Static Tidal Love Numbers in Compact Objects

• Static tidal Love numbers are dimensionless coefficients quantifying the induced multipolar moments of compact objects in response to an external tidal field.
• For neutron stars, nonzero TLNs depend on the internal structure and equation of state, affecting gravitational-wave phasing, while GR black holes exhibit vanishing TLNs due to hidden symmetries.
• Modified gravity and quantum models predict nonzero TLNs for black holes, providing potential observational signatures that deviate from four-dimensional general relativity.

Static tidal Love numbers (TLNs) quantify the conservative, static response of a compact gravitating object—such as a neutron star or black hole—to an external, static, long-wavelength tidal field. TLNs appear as dimensionless coefficients encoding the induced multipolar moments per unit applied tidal field in the asymptotic expansion of metric or curvature perturbations. The existence, magnitude, and sign of TLNs are sharply sensitive to the object’s internal structure and to the underlying gravitational theory: neutron stars exhibit generically nonzero TLNs, while in four-dimensional general relativity, Schwarzschild and Kerr black holes have vanishing static TLNs—a property rooted in hidden symmetries of the perturbation equations. TLNs thus bridge general relativity, effective field theory, symmetry considerations, gravitational-wave astronomy, and tests of quantum or high-curvature corrections in strong gravity.

1. Mathematical Definition and Physical Interpretation

For a static, spherically symmetric background metric, static TLNs are defined via the asymptotic (large-$r$) behavior of the field or metric perturbation sourced by an external tidal field of multipolar order $\ell$. For a representative radial master function $\Psi(r)$ (arising from scalar, electromagnetic, or gravitational perturbations), the large-$r$ expansion reads

$$\Psi(r) \sim C_\text{out} \ r^{\ell+1} [1+\ldots] + C_\text{in} \ r^{-\ell} [1+\ldots].$$

The $r^{\ell+1}$ term represents the imposed (external) field; the $r^{-\ell}$ term is the object's (induced) response. The dimensionless TLN is the ratio of the response to the source: $$\kappa_\ell = \left.\frac{C_\text{in}}{C_\text{out}}\right|_\text{reg.~at horizon} \sim \text{"response/source"}.$$ In relativistic language, these coefficients map directly to induced multipole moments, e.g., for the mass quadrupole in an external quadrupolar tidal field,

$$Q_{ij} = -\lambda_2 \mathcal{E}_{ij}, \quad \text{with} \quad \lambda_2 = \frac{2}{3} k_2 R^5$$

where $k_2$ is the quadrupolar TLN and $R$ a reference radius.

In general, nonzero positive TLNs ($k_\ell > 0$) indicate bulging along the tidal field; negative TLNs signify "squeezing." TLNs can also exhibit logarithmic "running" with radius or mass, manifesting as additional log terms in the asymptotic expansion.

2. TLNs for Neutron Stars and Compact Objects

For neutron stars, TLNs are nonzero and encode the star’s equation of state. For a relativistic star of compactness $C = M/R$ and polytropic index $n$, $k_2$ can be computed by integrating the perturbed Einstein equations and matching at the stellar surface: $$k_2 = \frac{8C^5}{5}(1-2C)^2 [2+2C(y-1)-y] / \mathcal{D}(C,y)$$ where $y=R H'(R)/H(R)$ is the "logarithmic derivative" of the even-parity perturbation at the stellar surface. Representative values for polytropic neutron-star models show $k_2 \in [0.05,0.25]$ for typical $C$ and $n$ (0711.2420). This relativistic suppression, relative to Newtonian values, is $\mathcal{O}(20\text{--}30\%)$ for moderate compactness.

The effect of TLNs is imprinted in the 5PN-order phasing of gravitational waves from inspiraling binaries, making them key observables for constraining stellar structure and internal composition.

3. Vanishing TLNs for Four-Dimensional Black Holes: Symmetry Origins

A central property of four-dimensional asymptotically flat Schwarzschild and Kerr black holes in general relativity is the exact vanishing of all static TLNs: $k_\ell^\text{BH} = 0 \qquad \text{for all spin and %%%%17%%%%}.$ This is not accidental but is forced by intrinsic symmetries of the static perturbation equations:

• In Schwarzschild, all static (spin-0, 1, 2) perturbations admit an $\mathrm{SO}(3,1)$ symmetry in the optical 3-geometry, with even/odd sectors linked by an $\mathrm{SO}(2)$ duality (Chandrasekhar duality) (Berens et al., 21 Oct 2025).
• The mode decomposition exhibits "ladder operators" $D_\ell^\pm$ satisfying intertwining relations, enabling one to build all higher-$\ell$ master solutions by acting on a unique regular ground state.
• Regularity at the horizon singles out the purely growing polynomial solution, which contains no decaying ($1/r^{\ell+1}$) piece: hence, $k_\ell=0$ for all allowed multipoles (Hui et al., 2021, Sharma et al., 12 Nov 2025).
• The vanishing of worldline EFT Wilson coefficients for Love operators is enforced by the special-conformal part of the symmetry, which forbids nontrivial tidal couplings under the $\mathrm{SO}(3,1)$ (or $SL(2,\mathbb{R})$) symmetry acting in the near zone (Charalambous, 12 Feb 2024, Berens et al., 21 Oct 2025).

