External Tidal Fields and Induced Multipoles

An external tidal field is not the same object as the multipole that a body develops in response. The distinction is cleanest in a buffer region: far enough from the compact body that its field has a multipole expansion, but well inside the curvature scale of the distant sources. There the growing radial branch is applied data and the decaying branch is induced data only after their gauge, normalization, and matching prescription have been fixed. This page treats linear spin-2 response in four-dimensional general relativity; material interiors, orbital evolution, and waveform inference are left to their dedicated treatments.

Required background. Presymplectic Potentials and Currents supplies the boundary-sensitive distinction between a field variation and a physical response.

Helpful background. Black-Hole Geometry and Mechanics supplies regular Schwarzschild and Kerr horizons; PDEs and Green Functions for Gravitation supplies homogeneous radial bases and retarded boundary data; Formulations of General Relativity supplies gauge transformations and constraint variables; Relativistic Two-Body Dynamics supplies the buffer-zone use of STF moments; and Black-Hole Perturbation Theory supplies the shared Fourier and master-field conventions.

Buffer-zone tidal data and multipoles

Let the compact object’s gravitational radius be $r_g=GM$ and the external curvature scale be $\mathcal R$. A matched expansion requires

$$r_g\ll r\ll \mathcal R,$$

in a local asymptotic rest frame centered on the object. The electric and magnetic tidal moments are symmetric trace-free (STF) tensors $\mathcal E_L(t)$ and $\mathcal B_L(t)$, with $L=i_1\cdots i_\ell$ and $\ell\geq2$. They are projections of the external Weyl tensor and its spatial derivatives. The induced mass and current moments are denoted $Q_L$ and $S_L$. For a spherical body, each irreducible multipole has a scalar linear response coefficient:

$$Q_L(\omega)=-\lambda^E_\ell(\omega)\mathcal E_L(\omega), \qquad S_L(\omega)=-\lambda^B_\ell(\omega)\mathcal B_L(\omega),$$

For a rotating body the response is generally a map between tensor components, or between angular modes. A scalar coefficient then refers to a specified diagonal mode or sector, not to an isotropic response for all $m$. For the nonrotating conservative electric static coefficient, the Binnington–Poisson dimensionless convention is

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

Here $R$ is the reference radius used by the matching convention; for a material body it is naturally the surface radius, while for a black hole it must be declared rather than inferred. These STF definitions and the electric/magnetic parity split are developed explicitly in Binnington and Poisson 2009, Eqs. (1.1)–(1.2) and (3.12)–(3.22), pp. 1–2 and 10–11 (PDF).

The response considered so far may be static and conservative. Allowing $\omega\neq0$ makes it dynamic; the time-even real part of a retarded coefficient is conservative, while its absorptive imaginary part is dissipative in the passive, nonrotating convention. All statements here use a four-dimensional spin-2 field. For Kerr, mode labels and the horizon-frame frequency $\omega-m\Omega_H$ must also be retained.

Applied and induced fields

The Newtonian quadrupole

The normalization can be checked without solving a relativistic radial equation. Use the positive Newtonian potential, so that $g_{tt}=-1+2U$ at leading weak-field order. For a localized density $\rho$,

$$Q_{ij}=\int \rho(\mathbf x) \left(x_i x_j-\frac13\delta_{ij}|\mathbf x|^2\right)d^3x, \qquad \mathcal E_{ij}=-\partial_i\partial_j U_{\mathrm{ext}}.$$

The STF part of the second derivative is understood; the buffer is vacuum. Taylor expansion of the external potential and the body’s multipole field give, after removing the monopole and center-of-mass dipole,

$$\begin{aligned} U_{\mathrm{pert}} &=-\frac12\mathcal E_{ij}x^ix^j +\frac{G}{2}Q_{ij}\partial_i\partial_j\frac1r\\ &=-\frac12r^2\mathcal E_{nn} +\frac{3G}{2r^3}Q_{nn}. \end{aligned}$$

Here $n^i=x^i/r$ and $\mathcal E_{nn}=\mathcal E_{ij}n^in^j$. The factor three follows from $\partial_i\partial_j(1/r)=(3n_i n_j-\delta_{ij})/r^3$ and the vanishing trace of $Q$. Thus $Q=-\lambda_E\mathcal E$ gives $\lambda_E=2k^E_2R^5/(3G)$, not a coefficient defined only by a falling power of a master field. These are the physical quadrupole and Newtonian limit of Binnington and Poisson 2009, Eqs. (1.1)–(1.2), p. 1 (PDF).

