Tides and Finite-Size Effects

Finite size turns a compact binary from two structureless trajectories into a coupled orbital–internal system. The companion’s curvature polarizes each body; internal modes store and return energy; spin changes the spectrum and the allowed tensor couplings; horizons and material processes absorb energy; and sufficiently strong deformation destroys the scale separation behind a worldline model. This chapter develops one response framework that distinguishes those effects instead of compressing them into a single “tidal correction.”

Helpful background. The Einstein Equation supplies the curvature dynamics; Weak Metric Perturbations and Gauge Freedom supplies perturbative response and gauge-invariant observables; Presymplectic Potentials and Currents supplies variational energy transfer; and Relativistic Matter and Compact Objects supplies equilibrium structure, mode spectra, and material properties.

Finite-size response as orbital physics

Choose a representative worldline with unit tangent $u^\mu$ and an orthonormal spatial frame carried by the body. The external Weyl curvature defines symmetric tracefree electric and magnetic tidal moments $\mathcal E_{ab}$ and $\mathcal B_{ab}$. The body’s response is encoded in mass-type and current-type multipoles such as $Q_{ab}$ and $S_{ab}$. At electric quadrupole order, this chapter uses

$$L_{\rm int}=-\frac12Q^{ab}\mathcal E_{ab}, \qquad Q_{ab}(\omega)=-\lambda_E^{\rm ret}(\omega)\mathcal E_{ab}(\omega).$$

The sign is a convention only after the coupling, curvature definition, Fourier transform, and response definition have all been specified together. With the convention above, an ordinary stable, nonrotating material body has a real positive deformability $\lambda_E(0)$ and an on-shell tidal Hamiltonian

$$\Delta H_{\rm tidal} =-\frac{\lambda_E(0)}{4} \mathcal E_{ab}\mathcal E^{ab}.$$

Thus polarization makes the fixed-separation potential more attractive. That statement is more portable than the sign of $Q_{ab}$ by itself. The same action normalization and induced quadrupole are used in the leading relativistic inspiral calculation of Flanagan and Hinderer 2008, Eqs. (3)–(8).

A retarded response function separates three physically different notions:

$$\lambda_E^{\rm ret}(\omega) =\lambda_E(0) +\Delta\lambda_E^{\rm dyn}(\omega) +i\,\lambda_E^{\rm diss}(\omega).$$

The first term is static: a real zero-frequency coefficient obtained after the body has relaxed within the assumed state. The real frequency-dependent term is dynamical: it contains dispersive corrections and poles associated with internal modes, even when the evolution is conservative. The imaginary term measures power transferred from periodic driving and is commonly called dissipative when that power reaches an irreversible sink. A fourth regime, disruptive, is not another coefficient in this linear decomposition. It occurs when deformation, mass shedding, or loss of spatial scale separation invalidates the response expansion itself.

These distinctions are operational. A real constant Love number cannot generate resonance or heating. A conservative mode has real dispersive response away from its poles, but its retarded response also has delta-function spectral weight at the poles: resonant transfer into a coherent mode does not itself produce entropy. The dynamical-tide page separates this storage from subsequent damping. A complex response can encode linear absorption, but it remains a perturbation about an identifiable body and therefore cannot follow matter streams after disruption. Goldberger and Rothstein formulate this hierarchy as matched worldline operators and dynamical correlators in Goldberger and Rothstein 2006, §§ II–IV and Goldberger and Rothstein 2006, § II, Eqs. (19)–(27).

A six-stage route through the chapter

The pages are ordered so that every new approximation has a defined input and a visible failure condition.

1. Relativistic Tidal Moments and Body Multipoles constructs $\mathcal E_{ab}$, $\mathcal B_{ab}$, and higher symmetric tracefree moments in a body frame, then separates applied growing fields from induced decaying fields in a buffer region. It explains why a Love number is not meaningful without a radial basis, a multipole normalization, and a state prescription. The output is the response data that orbital dynamics may use; object-specific stellar or black-hole boundary-value calculations remain with the corresponding compact-object subjects.

