Horizon Absorption and Tidal Heating

A time-dependent tidal field drives shear on a black-hole horizon. The shear increases the horizon area and, mode by mode, transfers energy and angular momentum according to the horizon generator. For Schwarzschild this is ordinary absorption and heating. For Kerr, the flux of energy measured at infinity may be negative in a superradiant sector even though the generator-energy flux and area production remain nonnegative. That sign distinction is the clean way to discuss dissipation without contradicting the vanishing conservative static Love numbers.

Required background. Presymplectic Potentials and Currents supplies the oriented boundary flux and balance-law framework.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon generator, surface gravity, and area law; PDEs and Green Functions for Gravitation supplies ingoing radial conditions and conserved Wronskians; Formulations of General Relativity supplies regular horizon gauges; Relativistic Two-Body Dynamics supplies slowly varying tidal tensors; and Black-Hole Perturbation Theory supplies the shared mode and flux normalizations.

Horizon generator and fluxes

For Kerr, let

$$\chi^a=t^a+\Omega_H\phi^a, \qquad k_H=\omega-m\Omega_H,$$

and use modes $e^{-i\omega t+im\phi}$. We orient $\dot M_H$ and $\dot J_H$ as fluxes into the future horizon. The gravitational fluxes are leading quadratic quantities in the amplitude of the linear perturbation. For a slowly evolving hole, use rates averaged over a time long compared with $\kappa^{-1}$ but short compared with the secular area-evolution time, or total changes between initially and finally settled stationary states. Under these qualifications, the first-law combination is

$$\langle\dot M_H\rangle-\Omega_H\langle\dot J_H\rangle =\frac{\kappa}{8\pi G}\langle\dot A_H\rangle\geq0.$$

Subsequent modal rates have this averaged meaning. This is the flux of the horizon-generator charge. A pure real-frequency mode also satisfies $\dot M_H=(\omega/m)\dot J_H$ when $m\neq0$. Combining the two relations shows that the signs of the asymptotic-time energy and the area need not coincide. Poisson derives the horizon shear, area balance, and modal energy–angular-momentum relations in Poisson 2004, §§ IV.B–IV.C, Eqs. (4.9)–(4.24), pp. 11–12 (PDF).

The declaration here is dynamic and dissipative. We work in four-dimensional vacuum general relativity with spin-2 perturbations of Schwarzschild or Kerr. The applied field is the normalized electric/magnetic STF tidal tensor; the induced dissipative moment is normalized through the retarded relation $Q_L=-\lambda^E_\ell\mathcal E_L$ and its magnetic analog. A horizon-regular ingoing Kerr or Regge–Wheeler–Zerilli variable fixes the flux; buffer matching fixes the applied amplitude. Local conservative contact subtractions do not alter the horizon flux. The low-frequency limit is taken at fixed subextremal $(M,a,\ell,m)$ after imposing the ingoing condition; it is not interchanged with extremality.

From tidal shear to Schwarzschild heating

For a nonrotating hole, $\Omega_H=0$, so the area law makes $\langle\dot M_H\rangle\geq0$ under the averaging conditions above. In the small-hole/slow-motion regime, the external curvature scale $\mathcal R$ is much larger than $GM$ and the electric and magnetic quadrupolar tidal tensors vary slowly along the hole’s worldline. Their components are measured in a parallel-transported local rest-frame triad. Dots denote differentiation along that worldline, identified with the slow advanced-time driving in the local black-hole expansion. Assume weak tides, $GM|\omega|\ll1$ for their driving frequencies, and retain the leading quadratic tidal order and leading slow-driving order. Restoring $G$ in Poisson’s result gives

$$G\left\langle\dot M_H\right\rangle =\frac{16}{45}(GM)^6 \left\langle \dot{\mathcal E}_{ab}\dot{\mathcal E}^{ab} +\dot{\mathcal B}_{ab}\dot{\mathcal B}^{ab} \right\rangle .$$

The right side is nonnegative and dimensionless in units $c=1$: $\mathcal E_{ab}$ and $\mathcal B_{ab}$ have dimension length$^{-2}$, so $(GM)^6\dot{\mathcal E}^{2}$ is dimensionless. The derivatives explain why a strictly static Schwarzschild tide produces no leading heating even though a time-dependent tide does. The shear calculation and its invariant form are given in Poisson 2004, §§ VIII.E–VIII.G, Eqs. (8.35)–(8.45), pp. 24–25 (PDF).

