Spin–Tide Couplings

Spin changes tidal response in two related but distinct ways. Even at zero forcing frequency, it permits new rotational-tidal response coefficients and mixes irreducible electric and magnetic structures. At nonzero frequency, Coriolis and frame-dragging effects split mode frequencies and move resonances. Neither effect is the body’s permanent spin-induced quadrupole, and neither is automatically dissipative. A clean calculation must fix the representative worldline, spin convention, body frame, Fourier sign, and order in spin before comparing coefficients.

Helpful background. The Einstein Equation supplies the exterior field equations; Weak Metric Perturbations and Gauge Freedom supplies slow-rotation perturbation theory; Presymplectic Potentials and Currents supplies action and angular-momentum currents; Relativistic Matter and Compact Objects supplies rotating stellar equilibria; Relativistic Two-Body Dynamics supplies spin supplementary conditions and orbital frames; and Black-Hole Perturbation Theory supplies Kerr response.

Spin conventions and irreducible tidal products

Let $S^\mu u_\mu=0$ be the spin vector associated with a fixed spin supplementary condition and representative worldline. In the body tetrad define

$$\chi^a=\frac{S^a}{Gm^2}, \qquad |\chi|\ll1.$$

The small-spin assumption applies to the static construction below. A shift of worldline accompanied by a change of spin supplementary condition can reshuffle spin and multipole terms; quoted response coefficients are comparable only after that choice and the multipole normalization are translated.

For quadrupolar electric and magnetic moments $\mathcal E_{ab}$ and $\mathcal B_{ab}$, angular-momentum addition gives

$$1\otimes2=1\oplus2\oplus3.$$

The corresponding irreducible products can be written

$$\begin{aligned} \mathcal F_a&=\mathcal E_{ab}\chi^b, &\mathcal K_a&=\mathcal B_{ab}\chi^b,\\ \widehat{\mathcal E}_{ab} &=2\chi^c\epsilon_{cd(a}\mathcal E_{b)}{}^d, &\widehat{\mathcal B}_{ab} &=2\chi^c\epsilon_{cd(a}\mathcal B_{b)}{}^d,\\ \mathcal F_{abc}&=\mathcal E_{\langle ab}\chi_{c\rangle}, &\mathcal K_{abc}&=\mathcal B_{\langle ab}\chi_{c\rangle}. \end{aligned}$$

The brackets mean symmetric tracefree projection. These tensors make the selection rules visible: a spin coupled to an $\ell=2$ tide can drive dipolar, quadrupolar, or octupolar sectors, subject to parity and the field equations. In the orientation convention used here, $\mathcal E_{ab}$ is parity even, $\mathcal B_{ab}$ is parity odd, and $\chi^a$ is axial. Landry and Poisson list the same irreducible products and their harmonic parity in Landry and Poisson 2015, § II, Eqs. (2.1)–(2.6).

Two elementary tests follow immediately. Every term linear in spin must reverse sign under $\chi^a\rightarrow-\chi^a$. Every rotational-tidal term must vanish at $\chi^a=0$, leaving the ordinary electric and magnetic Love response. A term that survives this limit has been misclassified or double counted.

Static rotational-tidal response

The exterior metric of a slowly rotating body in a weak, slowly varying quadrupolar tide contains the ordinary electric and magnetic Love coefficients plus response amplitudes multiplying spin–tide structures. In one widely used basis, two amplitudes accompany the quadrupolar structures and two accompany octupolar structures. The radial solutions separate an applied piece from a decaying response piece, just as in the nonrotating problem, but the new coefficients depend on the basis and normalization chosen for the spin-coupled tidal potentials.

Landry and Poisson’s exterior construction keeps all terms linear in $\chi^a\mathcal E_{ab}$ and $\chi^a\mathcal B_{ab}$ and identifies four rotational-tidal Love numbers in Landry and Poisson 2015, § I and §§ III–IV. Their result assumes:

• a spherical background corrected only to first order in spin;
• weak response, linear in the applied quadrupolar tide;
• a tidal field varying slowly enough to be treated as stationary;
• a vacuum exterior and a specified light-cone or Regge–Wheeler gauge basis.

