Horizon Absorption and Material Dissipation in Binaries

Dissipative finite-size effects remove mechanical energy and angular momentum from the orbit, but the sink depends on the body. A black-hole horizon absorbs or superradiantly returns flux; a material body can damp modes through shear viscosity, bulk viscosity, diffusion, elasticity, or nonlinear couplings. All these channels can be represented by retarded response functions, yet their coefficients and even their flux signs are not interchangeable. The right comparison is made at the same body-frame forcing frequency, with the same multipole normalization, and inside the validity domain of each microphysical or horizon calculation.

Helpful background. The Einstein Equation supplies flux conservation; Weak Metric Perturbations and Gauge Freedom supplies linear response; Presymplectic Potentials and Currents supplies canonical energy and horizon fluxes; Relativistic Matter and Compact Objects supplies viscosity and material modes; Relativistic Two-Body Dynamics supplies balance evolution; and Black-Hole Perturbation Theory supplies ingoing-horizon boundary conditions.

Retarded response and absorbed power

Use the Fourier convention $e^{-i\omega\tau}$ and the quadrupole coupling

$$L_{\rm int} =-\frac12Q^{ab}\mathcal E_{ab} -\frac12J^{ab}\mathcal B_{ab}.$$

For a nonrotating, isotropic, parity-invariant body,

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

At positive frequency, the time-averaged mechanical power deposited in a passive body is

$$\left\langle P_{\rm body}\right\rangle =\frac{\omega}{4} \left[ \operatorname{Im}\lambda_E^{\rm ret} \mathcal E_{ab}^*\mathcal E^{ab} +\operatorname{Im}\lambda_B^{\rm ret} \mathcal B_{ab}^*\mathcal B^{ab} \right]\ge0.$$

The real part of $\lambda$ changes conservative dynamics. Its imaginary part determines dissipation. A static Love number is therefore insufficient to calculate heating, and a vanishing static Love number does not imply vanishing absorption.

For a rotating body, decompose into azimuthal harmonics $e^{-i\omega t+im\phi}$ and replace the inertial frequency in the local constitutive law by the co-rotating frequency

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

Passivity constrains heat or entropy production with $\widehat\omega$, not necessarily the sign of energy measured at infinity. This distinction is what permits black-hole superradiance and analogous rotational energy extraction.

Horizon absorption and superradiance

For a perturbed Kerr horizon, the first law gives

$$\dot M_H-\Omega_H\dot J_H =\frac{\kappa}{8\pi G}\dot A_H\ge0,$$

where $\Omega_H$, $\kappa$, and $A_H$ are the horizon angular velocity, surface gravity, and area. For a monochromatic $m\ne0$ mode, energy and angular-momentum fluxes satisfy $\dot M_H=(\omega/m)\dot J_H$. Hence the area can increase while $\dot M_H\lt 0$. In the superradiant range

$$0\lt \omega\lt m\Omega_H,$$

the outgoing wave and orbit can gain energy at the expense of the hole’s rotational energy. Calling this “negative dissipation” would obscure the invariant statement: horizon entropy production remains nonnegative.

Poisson derives practical time-domain mass, angular-momentum, and area flux formulae for perturbed horizons in Poisson 2004, §§ IV C–V D, and their small-hole/slow-motion tidal limits in Poisson 2004, §§ VIII–IX. Those results require a perturbation of a stationary Kerr or Schwarzschild background and an averaging interval long compared with the hole’s scale. They are not a nonlinear merger-horizon prescription.

The low-frequency Kerr response makes the static/dissipative distinction explicit. Its conservative Love part vanishes in the stated four-dimensional Kerr problem, while its imaginary response contains a factor proportional to the horizon-frame frequency and changes sign at superradiance. Chia obtains this response for arbitrary Kerr spin and multipole order within the weak, adiabatic perturbation problem in Chia 2021, §§ V–VII, especially Eqs. (15)–(16).

In standard quasicircular post-Newtonian counting, horizon absorption by nonspinning black holes first corrects the leading flux at relative 4PN order, while spin-dependent absorption or superradiance enters at relative 2.5PN order. These order labels apply to the low-frequency binary expansion; they do not rank horizon absorption against a resonant material mode in every system. The matching and power counting are given in Porto 2016, § 7.8, pp. 66–68, and § 8.4, pp. 84–86.

