Relativistic Tidal Moments and Body Multipoles

A compact body does not respond directly to a coordinate separation or to a Newtonian-looking potential. It responds to the curvature and its derivatives measured on a representative worldline. Those local tidal moments drive mass and current multipoles, whose response coefficients can be matched to an exterior solution and then inserted into binary dynamics. This page fixes one explicit response convention, identifies the combinations that survive changes of notation, and keeps three logically different objects separate: a real zero-frequency susceptibility, a frequency-dependent conservative response, and an absorptive response.

Helpful background. The Einstein Equation supplies the vacuum curvature equations used in the buffer region; Weak Metric Perturbations and Gauge Freedom supplies linearized gauge transformations; Presymplectic Potentials and Currents motivates extracting response from an action; Relativistic Matter and Compact Objects supplies stellar structure and compactness; Relativistic Two-Body Dynamics supplies the worldline description; and Black-Hole Perturbation Theory supplies the exterior boundary-value problem.

Curvature in the body frame

Let $z^\mu(\tau)$ be a timelike representative worldline, with unit tangent $u^\mu$ and $u^\mu u_\mu=-1$. The spatial projector is

$$P_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu .$$

Choose an oriented orthonormal tetrad $(u^\mu,e^\mu{}_a)$ transported along the worldline. With $\epsilon_{\mu\nu\rho\sigma}$ oriented once and for all, define the electric and magnetic parts of the Weyl tensor by

$$\mathcal E_{\mu\nu} =C_{\mu\alpha\nu\beta}u^\alpha u^\beta, \qquad \mathcal B_{\mu\nu} =\frac12\epsilon_{\mu\alpha\rho\sigma} C^{\rho\sigma}{}_{\nu\beta}u^\alpha u^\beta .$$

This equation fixes the sign of $\mathcal B_{\mu\nu}$; reversing the spacetime orientation reverses every magnetic-type quantity below. In a vacuum neighborhood of the worldline, both tensors are spatial, symmetric, and tracefree:

$$\mathcal E_{\mu\nu}u^\nu=\mathcal B_{\mu\nu}u^\nu=0, \qquad \mathcal E^\mu{}_\mu=\mathcal B^\mu{}_\mu=0 .$$

Their body-frame components are $\mathcal E_{ab}=e^\mu{}_a e^\nu{}_b\mathcal E_{\mu\nu}$ and similarly for $\mathcal B_{ab}$. Electric moments are parity even; magnetic moments are parity odd. Higher moments are projected symmetric-tracefree derivatives, schematically

$$\mathcal E_{a_1\cdots a_\ell} \sim \nabla_{\langle a_1}\cdots\nabla_{a_{\ell-2}} \mathcal E_{a_{\ell-1}a_\ell\rangle}, \qquad \ell\ge2 ,$$

with corresponding magnetic moments. Precise numerical factors are conventional and must be carried through any comparison. Binnington and Poisson construct the electric and magnetic STF potentials and their exterior metric realization explicitly in Binnington and Poisson 2009, §§ I–III, especially Eqs. (1.3)–(1.5).

The Weyl definition assumes a vacuum buffer region around the body. If external matter intersects that neighborhood, Ricci curvature can provide additional local forcing; one cannot silently absorb it into the vacuum tidal moments. Likewise, the worldline and tetrad must be specified consistently. A shift of the representative center moves multipole strength between neighboring orders, while a rotation of the spatial triad rotates all STF components without changing scalar contractions.

Buffer-zone matching and body multipoles

Suppose a body of characteristic radius $R$ belongs to a binary with separation $b$ and external curvature radius $\mathcal R$. A controlled matched description requires an overlap region

$$R\ll r\ll \min(b,\mathcal R),$$

where $r$ is distance from the representative worldline. The internal solution may then be expanded for $r\gg R$, while the external binary solution is expanded for $r\ll b$. At each multipolar order, the growing radial solution represents the applied tidal field and the decaying solution represents the body’s induced multipole. Their ratio, after fixing the radial basis and multipole normalization, is response data. The separation of scales and matching logic are the organizing principles of the worldline effective theory developed in Goldberger and Rothstein 2006, §§ II–IV.