2. Adiabatic Tidal Dynamics inserts the real zero-frequency response into the binary action. It derives the $r^{-6}$ quadrupolar potential, the circular-orbit correction, the leading 5PN relative scaling, and the need to modify both the conservative energy and radiative flux. “Adiabatic” here means that the internal state tracks the forcing, not that the orbit is exactly stationary.

3. Dynamical Tides and Mode Resonances promotes the induced multipole to an oscillator or, equivalently, retains the poles of the retarded response. It shows why algebraic elimination fails near a swept resonance, why a causal solution depends on forcing history, and how a residual contact term prevents double counting when only selected modes are explicit.

4. Spin–Tide Couplings decomposes products of spin and tidal tensors into definite parity and multipolarity. It keeps separate the spin correction to an equilibrium response coefficient, the Coriolis splitting of dynamical poles, and the torque exerted by a nonspherical response. The page also states which conclusions rely on slow rotation, a chosen fluid state, or alignment.

5. Horizon Absorption and Material Dissipation in Binaries compares two physical realizations of the imaginary response. Horizon regularity fixes black-hole fluxes, while viscosity, diffusion, reactions, or damped modes determine material absorption. Superradiant orbital energy extraction is distinguished from horizon entropy decrease: the area law involves the horizon-frame combination of energy and angular momentum.

6. Tidal Disruption and Breakdown of the Worldline Expansion identifies separate size, deformation, and frequency diagnostics; compares scale estimates, nonlinear extensions, calibrated fits, and relativistic hydrodynamics; and states when a single worldline ceases to represent the matter. Because threshold modeling remains sensitive to calibration and equation of state, that page gives dated results, competing interpretations, uncertainties, and tests that could change the conclusion.

This order should also govern model construction. One first fixes tidal and multipole conventions, then selects static or dynamical conservative response, adds spin consistently, supplies the appropriate absorptive channel, and finally checks whether the assumed body and scale hierarchy still exist. Starting from a calibrated end-stage fit and reading it backward as a universal Love response would reverse that logic.

One response, several energy accounts

For a linear quadrupole subject to a nearly monochromatic real field, write the positive-frequency amplitude as $\mathcal E_{ab}(\omega)$. In the convention above, the cycle-averaged power delivered to a nonrotating passive body is

$$\left\langle\dot E_{\rm body}\right\rangle =\frac{\omega}{4} \operatorname{Im}\lambda_E^{\rm ret}(\omega) \mathcal E_{ab}^{*}(\omega)\mathcal E^{ab}(\omega) \ge0 \qquad (\omega\gt 0).$$

The real part instead changes the stored energy and conservative force. Near an isolated weakly damped mode,

$$\lambda_E^{\rm ret}(\omega) \simeq \lambda_{\rm contact} +\frac{\lambda_f\omega_f^2} {\omega_f^2-\omega^2-2i\gamma_f\omega},$$

where $\gamma_f$ is the amplitude-damping rate. Thus dispersion and absorption are two parts of the same causal response. Far below the pole, the expansion reduces to a real static coefficient plus derivative corrections. Near the pole, retaining the mode as a dynamical variable avoids a singular instantaneous approximation. If forcing becomes nonlinear, additional operators or coupled modes may extend the model, but only while deformation remains perturbative.