Material damping and thermal feedback

For a damped material mode,

$$\ddot Q_{ab}+2\gamma_\alpha\dot Q_{ab} +\omega_\alpha^2Q_{ab} =-\lambda_\alpha\omega_\alpha^2\mathcal E_{ab},$$

the retarded response is

$$\lambda_\alpha^{\rm ret}(\omega) =\frac{\lambda_\alpha\omega_\alpha^2} {\omega_\alpha^2-\omega^2-2i\gamma_\alpha\omega}.$$

The damping rate $\gamma_\alpha$ may contain several contributions,

$$\gamma_\alpha =\gamma_\alpha^{\rm shear} +\gamma_\alpha^{\rm bulk} +\gamma_\alpha^{\rm diffusion} +\gamma_\alpha^{\rm GW} +\cdots .$$

These rates depend on temperature, composition, superfluid gaps, magnetic field, mode eigenfunction, and frequency. They cannot be inferred from the real static deformability. Near resonance, even a small $\gamma_\alpha$ can produce appreciable heating because the mode amplitude is enhanced; far from resonance, the same mode may be negligible.

The mechanical mode energy obeys

$$\dot E_\alpha =-\frac12\mathcal E_{ab}\dot Q^{ab} -\frac{\gamma_\alpha} {\lambda_\alpha\omega_\alpha^2} \dot Q_{ab}\dot Q^{ab}.$$

The final term is lost from the coherent mode, but its destination depends on the damping channel. Viscous damping produces heat; the $\gamma_\alpha^{\rm GW}$ contribution produces outgoing radiation. Only the thermal part sources a heating equation. If the temperature change modifies $\gamma_\alpha$, the coupled response is nonlinear even when the instantaneous mode equation is linear. One must then evolve a thermal equation alongside the orbit and mode instead of inserting a fixed quality factor.

As a concrete, model-limited example, Ghosh et al. compute bulk-viscous damping of an $f$-mode from hyperonic weak reactions and propagate it into inspiral balance in Ghosh et al. 2025, § 2, Eqs. (4)–(11), and § 3.1. Their large effect applies to the assumed hyperonic equation of state, temperature evolution, equal-mass systems, and approximate stellar-mode treatment. It is evidence that material damping can matter in a stated regime, not a universal claim that bulk viscosity dominates every neutron star.

Mode damping also differs from direct absorption by a broad continuum. A narrow mode first stores coherent energy and can return part of it to the orbit; damping subsequently degrades that energy. A broad, nearly local spectral response may deposit energy without a separately resolved oscillation. Both appear in $\operatorname{Im}\lambda^{\rm ret}$, but their time-domain dynamics and initial-data dependence differ.

Worldline operators for dissipation

A purely local real worldline action such as

$$S_{\rm cons}\supset \int d\tau\, \left(C_E\mathcal E_{ab}\mathcal E^{ab} +C_B\mathcal B_{ab}\mathcal B^{ab}\right)$$

describes conservative finite-size response. It cannot generate irreversible absorption. The effective description must retain internal operators,

$$S_{\rm int} =-\int d\tau\, \left(Q_E^{ab}\mathcal E_{ab} +Q_B^{ab}\mathcal B_{ab}\right),$$

whose retarded two-point functions supply the spectral densities. Here the factor of one half used in the induced-multipole coupling earlier on the page has been absorbed into $Q_E^{ab}$ and $Q_B^{ab}$, so these internal operators have expectation values equal to half the corresponding induced multipoles. If $\langle Q_E^{ab}\rangle=-\chi_E^{ab,cd}\mathcal E_{cd}$ in this normalization, then $\lambda_E^{\rm ret}=2\chi_E$ in the isotropic scalar sector, and similarly for the magnetic response. Those correlators are fixed by matching to a physical absorption cross section or a material calculation. Goldberger and Rothstein introduce this construction and match the low-frequency Schwarzschild spectral function in Goldberger and Rothstein 2006, § II, Eqs. (19)–(27).

There is a formalism boundary here. The imaginary part of an in–out effective action diagnoses loss and is useful for cross sections, but varying it does not by itself give causal real-time equations for expectation values. A dissipative orbital force is most cleanly obtained with an in–in, doubled-history construction or with explicit damped degrees of freedom. Retarded rather than time-ordered correlators then enter the equations of motion. Mixing these prescriptions can produce advanced terms or an incorrect sign.