Matching a material interior determines which combinations depend on the equation of state. For barotropic perfect-fluid models, Landry found two independent internal-structure-dependent rotational-tidal coefficients while two other exterior constants take fixed values within that fluid-state prescription; see Landry 2017, §§ I A–I B and §§ IV–V. This is not a universal statement for arbitrary elastic, stratified, magnetized, superfluid, or rapidly rotating matter. Even the gravitomagnetic response depends on whether the fluid perturbation is constrained to be strictly static or allowed to be irrotational.

It is also essential to distinguish rotational-tidal response from a permanent spin-induced multipole. A stationary axisymmetric body already has, schematically,

$$Q^{\rm spin}_{ab} \propto-\frac{C_{ES^2}}{m} S_{\langle a}S_{b\rangle},$$

which is quadratic in spin and exists without an external tide. A rotational-tidal multipole is linear in both $S$ and $\mathcal E$ or $\mathcal B$ and vanishes when either is removed. Mixing the two contaminates both the inferred Love coefficient and the point-particle spin sector.

Within the four-dimensional, slowly rotating, stationary construction of Landry and Poisson, the corresponding black-hole Love coefficients vanish. That statement does not set the Kerr response function to zero at nonzero frequency: rotating horizons have an absorptive response, and the co-rotating frequency controls its sign.

Coriolis splitting of dynamical tides

For a rotating star, the relevant forcing frequency is measured relative to the body. In an axisymmetric Newtonian description,

$$\widehat\omega=\omega-m\Omega_*,$$

where $\Omega_*$ is the stellar spin frequency and $m$ labels the azimuthal harmonic. Consequently a tide that is stationary in an inertial coordinate system need not be stationary relative to the fluid.

At linear order in spin, the leading Coriolis interaction of a dynamical quadrupole has the structure

$$L_{SQ} \sim C_{SQ}S^i\epsilon_{ijk} Q^{j\ell}\dot Q^k{}_\ell .$$

It is parity even, rotationally invariant, and odd under spin reversal. In a mode basis it shifts the $m$ and $-m$ poles in opposite directions. Steinhoff, Hinderer, Dietrich, and Foucart derive the selection rules and the unique electric-quadrupole Coriolis term retained in their model in Steinhoff et al. 2021, § II B, Eqs. (2.19)–(2.23), and § III, Eqs. (3.4)–(3.14).

For an aligned, circular binary their response takes the illustrative form

$$\lambda_m(\omega) =\frac{z^2\omega_0^2\lambda_0} {z^2\omega_0^2- \left[\omega-m\Omega_{\rm FD} +mz\overline C_{SQ}S\right]^2}.$$

Here $z$ is a redshift factor, $\Omega_{\rm FD}$ is the frame-dragging angular velocity of the local frame, and $\overline C_{SQ}$ is a matched Coriolis coefficient. This equation is model-specific, but its structure is general: redshift, frame dragging, and intrinsic stellar rotation all move the response poles. The derivation and pole matching appear in Steinhoff et al. 2021, §§ IV–V, Eqs. (4.9)–(5.7).

The symmetry check is

$$\lambda_m(\omega;S) =\lambda_{-m}(\omega;-S)$$

when every other orientation is reversed consistently. An aligned spin can move a particular prograde resonance one way while an antialigned spin moves it the other; a statement that “spin raises the mode frequency” without naming frame, branch, and sign convention is incomplete.

Resonance, torque, and dissipation

Mode splitting is conservative when the poles lie on the real axis and the response has no absorptive part. Damping moves poles into the lower half-plane for the $e^{-i\omega t}$ convention and gives $\operatorname{Im}\lambda_m\ne0$. Rotation also permits angular-momentum exchange even in a conservative resonance: the mode carries a tidal spin, and the orbit feels a torque.

This distinction limits “effective Love number” substitutions. A real scalar inferred from $\mathcal E_{ij}Q^{ij}$ can reproduce a radial conservative force along some circular trajectories, but it does not in general encode the tangential torque. For rapidly spinning, antialigned neutron stars near an $f$-mode resonance, Yu, Arras, and Weinberg show explicitly that the tidal torque dominates the orbital backreaction and requires an additional response factor in Yu, Arras, and Weinberg 2024, § III, especially Eqs. (53)–(60). Their large resonant effects concern rapidly spinning configurations and should not be imported into a slow-spin expansion.

