Relativistic Love Numbers and Gauge Invariance

A relativistic Love number is meaningful only after the applied field, induced multipole, gauge-invariant observable, and subtraction convention have been identified together. The robust object is not an arbitrary coefficient in a large-radius expansion but a matched response: the same physical solution is expressed in a chosen buffer basis and compared with a normalized multipole or effective interaction. This page defines that object for linear gravitational response in four spacetime dimensions and explains which parts are invariant, which are conventional, and which become dissipative away from the static limit.

Required background. Presymplectic Potentials and Currents supplies the boundary terms needed to distinguish physical variations from degenerate gauge directions.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon-regular coordinate systems; PDEs and Green Functions for Gravitation supplies retarded solutions and response kernels; Formulations of General Relativity supplies gauge-invariant perturbation variables; Relativistic Two-Body Dynamics supplies STF multipole normalizations; and Black-Hole Perturbation Theory supplies the shared mode convention.

Matched tidal response

Define the electric response of a spherical body, or a specified diagonal angular sector, by

$$Q_L(\omega)=-\lambda^E_\ell(\omega)\mathcal E_L(\omega).$$

For a rotating body the general response is a tensor or mode-space map; a single scalar need not describe all components. The magnetic relation is analogous once the current-moment normalization is fixed. The induced moment $Q_L$ is defined by matching in $GM\ll r\ll\mathcal R$, not by choosing one metric component. For a nonrotating static conservative response and fixed reference radius $R$, the dimensionless electric convention is

$$k^E_\ell =\frac{(2\ell-1)!!}{2(\ell-2)!} \frac{G\lambda^E_\ell(0)}{R^{2\ell+1}}.$$

The declaration for this page is precise: static means the conservative $\omega\to0$ coefficient; dynamic means finite-frequency retarded response; For a nonrotating passive sector, real time-even terms are conservative and the absorptive part is dissipative. With rotation, use the symmetric and antisymmetric parts of the response on real STF components rather than applying that scalar rule to a single complex $m$ component. The background is four-dimensional, the field has spin two, and the applied/induced split is made with STF electric or magnetic moments. The buffer calculation may use light-cone gauge for Schwarzschild or a curvature-based Teukolsky variable for Kerr, but the final definition is a matched STF or worldline observable. Local analytic pieces are quoted in a declared subtraction scheme and scale. The limit is taken at fixed background parameters and fixed $(\ell,m)$ after matching the near and far regions; extremality, $\omega\to0$, and $r\to\infty$ are not silently interchanged.

Binnington and Poisson demonstrate the construction in light-cone gauge and show that their electric and magnetic coefficients occur in residual-gauge invariant combinations (Binnington and Poisson 2009, § III, especially eqs. (3.9)–(3.13) and (3.19)–(3.22), pp. 9–11 (PDF)). That result does not say that every ratio of master-field coefficients is invariant.

Three descriptions of one response

The same low-frequency physics can be encoded in three languages.

Description	Applied datum	Induced datum	Convention that must be supplied
Buffer-zone metric or curvature	Coefficient of the normalized growing solution	Coefficient of the normalized falling solution	Gauge, radial coordinate, master-field normalization
Retarded response	Source $\mathcal E_L(\omega)$	$Q_L(\omega)=-\lambda_\ell(\omega)\mathcal E_L(\omega)$	Fourier sign, causal prescription, contact subtraction
Worldline effective theory	External Weyl tensor on the worldline	Functional derivative of the finite-size action	Operator basis, renormalization scheme and scale

Equivalence means that an on-shell quantity—such as a low-frequency scattering amplitude or absorption probability—agrees after the dictionaries are applied. A local field redefinition can shift a polynomial $c_0+c_2\omega^2+\cdots$ between the bulk solution and a worldline contact term. It cannot remove the same physical phase shift while leaving the rest of the matching unchanged. In the static Schwarzschild construction, horizon regularity selects the growing polynomial and sets the conservative response branch to zero. In a rotating problem, a response may retain an imaginary piece because a field stationary at infinity need not be stationary with respect to the horizon generator.

