Horizon Absorption at Finite Amplitude

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At finite amplitude, absorption by a black hole is best measured as a quasi-local change of horizon geometry, not as a perturbative correction assigned to a coordinate worldline. On a dynamical horizon, exact balance laws relate area growth, matter flux, shear, and angular-momentum transfer. Their perturbative tidal-heating limit is a check; it is not the definition used during a close nonlinear encounter.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon mechanics; Black-Hole Perturbation Theory supplies weak-field absorption; Formulations of General Relativity supplies hypersurface geometry; Relativistic Two-Body Dynamics supplies encounter kinematics; Computational Relativity supplies horizon diagnostics.

Exact area balance

Let $\mathcal H$ be a smooth spacelike dynamical horizon with spherical leaves $S$ satisfying $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$ , in four-dimensional general relativity with $\Lambda=0$ . Let $\widehat\tau^a$ be its future unit timelike normal and $\widehat r^a$ its unit tangent normal to the leaves, pointing toward increasing area. Fix the outgoing normalization by $\ell^a=\widehat\tau^a+\widehat r^a$ ; choose $n^a=(\widehat\tau^a-\widehat r^a)/2$ , so $\ell\mathbin{\cdot}n=-1$ . The inner product alone would leave a boost freedom and would not fix the coefficients in the flux law. Let

A=4\pi R^2

define the area radius. With the lapse $N_R=|D R|$ on $\mathcal H$ , the balance between leaves $S_1$ and $S_2$ has the form

\frac{R_2-R_1}{2G} =\int_{\Delta\mathcal H} T_{ab}\,\widehat\tau^a\xi_{(R)}^b\,d^3V +\frac{1}{16\pi G} \int_{\Delta\mathcal H} N_R\left( |\sigma^{(\ell)}|^2+2|\zeta|^2 \right)d^3V.

Here $D$ is the derivative intrinsic to $\mathcal H$ , $\xi_{(R)}^a=N_R\ell^a$ , and the shear is the trace-free part of the projected $\nabla_a\ell_b$ . With leaf projector $\widetilde q_a{}^b$ , the second term uses $\zeta_a=\widetilde q_a{}^b\widehat r^c\nabla_c\ell_b$ . The gravitational integrand is nonnegative. For matter obeying the dominant energy condition, the matter term is also nonnegative, so $R_2\ge R_1$ . The assumptions on topology and $\Lambda$ matter: they are what reduce the integrated intrinsic-curvature term to the displayed $R/(2G)$ .

The precise derivation, normalization, and vacuum gravitational flux are given in Ashtekar and Krishnan 2004, §3.2, Eqs. (13)–(25), and §3.3, Eq. (26), pp. 16–24. Rescaling $\ell$ and $n$ changes intermediate quantities; the balance law is used with its declared foliation and normalization.

Spin and energy exchange

For an axial vector field $\varphi^a$ tangent to a leaf, define quasi-local angular momentum using the horizon connection or, in a spatial slice, its extrinsic curvature. If the slice convention is

K_{ab}=-h_a{}^c h_b{}^d\nabla_c\widehat\tau_d,

then one standard expression is

J_S[\varphi] =\frac{1}{8\pi G}\oint_S K_{ab}\varphi^a\widehat r^b\,d^2V,

with $\widehat r^a$ the outward unit normal to $S$ within the slice. Reversing the sign convention for $K_{ab}$ reverses the displayed prefactor. Approximate axial symmetry requires stating how $\varphi^a$ is selected.

The change in $J_S$ equals matter and gravitational angular-momentum flux through the horizon. Combining $A$ and $J$ with a Kerr-inspired quasi-local mass,

M_H^2 =\frac{A}{16\pi G^2} +\frac{4\pi J_H^2}{A},

provides a nonlinear energy diagnostic for an uncharged near-axisymmetric horizon. The angular-momentum and energy balance laws are developed in Ashtekar and Krishnan 2004, §4.2.1, Eqs. (42)–(46), and §4.2.2, Eqs. (47)–(50), pp. 31–36.

From tidal heating to strong absorption

In the slow, weak-field limit, horizon shear is driven linearly by the external Weyl curvature, so the area flux is quadratic in the tidal field. This reproduces perturbative tidal heating and torque. During a close encounter, however, the response can be nonadiabatic and the horizon can change slicing, develop large multipoles, or join a common horizon. The exact balance law remains meaningful on a smooth spacelike dynamical-horizon segment even when a low-order absorption expansion fails.

For a numerical measurement:

locate and track all relevant marginal surfaces rather than only the outermost one;
compute area and angular momentum with one orientation and extrinsic-curvature convention;
integrate matter, shear, and connection flux over the same horizon tube;
compare $\Delta A$ , $\Delta J$ , and $\Delta M_H$ with asymptotic losses;
report errors from horizon finding, approximate symmetry, slicing, and temporal integration.

A jump between different outermost marginal surfaces is not a burst of physical flux through one smooth tube. Branch identification must precede budget interpretation.

Independent checks and limitations

A nontrivial spacelike dynamical-horizon segment cannot have identically vanishing vacuum gravitational flux: the displayed law would give constant area, contradicting its increasing area radius. An isolated horizon is a null zero-flux limit and requires the corresponding null-horizon formulation. Every term has dimensions of length divided by $G$ in $c=1$ units. At weak amplitude, the flux is second order in the first-order shear. The balance law does not turn an apparent horizon into the event horizon, and it does not define a unique quasi-local spin without a specified axial field.

Check the area law

On a vacuum dynamical-horizon segment, suppose $\zeta=0$ and the integral of $N_R|\sigma^{(\ell)}|^2$ is $8\pi L$ in length units. What is $R_2-R_1$ ?

Solution

The balance law gives

\frac{R_2-R_1}{2G} =\frac{8\pi L}{16\pi G} =\frac{L}{2G},

so $R_2-R_1=L$ . The positive result independently checks the shear-flux sign.

References

Ashtekar, Abhay, and Badri Krishnan, “Isolated and Dynamical Horizons and Their Applications”, Living Reviews in Relativity 7, 10, 2004. Open PDF.