One normalized electric shear calculation

Let $r_+=2GM$, and expand $\mathcal E_{ij}n^in^j=\sum_m\mathcal E_mY_{2m}$ using unit-$L^2$ scalar harmonics. The physical even master amplitude on the horizon is

$$\psi_{\mathrm e,H}^{2m}=-\frac{r_+^3}{6}\mathcal E_m.$$

This is Poisson 2004, Eq. (8.33), p. 24 (PDF); both coefficients transform the same way when the scalar harmonics are normalized. The metric-to-shear relation then gives

$$\begin{aligned} p_{AB}^{\mathcal H,\mathrm e} &=r_+\sum_m\psi_{\mathrm e,H}^{2m}Y_{AB}^{2m},\\ \sigma_{AB}^{(1),\mathrm e} &=\frac12\partial_vp_{AB}^{\mathcal H,\mathrm e} =-\frac{r_+^4}{12}\sum_m\dot{\mathcal E}_mY_{AB}^{2m}. \end{aligned}$$

Use $q_{AB}^{(0)}=r_+^2\Omega_{AB}$ to raise horizon indices and $dA_0=r_+^2d\Omega$. The unit-sphere tensor norm is $\int Y_{AB}^{2m}\overline Y^{2m'AB}d\Omega=12\delta_{mm'}$. Consequently the positive quadratic shear flux is

$$\mathcal F_H^{\mathrm e} =\frac1{8\pi G}\int \sigma_{AB}^{(1),\mathrm e}\sigma^{(1),\mathrm e\,AB}dA_0 =\frac{3}{8\pi G}\sum_m |\partial_v\psi_{\mathrm e,H}^{2m}|^2.$$

The real tidal tensor requires the complete conjugate mode pairs; do not add another factor of two after summing all $m$. The STF sphere identity

$$\int n_i n_j n_k n_l\,d\Omega =\frac{4\pi}{15} (\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$$

gives $\sum_m|\dot{\mathcal E}_m|^2=(8\pi/15) \dot{\mathcal E}_{ij}\dot{\mathcal E}^{ij}$. Thus

$$\mathcal F_H^{\mathrm e} =\frac{3}{8\pi G}\frac{r_+^6}{36}\frac{8\pi}{15} \dot{\mathcal E}_{ij}\dot{\mathcal E}^{ij} =\frac{16(GM)^6}{45G} \dot{\mathcal E}_{ij}\dot{\mathcal E}^{ij}.$$

This explicitly recovers the electric coefficient from the normalized master field, physical area measure and shear. The magnetic contribution in the full formula above follows Poisson’s odd calculation; the electric derivation is not a license to ignore magnetic harmonic or current-moment normalization.

Finite cuts and a mass interpretation

The instantaneous $\mathcal F_H$ is the positive physical shear flux, not by itself an instantaneous horizon mass derivative on arbitrary cuts. To display the distinction, write $\sigma=\epsilon\sigma^{(1)}+\cdots$ and $\theta=\epsilon^2\theta^{(2)}+\cdots$. With $v$ normalized to asymptotic time and the final stationary event-horizon condition, the vacuum equation gives

$$\begin{aligned} (\partial_v-\kappa)\theta^{(2)} &=-\sigma_{AB}^{(1)}\sigma^{(1)AB},\\ \kappa\Delta A^{(2)} &=8\pi G\int_{v_0}^{v_1}\mathcal F_H\,dv +\left[\int\theta^{(2)}dA_0\right]_{v_0}^{v_1}. \end{aligned}$$

The second line integrates over the horizon portion between cuts $v_0,v_1$ and over the background spherical sections, not over a spacelike exterior slice. Only with vanishing endpoint expansion, or the specified coarse-grained slow evolution, can the expansion term be dropped for the physical-process mass/area interpretation. This is the same finite-cut identity established on the metric-reconstruction page.

Matching shear to retarded response

For the physical STF moments the work is

$$\langle P\rangle =\frac12\langle Q^{ij}\dot{\mathcal E}_{ij}\rangle +\frac23\langle J^{ij}\dot{\mathcal B}_{ij}\rangle.$$

The leading induced moments $Q_{ij}=32(GM)^6\dot{\mathcal E}_{ij}/(45G)$ and $J_{ij}=8(GM)^6\dot{\mathcal B}_{ij}/(15G)$ reproduce the shear heating above. They are Poisson 2004, Eqs. (8.43)–(8.45), p. 25 (PDF).

For a monochromatic quadrupolar tide, $\mathcal E_{ab}(t)=\operatorname{Re}(\mathcal E^0_{ab}e^{-i\omega t})$, time averaging gives

$$\left\langle \dot{\mathcal E}_{ab}\dot{\mathcal E}^{ab} \right\rangle =\frac{\omega^2}{2} \mathcal E^0_{ab}\mathcal E^{0*ab},$$

and likewise for $\mathcal B_{ab}$. Since $Q=-\lambda_E\mathcal E$ and $J=-\lambda_B\mathcal B$, this fixes

$$\begin{aligned} \operatorname{Im}\lambda_E&=\frac{32(GM)^6}{45G}\omega,& \langle P_E\rangle&=\frac{\omega}{4}\operatorname{Im}\lambda_E \mathcal E^0_{ij}\overline{\mathcal E^{0ij}},\\ \operatorname{Im}\lambda_B&=\frac{8(GM)^6}{15G}\omega,& \langle P_B\rangle&=\frac{\omega}{3}\operatorname{Im}\lambda_B \mathcal B^0_{ij}\overline{\mathcal B^{0ij}}. \end{aligned}$$