This last distinction matters. Love numbers are conservative coefficients; absorption belongs to the dissipative response. Treating an imaginary zero-frequency-in-$t$ Kerr coefficient as a static conservative Love number mixes frames and time-reversal sectors. Charalambous, Dubovsky, and Ivanov formulate the coefficients as effective-theory Wilson coefficients precisely to separate those sectors (Charalambous, Dubovsky, and Ivanov 2021, §§ 2–3, especially Eqs. (2.12)–(2.14), pp. 7–8 (PDF)).

An explicit quadrupolar dictionary

Take the leading vacuum-buffer quadrupole of External Tidal Fields. Subtract the background monopole and hold the same applied $\mathcal E_{ij}$ and physical radius fixed. The following descriptions give the same response, with $Q_{ij}$ the physical STF mass quadrupole:

$$\begin{aligned} Q_{ij}&=-\lambda_E\mathcal E_{ij},\\ U_{\mathrm{pert}} &=-\frac12 r^2\mathcal E_{nn} -\frac{3G\lambda_E}{2r^3}\mathcal E_{nn},\\ L_{\mathrm{eff}}&=\frac{\lambda_E}{4}\mathcal E_{ij}\mathcal E^{ij}, \qquad Q_{ij}=-2\frac{\partial L_{\mathrm{eff}}}{\partial\mathcal E^{ij}}. \end{aligned}$$

The action line describes a real conservative static coefficient. A complex retarded response requires a causal kernel, not substitution into this single-history local action. The common measured quantity is

$$\frac{C_{\hat0\hat r\hat0\hat r}^{\mathrm{pert}}}{\mathcal E_{nn}} =1+\frac{18G\lambda_E}{r^5}, \qquad \mathcal E_{nn}\ne0.$$

The quotient is only a convenient way to display the linear relation; at a node of $\mathcal E_{nn}$ use the undivided equation. It is a leading weak-field statement, not an exact strongly curved finite-radius formula. The preceding page verifies it under a coordinate change with the transformed radial metric and connection; the EFT page derives the same induced potential from its field equation.

For this electric convention, dimensional and dimensionless coefficients are related by

$$\lambda^E_{2m}=\frac{2R^5}{3G}\widehat k^{E,\mathrm{BP}}_{2m} =\frac{r_s^5}{3G}\mathcal K^{(2)}_{2m}, \qquad r_s=2GM.$$

Here $\widehat k^{E,\mathrm{BP}}$ is a complex stationary response expressed in BP’s radius and moment normalization; it is not itself a conservative Love number. For the diagonal Kerr modes considered here,

$$k^{E,\mathrm{Love}}_{2m} =\operatorname{Re}\widehat k^{E,\mathrm{BP}}_{2m} =\frac{r_s^5}{2R^5}\operatorname{Re}\mathcal K^{(2)}_{2m}.$$

Thus CDI’s dimensionless curvature-response coefficient $\mathcal K$ is twice the BP-normalized complex response when $R=r_s$, not the same symbol in another notation. Taking the symmetric conservative part is an additional step. For the static nonrotating conservative case, $\widehat k$ reduces to the usual real BP $k^E$. CDI also uses the opposite sign for its mass-moment response coefficient: $\lambda_E=-\lambda^{\mathrm{CDI}}/G$ after restoring physical mass units. This electric translation follows from Charalambous, Dubovsky, and Ivanov 2021, Eqs. (2.12)–(2.14) and footnote 6, pp. 7–8 (PDF). It is not a conversion formula for every magnetic current convention. The static calculation will derive $\mathcal K$ and specify its integer-multipole prescription.

A compensated change of basis

The distinction between a physical moment and a contact coefficient can be made algebraic. For one angular direction write

$$U_{\mathrm{pert}}=A S+B D,\qquad S=-\frac{r^2}{2},\quad D=-\frac{3G}{2r^3}, \quad A=\mathcal E_{nn},\quad B=\lambda_E A.$$

For any fixed real constant $\eta$ with the dimensions of $\lambda_E$, change the applied basis and its quoted response coefficient together:

$$S'=S+\eta D,\qquad D'=D,\qquad A'=A,\qquad B'=B-\eta A.$$

Then $A'S'+B'D'=AS+BD$ identically. The coefficient read in the new basis is $B'/A=\lambda_E-\eta$, but its complete curvature contains $18G[(B'/A)+\eta]/r^5=18G\lambda_E/r^5$, as before. In the effective description this is a shift of the quoted local coefficient accompanied by an equal compensating term in the rest of the matched description. The physical $Q$ relative to the original measured applied tide has not changed. Shifting $\lambda_E$ alone while retaining $S$, the remaining action and the external data would change the curvature and describe a different response, not the same theory in a new scheme.