The word “induced” matters. A rotating body can possess permanent spin-induced multipoles even when the external tide is absent. Those background moments must be subtracted before defining the part linear in $\mathcal E_L$ or $\mathcal B_L$. At quadrupolar order, write the induced mass and current multipoles as $Q_{ab}$ and $J_{ab}$. We define their coupling convention by

$$L_{\rm int} =-\frac12 Q^{ab}\mathcal E_{ab} -\frac12 J^{ab}\mathcal B_{ab}.$$

The factors of one half are part of the definition. A paper using a different current-multipole normalization will quote a different magnetic susceptibility even when it predicts the same observable.

For an isotropic, nonrotating, parity-invariant body in the linear regime, the causal response is diagonal in the electric and magnetic sectors:

$$Q_{ab}(\omega)=-\lambda_E(\omega)\mathcal E_{ab}(\omega), \qquad J_{ab}(\omega)=-\lambda_B(\omega)\mathcal B_{ab}(\omega).$$

Use the Fourier time dependence $e^{-i\omega\tau}$. For a monochromatic real field, $\mathcal E_{ab}(\tau)=\operatorname{Re} [\mathcal E_{ab}(\omega)e^{-i\omega\tau}]$; this convention fixes the factor in the averaged power below. Reality requires

$$\lambda_{E,B}(-\omega)=\lambda_{E,B}(\omega)^* .$$

The response contains three regimes that should not be conflated. The static deformability is the real zero-frequency limit $\lambda_E(0)$. A dynamical response has frequency dependence and poles associated with internal modes. Away from its poles, an undamped mode has a real response; the retarded prescription also supplies spectral delta functions at the poles, representing coherent excitation. A damped response broadens those poles and transfers coherent mechanical energy into other channels. Thus an imaginary part diagnoses transfer from the driving field, but identifying it as heat requires specifying where that energy goes. The static replacement $Q_{ab}=-\lambda_E(0)\mathcal E_{ab}$ is justified only when the forcing frequency is well below every relevant mode frequency and damping is treated consistently.

Love numbers and convention-independent content

For a spherical body with areal radius $R$, our electric quadrupolar deformability is related to the commonly used dimensionless Love number by

$$\lambda_E(0)=\frac{2}{3G}k_2 R^5, \qquad \Lambda=\frac{2}{3}k_2 C^{-5}, \qquad C=\frac{Gm}{R}.$$

Thus $k_2$, $\lambda_E$, and $\Lambda$ are not interchangeable: $k_2$ is dimensionless but depends on the radius convention used to make the exterior coefficient dimensionless; $\lambda_E$ is the coefficient in a specified multipole coupling; and $\Lambda$ combines the response with the body’s mass. The normalization $Q_{ab}=-(2/3)k_2R^5\mathcal E_{ab}$ in $G=1$ units appears already in the Newtonian and relativistic construction of Binnington and Poisson 2009, § I, Eqs. (1.1)–(1.2).

A convenient auxiliary action reproducing our static electric response:

$$L_Q =-\frac12 Q^{ab}\mathcal E_{ab} -\frac{1}{4\lambda_E}Q^{ab}Q_{ab}.$$

Varying $Q_{ab}$ gives $Q_{ab}=-\lambda_E\mathcal E_{ab}$. Eliminating it gives

$$L_Q^{\rm on\ shell} =\frac{\lambda_E}{4}\mathcal E^{ab}\mathcal E_{ab}, \qquad \Delta H_{\rm tidal} =-\frac{\lambda_E}{4}\mathcal E^{ab}\mathcal E_{ab}$$

for a time-independent configuration. This is why a sign quoted for $Q_{ab}$ alone is not portable: the sign of the coupling and the definition of $\mathcal E_{ab}$ matter together. The invariant check is the sign and magnitude of an observable such as the conservative binding-energy shift after all definitions have been translated. Porto gives the corresponding background-versus-response and matching discussion in Porto 2016, § 6.5, pp. 30–37.

At nonzero frequency, an on-shell action written only with a real local coefficient cannot describe absorption. A retarded kernel is needed. For $\omega\gt 0$, passivity in the present convention gives

$$\left\langle \dot E_{\rm body}\right\rangle =-\frac12\left\langle \mathcal E_{ab}\dot Q^{ab} \right\rangle =\frac{\omega}{4} \operatorname{Im}\lambda_E(\omega) \mathcal E_{ab}^*(\omega)\mathcal E^{ab}(\omega) \ge0 ,$$

for a nonrotating body in its ground state. Rotation can invalidate this simple positivity statement because the horizon or material mode exchanges angular momentum with the orbit; the relevant co-rotating frequency then replaces $\omega$.