Orbital evolution must use a complete energy account. Schematically,

$$\frac{d}{dt} \left( E_{\rm orb}+E_{\rm modes}+E_{\rm spin} \right) =-\mathcal F_\infty -\mathcal F_{\rm body},$$

Here $E_{\rm orb}$ includes the orbit–mode interaction energy, $E_{\rm spin}$ includes any explicitly evolved rotational energy, and $\mathcal F_\infty$ is radiation carried to infinity. The body flux $\mathcal F_{\rm body}$ contains only energy transferred to degrees of freedom absent from the left-hand side: for example, heat or horizon mass when that mass is held fixed in the orbital model. If the evolving horizon masses are instead included in the total energy, the horizon transfer is internal and cancels from its balance. Conservative tidal energy belongs on the left, not inside a dissipative flux. A resonantly excited mode also belongs on the left until it damps; material damping produces heat, while gravitational-wave damping contributes to the radiative loss. Each channel must appear once. For rotation, the useful first-law combination is $dE-\Omega_{\rm spin}dJ$, because energy and angular momentum are transferred together.

For a Kerr horizon, the horizon-frame frequency is $\widetilde\omega=\omega-m\Omega_H$. A mode with $\widetilde\omega\lt 0$ can carry negative Killing energy into the horizon, increasing the orbital or outgoing energy. The corresponding entropy production remains nonnegative because the area balance is governed by $\dot M_H-\Omega_H\dot J_H$. Poisson derives the time-domain horizon mass, angular-momentum, and area fluxes in Poisson 2004, §§ IV C–V D. This is the cleanest example of why “dissipation” must name the energy and frame being discussed.

Validity map

No single small parameter controls every finite-size effect. A useful minimum set for body $A$ is

$$\epsilon_{\rm size} \sim\frac{R_A}{\mathcal L_{\rm ext}}, \qquad \epsilon_{\rm lin} \sim\frac{|Q_A|}{m_AR_A^2}, \qquad \epsilon_{\rm ad} \sim\frac{|\omega_{\rm tide}|}{\omega_{\rm mode}}, \qquad \epsilon_{\rm spin} \sim\frac{|\Omega_{\rm spin}|}{\omega_{\rm mode}}.$$

Here $\mathcal L_{\rm ext}$ is the smaller relevant curvature or curvature-gradient scale. The local worldline expansion asks for $\epsilon_{\rm size}\ll1$. Linear response separately asks for $\epsilon_{\rm lin}\ll1$. Static elimination asks for $\epsilon_{\rm ad}\ll1$, while a slow-rotation truncation asks for $\epsilon_{\rm spin}\ll1$. A mode model can remain useful when $\epsilon_{\rm ad}$ is order unity, but it does not thereby repair large $\epsilon_{\rm lin}$. Conversely, a strongly deformable star at large mass ratio can approach material failure even while $R_A/r$ remains small enough for a spatial multipole expansion.

The response also depends on assumptions not represented by a magnitude alone:

• the equilibrium state, including temperature, composition, stratification, magnetic field, superfluid components, and circulation;
• the basis and matching convention used to divide applied and induced fields;
• the boundary condition, such as regularity at a center, outgoing radiation, or ingoing regularity at a horizon;
• the number of explicit modes and nonlinear operators retained;
• the orbital regime, including eccentricity, precession, and rate of frequency sweep;
• the calibration region of any phenomenological merger or disruption fit.

A calculation should therefore report both its coefficients and its domain. A neutron-star Love number computed for a cold barotropic equilibrium is not automatically the coefficient for a hot post-merger remnant. A Kerr low-frequency response is not a prescription for a nonlinear common horizon. A black-hole–neutron-star remnant-mass fit is not an invariant disruption criterion outside its mass-ratio, spin, compactness, and equation-of-state coverage.

Cross-regime consistency checks

Several checks expose most convention and modeling errors before a long waveform calculation.

Dimensions. In units with $c=1$, curvature has dimension $L^{-2}$ and a mass quadrupole has dimension $ML^2$, so $\lambda_E$ has dimension $ML^4=L^5/G$. The static quadrupole energy therefore scales as $\lambda_E(Gm/r^3)^2$, which has the dimension of energy.