Similar logic extends beyond linear order: static TLNs vanish identically at quadratic (and higher) order in the applied tidal field amplitude (Riva et al., 2023).

4. TLNs in Modified Gravity, Quantum Gravity, and Beyond

4.1 Scalar-Tensor and Horndeski Theories

In scalar-tensor theories (including the most general Horndeski class), static TLNs generically do not vanish. For neutron stars in these models, the asymptotic expansion of the even-parity perturbations involves additional $1/r^3$ terms independent of the actual tidal deformability—contaminated by the effects of scalar "hair" (scalar charge) (Diedrichs et al., 14 Jan 2025). The correct extraction of TLNs requires matching to the effective worldline action, separating the pure tidal contribution from hair-induced pieces. For certain models (e.g., the Damour–Esposito–Farèse model), naive methods can differ by up to 10% from the correct value.

4.2 Loop Quantum Black Holes

In loop-quantized black hole models, such as the Ashtekar–Olmedo–Singh (AOS) solution and covariant loop quantum variants (ZLMY-I, ZLMY-II, ABV), static TLNs are generically nonzero and negative for scalar, vector, and gravitational perturbations (Motaharfar et al., 15 Jan 2025, Motaharfar et al., 20 May 2025). The leading scaling is

$|\kappa_\ell^s| \sim (M_P/M)^{a}, \quad \text{with %%%%27%%%% or %%%%28%%%% depending on the model},$

so that TLNs vanish in the large-mass (classical) limit, but could be significant for Planck-scale black holes. For AOS, $|\kappa_2^{s=2}| \sim 10^{-3}$ at $M=10^4 M_\textrm{Pl}$, far below current detector thresholds for astrophysical black holes, but potentially observable for primordial black holes in the $10^4\text{--}10^8 M_\textrm{Pl}$ mass range.

4.3 Higher-Dimensional and Higher-Curvature BHs

In higher dimensions, Schwarzschild–Tangherlini and Myers–Perry black holes generically have nonvanishing static TLNs. The explicit analytic expressions exhibit towers of "magic zeros" (multipoles for which $k_\ell=0$) due to generalized ladder symmetries, but in general—outside $d=4$—the vanishing is not generic (Charalambous, 12 Feb 2024, Rodriguez et al., 2023, Singha et al., 20 Aug 2025). Similarly, higher-curvature theories (e.g., Einstein–Gauss–Bonnet gravity) yield nonzero TLNs for black holes, with explicit dependence on the Gauss–Bonnet coupling and spacetime dimension (Singha et al., 20 Aug 2025).

5. TLNs for Rotating Black Holes and Exotic Compact Objects

For Kerr black holes, the vanishing of static TLNs persists for all multipoles and all spin sectors, including non-extremal and extremal cases, under both axisymmetric and generic (non-axisymmetric) tidal perturbations (Bhatt et al., 19 Dec 2024, Ivanov et al., 2022, Tiec et al., 2020). The Teukolsky approach shows that the relevant response functions are purely imaginary, reflecting dissipative (absorptive) response but no conservative (static) tidal deformability. Only dynamical (frequency-dependent) TLNs can be nonzero in the presence of rotation, with model-dependent scaling and possible resonances if the object is not a true black hole (Chakraborty et al., 2023).

For exotic compact objects (ECOs) with reflective or partially absorptive surfaces near the would-be black hole horizon, TLNs are generally nonzero, with dependence on reflectivity and frequency. For perfectly reflecting ECOs, TLNs vanish logarithmically slowly as the surface approaches the event horizon, possibly leaving tiny but nonzero imprints even for Planck-scale deviations.

6. Analogues, Generalizations, and Methodological Aspects

Analog black holes in fluid mechanics (e.g., draining bathtub flows) exhibit similar behavior: their tidal response equations admit ladder symmetries leading to vanishing TLNs for special choices of multipole and dimension, with logarithmic running for critical ratios, closely mirroring the general relativistic situation (Luca et al., 11 Dec 2024).

Across all modern approaches, the extraction of TLNs employs analytic continuation in multipole index, identification of regular (horizon) and decaying (spatial infinity) branches, and careful matching of boundary conditions and gauge choices—crucial for distinguishing the physical induced multipole from background or coordinate artifacts (Ivanov et al., 2022, Barura et al., 17 May 2024, Diedrichs et al., 14 Jan 2025).

7. Observational Implications and Theoretical Significance

Static TLNs are measurable in principle through their effect on gravitational-wave phasing in the late inspiral of compact binaries, appearing at 5PN order in the phase. For neutron-star binaries, they encode the equation of state. For black holes, the null result in GR simplifies parameter estimation and waveform modeling, and any future measurement of nonzero static TLNs for a black hole binary would be a clear indicator of new physics (e.g., quantum gravity, modified gravity, or exotic compact object structure).

In summary, static tidal Love numbers serve as a sharp diagnostic of internal structure, symmetries, and possibly even quantum properties of compact objects in strong-field gravity. Their vanishing in 4d GR black holes is not generic, but is rather a reflection of an enhanced symmetry that is fragile to departures from Einstein's theory, finite spatial dimensions, or classical event horizon structure.