Newtonian and dimensional checks

Curvature has dimension $L^{-2}$, while a mass quadrupole has dimension $ML^2$. Consequently $\lambda_E$ has dimension $ML^4$, equivalently $L^5/G$ when $c=1$. This agrees with the $R^5/G$ scaling above. The magnetic tensor has the same dimension as the electric tensor but opposite parity, so a parity-invariant nonrotating body cannot have a linear cross-response $Q_{ab}\propto\mathcal B_{ab}$.

In the Newtonian limit, use a companion potential $\Phi=-Gm_B/r$ and the convention

$$\mathcal E_{ij}=\partial_i\partial_j\Phi =\frac{Gm_B}{r^3} \left(\delta_{ij}-3n_i n_j\right).$$

It is STF away from the companion and satisfies

$$\mathcal E_{ij}\mathcal E_{ij} =6\left(\frac{Gm_B}{r^3}\right)^2 .$$

Because $\mathcal E_{rr}=-2Gm_B/r^3$, the relation $Q_{rr}=-\lambda_E\mathcal E_{rr}\gt 0$ describes radial elongation for $\lambda_E\gt 0$. Substitution into the on-shell Hamiltonian gives the leading contribution of body $A$,

$$\Delta H_A =-\frac32\lambda_A \left(\frac{Gm_B}{r^3}\right)^2 .$$

This one calculation simultaneously checks the STF normalization, dimensions, response sign, and the expected $r^{-6}$ scaling. It is the starting point for adiabatic tidal dynamics, not a derivation of $\lambda_A$ from the body’s interior.

The construction fails in recognizable ways: the buffer region disappears when $R/b$ is not small; linear response fails when $|Q|/(mR^2)$ is not small; the adiabatic approximation fails near a mode pole; and a finite local derivative expansion fails when the forcing varies on the internal light-crossing or material timescale. These are distinct failure modes, so adding more post-Newtonian orders cannot repair a missing internal mode or an onset of mass shedding.

Exercises

1. Projected trace. Show directly that $\mathcal E^\mu{}_\mu=0$ in a vacuum region.

Solution

Contract the definition:

$$\mathcal E^\mu{}_\mu =g^{\mu\nu}C_{\mu\alpha\nu\beta}u^\alpha u^\beta .$$

Every contraction of the Weyl tensor vanishes by definition, so the result is zero. Orthogonality follows independently from antisymmetry in the second index pair:

$$\mathcal E_{\mu\nu}u^\nu =C_{\mu\alpha\nu\beta}u^\alpha u^\nu u^\beta=0,$$

because $u^\nu u^\beta$ is symmetric while $C_{\mu\alpha\nu\beta}$ is antisymmetric under $\nu\leftrightarrow\beta$. Symmetry of $\mathcal E_{\mu\nu}$ follows from the pair-exchange symmetry of the Weyl tensor.

2. Leading tidal energy. Derive the coefficient $-3/2$ in $\Delta H_A$ without choosing a coordinate axis.

Solution

Let $T_{ij}=\delta_{ij}-3n_i n_j$. Since $n_i n_i=1$,

$$T_{ij}T_{ij} =\delta_{ij}\delta_{ij} -6\delta_{ij}n_i n_j +9n_i n_j n_i n_j =3-6+9=6.$$

Therefore $\mathcal E^2=6(Gm_B/r^3)^2$. Inserting this into $\Delta H_A=-\lambda_A\mathcal E^2/4$ gives

$$\Delta H_A=-\frac{6}{4}\lambda_A \left(\frac{Gm_B}{r^3}\right)^2 =-\frac32\lambda_A \left(\frac{Gm_B}{r^3}\right)^2 .$$

References

• Binnington, Taylor, and Eric Poisson. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80, 084018 (2009). DOI; Open PDF.
• Goldberger, Walter D., and Ira Z. Rothstein. “An Effective Field Theory of Gravity for Extended Objects.” Physical Review D 73, 104029 (2006). DOI; Open PDF.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633, 1–104 (2016). DOI; Open PDF.