There are therefore three separate data sets:

1. static rotational response, the zero-frequency real coefficients of a specified material state;
2. dynamical conservative response, the spin-split real pole structure and mode residues;
3. dissipative response, the imaginary part and associated energy–angular-momentum flux.

The first does not determine the second, and neither alone determines the third. A waveform model should state which of these it retains.

Exchange and limiting checks

For a binary, construct two body-local copies. The tide on body 1 depends on the field sourced by body 2 and couples to $S_1$; exchanging labels $1\leftrightarrow2$ must map it into the corresponding term on body 2. A coupling such as $S_1\mathcal E_1^2$ that is copied without exchanging both the spin and the local tidal field fails this test.

Other useful limits are:

• $\chi_A\to0$: recover nonrotating static or dynamical tides;
• $\omega\to0$ before $\chi\to0$: recover the chosen stationary rotational response, provided the fluid-state limit exists;
• $R_A/r\to0$: suppress material tidal operators relative to point-particle spin interactions;
• reversing all spins and orbital angular momentum: preserve scalar observables while interchanging prograde and retrograde mode labels.

Near rapid rotation, avoided crossings, centrifugal deformation, precession, and nonlinear mode coupling can invalidate the linear-spin basis. Near disruption, no finite set of rotational Love coefficients guarantees error control. Those regimes require a rotating stellar perturbation calculation or relativistic hydrodynamics, not extrapolation of $\chi\ll1$ coefficients.

Exercises

1. Verify the rank decomposition. Show that $\widehat{\mathcal E}_{ab}$ is symmetric and tracefree.

Solution

Symmetrization is explicit:

$$\widehat{\mathcal E}_{ab} =\chi^c\left( \epsilon_{cda}\mathcal E_b{}^d +\epsilon_{cdb}\mathcal E_a{}^d \right).$$

Contracting $a$ and $b$ gives

$$\delta^{ab}\widehat{\mathcal E}_{ab} =2\chi^c\epsilon_{cda}\mathcal E^{ad}=0,$$

because $\epsilon_{cda}$ is antisymmetric in $d,a$ while $\mathcal E^{ad}$ is symmetric. Thus it belongs to the rank-2 STF part of $1\otimes2$.

2. Spin-reversal check. Use the response denominator to show how the pole set transforms under $S\to-S$.

Solution

The poles satisfy

$$\omega-m\Omega_{\rm FD} +mz\overline C_{SQ}S =\pm z\omega_0.$$

Under $m\to-m$, $S\to-S$, and a simultaneous reversal of the frame-dragging orientation $\Omega_{\rm FD}\to-\Omega_{\rm FD}$, the bracket is unchanged:

$$\omega-(-m)(-\Omega_{\rm FD}) +(-m)z\overline C_{SQ}(-S) =\omega-m\Omega_{\rm FD}+mz\overline C_{SQ}S.$$

The pole set is therefore unchanged without interchanging the two choices of $\pm$. Keeping $m$ fixed while reversing only $S$ instead measures the physical splitting between opposite spin orientations.

References

• Landry, Philippe, and Eric Poisson. “Tidal Deformation of a Slowly Rotating Material Body: External Metric.” Physical Review D 91, 104018 (2015). DOI; Open PDF.
• Landry, Philippe. “Tidal Deformation of a Slowly Rotating Material Body: Interior Metric and Love Numbers.” Physical Review D 95, 124058 (2017). DOI; Open PDF.
• Steinhoff, Jan, Tanja Hinderer, Tim Dietrich, and Francois Foucart. “Spin Effects on Neutron Star Fundamental-Mode Dynamical Tides: Phenomenology and Comparison to Numerical Simulations.” Physical Review Research 3, 033129 (2021). DOI; Open PDF.
• Yu, Hang, Phil Arras, and Nevin N. Weinberg. “Dynamical Tides during the Inspiral of Rapidly Spinning Neutron Stars: Solutions beyond Mode Resonance.” Physical Review D 110, 024039 (2024). DOI; Open PDF.