Static invariance and dynamical scheme dependence

At strictly static order, a chosen response convention can be tested by an allowed residual gauge transformation: the matched $k_\ell$ must remain unchanged even though individual metric functions move. At finite frequency, one also confronts local counterterms and logarithms. The renormalized response can be written schematically as

$$\lambda_\ell(\omega;\mu_{\mathrm{ren}}) =c_0(\mu_{\mathrm{ren}})+c_2(\mu_{\mathrm{ren}})\omega^2 +\beta_2\omega^2\log\!\left(\frac{-(\omega+i0)^2}{\mu_{\mathrm{ren}}^2}\right) +\lambda_{\mathrm{abs}}(\omega)+\cdots .$$

Here $\mu_{\mathrm{ren}}\gt0$ is a frequency scale. The logarithm is the boundary value of the retarded branch:

$$\log\!\left(\frac{-(\omega+i0)^2}{\mu_{\mathrm{ren}}^2}\right) =\log\!\left(\frac{\omega^2}{\mu_{\mathrm{ren}}^2}\right) -i\pi\operatorname{sgn}\omega,\qquad \omega\ne0.$$

It therefore obeys the required reality relation $\lambda(-\omega)=\overline{\lambda(\omega)}$ for real scalar response. Writing $-\omega^2-i0$ instead would choose the same imaginary sign at both frequencies and violate that relation. In this schematic expansion $\lambda_{\mathrm{abs}}$ denotes absorptive terms not already included in the logarithm; counting its discontinuity twice is incorrect.

The finite $c_n$ depend on the operator basis and subtraction condition. The coefficient of a logarithmic running and the complete on-shell observable do not. A 2026 MST-to-worldline matching analysis makes the finite-part issue especially explicit: the renormalization scheme and the initial condition of the flow must be fixed before a dynamical Love number is quoted (Kobayashi et al. 2026, § 3.2.3, especially eqs. (3.35)–(3.38)). This qualification applies to dynamic local coefficients; it does not undo the standard static four-dimensional vacuum result in its declared scheme.

Exercises

A calculation reports the static quadrupolar relation $Q_{ab}=-\lambda_2\mathcal E_{ab}$ with $\lambda_2=64(GM)^5/(3G)\times10^{-3}$. It uses the reference radius $R=2GM$. What is $k_2$ in the convention above, and what information is still missing before comparing a frequency-dependent coefficient?

Solution

For $\ell=2$,

$$k_2=\frac{3G\lambda_2}{2R^5} =\frac{3G}{2(2GM)^5}\, \frac{64(GM)^5}{3G}\,10^{-3}=10^{-3}.$$

The powers of $G$ and length cancel, so $k_2$ is dimensionless. A dynamic comparison additionally requires the Fourier convention, the definition of the retarded source and induced moment, the multipole/tetrad normalization, the gauge or on-shell matching observable, the subtraction scheme and scale, and the order of the low-frequency and extremal limits. Agreement of a raw $\omega^2$ coefficient without those data would not be a physical check.

We can now ask why the conservative static coefficient vanishes for the standard black-hole problem on the next page.

References

• Binnington, Taylor, and Eric Poisson. 2009. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80, 084018. DOI.
• Charalambous, Panagiotis, Sergei Dubovsky, and Mikhail M. Ivanov. 2021. “On the Vanishing of Love Numbers for Kerr Black Holes.” Journal of High Energy Physics 05, 038. DOI.
• Kobayashi, Hajime, Shinji Mukohyama, Naritaka Oshita, Kazufumi Takahashi, and Vicharit Yingcharoenrat. 2026. “Dynamical Tidal Response of Nonrotating Black Holes: Connecting the Mano–Suzuki–Takasugi Formalism and Worldline EFT.” Physical Review D 113, 084011. DOI.