Zero-frequency limit. A dynamical model must reproduce the chosen static coefficient:

$$\lim_{\omega\to0} \operatorname{Re}\lambda_E^{\rm ret}(\omega) =\lambda_E(0).$$

If explicit modes already saturate this limit, adding the full static contact coefficient double counts their contribution. Steinhoff et al. display the oscillator-to-response matching and residual contact term in Steinhoff et al. 2016, §§ I A–II A, Eqs. (1.1)–(2.2).

Passivity and rotation. For a nonrotating ground-state body, the spectral density must give nonnegative absorbed power for positive frequency. For a rotating body, apply this check in the co-rotating channel and track angular momentum; do not demand positivity of inertial-frame energy flux during superradiance.

Observable sign. Translate every curvature and multipole convention before comparing formulas. The final conservative binding energy, scattering angle, redshift invariant, or phase correction must agree; agreement of an isolated symbol is insufficient.

Matched truncation. Keep conservative energy, radiation to infinity, body absorption, and explicit internal energy at compatible formal accuracy. Adding a high-order tidal term to only one side of the balance equation does not produce a controlled high-order phase.

Breakdown behavior. A well-designed reduced model should signal rather than conceal loss of validity. Growing deformation, substantial power in omitted modes, loss of convergence under added operators, or disagreement with resolved hydrodynamics near a threshold should enlarge the uncertainty or trigger a change of description.

Solved synthesis check

Consider a nonrotating neutron star whose quadrupolar response is modeled by

$$\lambda^{\rm ret}(\omega) =\lambda_c+ \frac{\lambda_f\omega_f^2} {\omega_f^2-\omega^2-2i\gamma\omega}, \qquad \lambda_c,\lambda_f,\omega_f,\gamma\gt 0.$$

Classify the model in three frequency regimes, determine its static deformability, and state one condition under which it cannot describe the system even though the formula remains finite.

Solution

At $\omega\ll\omega_f$ and $\gamma\omega\ll\omega_f^2$,

$$\lambda^{\rm ret}(\omega) =\lambda_c+\lambda_f +O\!\left(\frac{\omega^2}{\omega_f^2}\right) +i\,O\!\left(\frac{\gamma\omega}{\omega_f^2}\right).$$

The static deformability is therefore $\lambda(0)=\lambda_c+\lambda_f$. Far below resonance the response is approximately adiabatic and conservative, with a small dissipative lag if $\gamma\ne0$.

When $|\omega-\omega_f|$ is comparable to the resonance width or to the frequency-sweep scale, the real response varies rapidly and the imaginary part is enhanced. This is a dynamical and dissipative regime. An instantaneous replacement $Q=-\lambda(0)\mathcal E$ fails; one should evolve the mode or use the full causal kernel with the forcing history.

Well above the pole but still inside the effective description, the oscillator contribution becomes dispersive and decreases approximately as $-\lambda_f\omega_f^2/\omega^2$. The contact term remains, but omitted higher modes and short-distance physics may become important, so the formula alone does not establish accuracy.

Finally, finiteness of $\lambda^{\rm ret}$ does not guarantee validity. If $|Q|/(mR^2)$ becomes order unity, if a surface begins to shed mass, or if the external curvature varies across the body on the scale $R$, the linear single-worldline response has failed. Damping keeps the algebraic pole finite but does not prevent nonlinear deformation or disruption.

References

• Flanagan, Éanna É., and Tanja Hinderer. “Constraining Neutron-Star Tidal Love Numbers with Gravitational-Wave Detectors.” Physical Review D 77, 021502(R) (2008). 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.
• Goldberger, Walter D., and Ira Z. Rothstein. “Dissipative Effects in the Worldline Approach to Black Hole Dynamics.” Physical Review D 73, 104030 (2006). DOI; Open PDF.
• Poisson, Eric. “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 (2004). DOI; Open PDF.
• Steinhoff, Jan, Tanja Hinderer, Alessandra Buonanno, and Andrea Taracchini. “Dynamical Tides in General Relativity: Effective Action and Effective-One-Body Hamiltonian.” Physical Review D 94, 104028 (2016). DOI; Open PDF.