Matching also prevents double counting. If a black-hole absorption coefficient has been fixed from the horizon cross section, one must not add a second phenomenological “tidal heating” flux representing the same horizon modes. If a stellar oscillator already contains $\gamma_\alpha^{\rm GW}$, its radiative loss must be reconciled with the system multipole flux rather than counted twice.

Comparing channels in a binary

At a fixed orbital state, define

$$\mathcal R_X(\Omega) =\frac{|\mathcal F_X(\Omega)|} {\mathcal F_\infty(\Omega)}$$

for $X=H$, shear, bulk, diffusion, or mode damping. A defensible dominance statement specifies masses, spins and orientations, eccentricity, temperature, equation of state, forcing harmonic, and frequency interval, then compares these ratios using the same energy-at-infinity convention.

Several qualitative regimes follow:

• For slowly moving, nonspinning black-hole binaries, horizon absorption is parametrically high order and usually subleading during early inspiral.
• For rapidly rotating holes, the lower-order spin channel can be larger and can reverse the sign of the horizon energy flux in the superradiant band.
• For material bodies far from resonance, viscosity may be extremely weak and conservative static tides dominate finite-size phasing.
• Near a coupled mode resonance, mode storage and damping can dominate locally even if their broadband contribution is small.
• Near merger, temperature and nonlinear hydrodynamics can invalidate a fixed linear response before any formal PN hierarchy settles the comparison.

The balance equation should therefore be written

$$\frac{d}{dt} \left(E_{\rm orb}+E_{\rm rot}+\sum_\alpha E_\alpha\right) =-\mathcal F_\infty -\sum_H\mathcal F_H -\sum_{\rm heat}\mathcal F_{\rm heat},$$

Here $E_{\rm orb}$ includes the orbit–mode interaction, $E_{\rm rot}$ is the explicitly evolved material rotational energy, and the horizon masses and material thermal energy are excluded from the left. All fluxes must be converted to the same asymptotic energy and time used by that left-hand side; local heat is nonnegative in the material frame, while horizon Killing-energy flux is signed. If the horizon masses or thermal energies are included among the resolved energies, their corresponding fluxes move inside the total balance. Likewise, holding material spin fixed requires accounting for its absorbed rotational work as an unresolved transfer. This specification prevents counting the same energy twice.

Exercises

1. Positivity of oscillator absorption. Compute $\operatorname{Im}\lambda_\alpha^{\rm ret}$ for $\omega\gt 0$ and $\gamma_\alpha\gt 0$.

Solution

Multiplying numerator and denominator by the complex conjugate gives

$$\operatorname{Im}\lambda_\alpha^{\rm ret} =\frac{ 2\lambda_\alpha\omega_\alpha^2\gamma_\alpha\omega }{ (\omega_\alpha^2-\omega^2)^2 +4\gamma_\alpha^2\omega^2 }.$$

For a stable mode with $\lambda_\alpha\gt 0$, this is nonnegative at $\omega\gt 0$. Substitution into the power formula gives nonnegative loss of driving energy into the damped response. It is heating only to the extent that the damping is thermal. The peak width is controlled by $\gamma_\alpha$, while its position approaches $\omega_\alpha$ for weak damping.

2. Superradiant sign. For $m\gt 0$ and $0\lt \omega\lt m\Omega_H$, show that $\dot A_H\ge0$ is compatible with $\dot M_H\lt 0$.

Solution

Use $\dot J_H=(m/\omega)\dot M_H$ in the first law:

$$\frac{\kappa}{8\pi G}\dot A_H =\dot M_H \left(1-\frac{m\Omega_H}{\omega}\right).$$

The parenthesis is negative in the superradiant range. Therefore the product is nonnegative when $\dot M_H\le0$. The hole loses mass and angular momentum while its area still grows.

References

• Chia, Horng Sheng. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104, 024013 (2021). DOI; Open PDF.
• Ghosh, Suprovo, Samanwaya Mukherjee, Sukanta Bose, and Debarati Chatterjee. “Tidal Dissipation in Binary Neutron Star Inspirals from Hyperon Bulk Viscosity: Phase Modelling and Parameter Estimation Bias.” Monthly Notices of the Royal Astronomical Society 543, 2987–2996 (2025). DOI; Open article.
• 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.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633, 1–104 (2016). DOI; Open PDF.