For the quadrupolar magnetic formulas in this chapter, write $J_{ij}=S_{ij}$ for the physical current quadrupole used by Poisson. The physical electric and magnetic couplings have different weights:

$$L_{\mathrm{int}} =-\frac12 Q^{ij}\mathcal E_{ij} -\frac23 J^{ij}\mathcal B_{ij}.$$

These weights matter when response is converted into absorbed power. They are consistent with the work formula in Poisson 2004, Eq. (8.43), p. 25 (PDF). A differently normalized worldline operator is permissible, but must not silently replace $Q_{ij}$ or $J_{ij}$. No higher-multipole magnetic normalization is inferred from this quadrupole dictionary.

Relativistic radial branches

For one parity and multipole, choose a potential-like, gauge-invariant radial amplitude $\Psi_{\ell m}$ whose leading static buffer equation is $[r^{-2}\partial_r(r^2\partial_r)-\ell(\ell+1)/r^2]\Psi_{\ell m}=0$. Other master fields, including the raw Teukolsky and Regge–Wheeler fields, have different leading powers and must be rescaled or matched with their own dictionary. At nonzero frequency this expansion also requires the near-zone condition $|\omega|r\ll1$, in addition to $GM\ll r\ll\mathcal R$. In that regime the leading structure is

$$\Psi_{\ell m}(r,\omega) =A_{\ell m}(\omega)r^\ell\bigl[1+O(r_g/r)\bigr] +B_{\ell m}(\omega)r^{-\ell-1}\bigl[1+O(r_g/r)\bigr].$$

Frequency-dependent corrections suppressed by the stated near-zone expansion are not displayed.

The applied amplitude is fixed by matching the growing branch to the distant-source solution. The induced amplitude is fixed by matching the decaying branch to a specified multipole observable. A bare ratio $B/A$ is therefore not yet a Love number: a change of radial variable, a change of master field, or a local counterterm can reshuffle analytic pieces. In the light-cone gauge, Binnington and Poisson instead identify combinations of metric functions that remain invariant under the residual gauge freedom (Binnington and Poisson 2009, § III.A, eqs. (3.6)–(3.13)). A curvature scalar or an on-shell scattering amplitude offers another matching observable.

The operational split is thus a three-step statement: specify the external solution and center-of-mass frame, choose the buffer gauge and radial basis, then map the two amplitudes to normalized STF moments. Only the complete map can be compared across calculations. The same discipline prevents a finite coefficient in one master variable from being mistaken for a physical induced multipole.

Gauge changes and what remains measurable

Consider an allowed asymptotic redefinition

$$r'=r\left[1+\epsilon\left(\frac{r_g}{r}\right)^{2\ell+1}\right].$$

To first order in $\epsilon$, a term $A r^\ell$ acquires a contribution proportional to $A\epsilon r_g^{2\ell+1}r'^{-\ell-1}$. A coefficient read from the decaying power alone therefore shifts. The external tidal tensor and the matched multipole—or, equivalently, the corresponding local worldline coefficient—must be transformed at the same time. A scattering phase, an absorption probability, or a force constructed from the fully matched fields does not change.

This is not a declaration that every response coefficient is arbitrary. Once the gauge, subtraction scale, response normalization, and boundary conditions are fixed, the coefficient is well defined. Nonanalytic frequency dependence, renormalization-group running, and horizon absorption can carry information that a local analytic redefinition cannot remove. Chia’s growing/decaying comparison in the Weyl scalar makes both the applied/induced split and the need for horizon and infinity conditions explicit (Chia 2021, §§ III–VI, eqs. (5)–(14)).

