeler gauge,

$r^\ell$0

while in the gauge-invariant Kodama–Ishibashi formalism,

$r^\ell$1

The two formalisms agree after explicit conversion formulas are applied, and in the large-$r^\ell$2 limit the results match the black-brane/Kovtun–Starinets response (Franzin et al., 2024).

Green-function methods provide the most general response-theory perspective. The retarded kernel contains both source propagation and object response, but for asymptotically flat black holes radiation-reaction and tail effects obstruct a direct identification of the full retarded Green’s function with the static Love tensor. BTZ black holes are special because the absence of radiative modes removes this contamination, allowing a direct correspondence between the tidal response coefficient and the relevant term in the Green’s function (Luca et al., 2024).

Taken together, these results establish the static tidal Love tensor as a unifying response concept with several inequivalent realizations: a scalar coefficient for ordinary nonspinning matter, a spin-dependent tensorial map for anisotropic or rotating objects, a matrix of mode-mixing coefficients in higher dimensions, and a set of EFT Wilson coefficients whose vanishing, running, or nonzero value depends sensitively on dimensionality, asymptotics, horizon regularity, field content, and the precise separation between conservative, dissipative, and dynamical response sectors.