Thus the Schwarzschild heating rate vanishes quadratically as $\omega\to0$ at fixed applied amplitude. This is an independent low-frequency check on the imaginary response: its leading odd power must combine with the explicit factor of $\omega$ in the power formula to reproduce the same $\omega^2$ scaling. It also exposes why setting $\omega=0$ before distinguishing source normalization from flux erases the dissipative information.

A plane-wave absorption check

Take a weak monochromatic plane gravitational wave in the local rest frame. If its complex strain amplitudes are $h_+^0,h_\times^0$, set $H^2=|h_+^0|^2+|h_\times^0|^2$. Its incident intensity and tidal contractions are

$$I_{\mathrm{inc}}=\frac{\omega^2H^2}{32\pi G},\qquad \langle\dot{\mathcal E}:\dot{\mathcal E}\rangle =\langle\dot{\mathcal B}:\dot{\mathcal B}\rangle =\frac{\omega^6H^2}{4}.$$

Here the electric wave curvature is $-\ddot h_{ij}^{\mathrm{TT}}/2$; the two polarization tensors each have norm two and the real-time average supplies another factor $1/2$. Substitution into the heating formula gives

$$\langle\dot M_H\rangle =\frac{8(GM)^6\omega^6H^2}{45G} =I_{\mathrm{inc}}\, \frac{4\pi(2GM)^6\omega^4}{45}.$$

The final factor is the leading gravitational absorption cross section, not the scalar area-limit cross section. It agrees with Goldberger and Rothstein 2006, Eqs. (23)–(27), p. 12 (PDF) after their unit-coupled operators are translated as $Q^E=Q/2$, $Q^B=2J/3$. The EFT page performs that translation explicitly. This comparison uses the sourced full-theory cross section; it is not a new numerical scattering solution.

Kerr: heating, torque, and superradiant extraction

For positive asymptotic frequency, $k_H>0$ describes ordinary absorption. If $0<\omega<m\Omega_H$, then $k_H<0$. The mode carries negative asymptotic-time energy into the horizon, so $\dot M_H<0$ and the outgoing wave is amplified. The area still grows because $\dot M_H-\Omega_H\dot J_H\geq0$; the hole loses sufficient angular momentum that the generator-energy balance remains positive. Calling every negative $\dot M_H$ “negative absorption” without this orientation would conceal the thermodynamic check.

Rotation also changes low-frequency power counting. A field stationary with respect to $t^a$ can have $k_H=-m\Omega_H$ and drive dissipation. This is why a Kerr response may possess an imaginary piece in a nominal $\omega=0$ limit while its conservative Love coefficient vanishes. Chia separates the real tidal deformation from this absorptive response in Chia 2021, §§ III and VI, eqs. (5)–(7) and (14)–(18).

The result does not cover material viscosity, orbital backreaction, waveform phasing, or nonlinear horizon evolution. Those are separate problems. Nor is the Schwarzschild coefficient above a universal Kerr formula: the small-spin and low-frequency limits require their own matched expansion.

Exercises

Take $\Omega_H\gt0$, $m=2$, $\omega=1.5\Omega_H$, and define the positive area-production rate

$$\mathcal D =\dot M_H-\Omega_H\dot J_H>0.$$

Find the signs and values of the two fluxes in terms of $\mathcal D$.

Solution

The horizon frequency is $k_H=1.5\Omega_H-2\Omega_H=-0.5\Omega_H$, so the mode is superradiant. The modal relation gives $\dot M_H=(\omega/m)\dot J_H=0.75\Omega_H\dot J_H$. Hence

$$\mathcal D =(0.75\Omega_H-\Omega_H)\dot J_H =-0.25\Omega_H\dot J_H,$$

and therefore

$$\dot J_H=-\frac{4\mathcal D}{\Omega_H}, \qquad \dot M_H=-3\mathcal D.$$

Both asymptotic-time charges leave the hole, but $\dot M_H-\Omega_H\dot J_H=\mathcal D>0$ exactly. Thus the result satisfies the future-horizon area law and the declared inward-flux orientation.

The causal object that packages this flux together with dispersive response is developed under frequency-dependent response and dissipation.

References

• Chia, Horng Sheng. 2021. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104, 024013. DOI.
• Goldberger, Walter D., and Ira Z. Rothstein. 2006. “Dissipative Effects in the Worldline Approach to Black Hole Dynamics.” Physical Review D 73, 104030. 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.