A curvature measurement in two coordinates

A falling coordinate coefficient is not itself a measurement. At leading weak-field order, a radially oriented orthonormal laboratory measures the electric Weyl component

$$\begin{aligned} C_{\hat0\hat r\hat0\hat r}^{\mathrm{pert}} &=-\partial_r^2U_{\mathrm{pert}} =\mathcal E_{nn}-\frac{18GQ_{nn}}{r^5}\\ &=\mathcal E_{nn}\left(1+\frac{18G\lambda_E}{r^5}\right). \end{aligned}$$

This calculation fixes the applied tide, subtracts the background monopole, and keeps the leading tidal and induced terms in the vacuum buffer. It is not an exact finite-radius observable in the strongly curved region. Relativistic corrections are organized separately in powers of $r_g/r$.

To see the coordinate cancellation explicitly, start with no physical quadrupole and make the small radial change

$$\begin{aligned} r&=r(R)=R-\epsilon\frac{r_g^5}{R^4},\\ U_{\mathrm{pert}} &=-\frac12\mathcal E_{nn}R^2 +\epsilon\frac{\mathcal E_{nn}r_g^5}{R^3} +O(\epsilon^2). \end{aligned}$$

Here $R$ is a new coordinate, not the reference radius used to define $k_2$. The apparent $R^{-3}$ coefficient is nonzero. But the Euclidean spatial metric is now $dr^2+r^2d\Omega^2=r'(R)^2dR^2+r(R)^2d\Omega^2$. The same orthonormal measurement uses its connection and radial unit vector:

$$C_{\hat0\hat r\hat0\hat r}^{\mathrm{pert}} =-\frac1{r'^2} \left(\frac{d^2U_{\mathrm{pert}}}{dR^2} -\frac{r''}{r'}\frac{dU_{\mathrm{pert}}}{dR}\right) =\mathcal E_{nn}.$$

The last equality follows directly from the chain rule for $U_{\mathrm{pert}}=-\mathcal E_{nn}r(R)^2/2$; it is exact for this coordinate change wherever $r'\ne0$, within the leading weak-field model. Reading only $-\partial_R^2U$ would falsely assign a response to a coordinate effect. With a genuine $Q$, the same calculation returns $\mathcal E_{nn}-18GQ_{nn}/r(R)^5$ at the same physical point.

The relativistic definition uses this same observable to compare response conventions. The effective theory will reproduce its induced potential from a normalized worldline term.

Exercises

Suppose $\Psi=A r^\ell$ in one buffer coordinate and use the transformation above. Does the absence of an $r^{-\ell-1}$ term survive as a statement about the raw coefficient?

Solution

Invert the transformation to first order: $r=r'[1-\epsilon(r_g/r')^{2\ell+1}]$. Hence

$$A r^\ell =A r'^\ell-\ell\epsilon A r_g^{2\ell+1}r'^{-\ell-1} +O(\epsilon^2).$$

The raw decaying coefficient has shifted from zero to $-\ell\epsilon A r_g^{2\ell+1}$. Dimensions check: both terms have the same dimension because $r_g^{2\ell+1}r'^{-\ell-1}$ scales as $r'^\ell$. This does not create a physical quadrupole. The coordinate definition of the tidal and multipole moments must be rematched, and the induced shift is then absorbed by the corresponding local convention. The exercise isolates why “coefficient of a falling power” is insufficient without a declared gauge and matching map.

The next page turns this operational split into gauge-controlled relativistic Love numbers.

References

• Binnington, Taylor, and Eric Poisson. 2009. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80, 084018. DOI.
• Chia, Horng Sheng. 2021. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104, 024013. DOI.
• Poisson, Eric. 2004. “Absorption of Mass and Angular Momentum by a Black Hole: Time-Domain Formalisms for Gravitational Perturbations, and the Small-Hole/Slow-Motion Approximation.” Physical Review D 70, 084044. DOI.