Quasi-Local and Event-Horizon Budgets

Quasi-local and event horizons support different kinds of budgets. A dynamical horizon gives an exact finite relation between charge changes and flux through a spacelike tube. An event horizon gives the invariant global black-hole boundary and an area theorem, but its teleological growth cannot generally be assigned to contemporaneous local influx. A complete collapse budget uses both without forcing one framework’s terms into the other.

Required background. Hypersurface normals and tangential projectors supplies the causal and surface geometry of both boundaries.

Helpful background. Black-Hole Geometry and Mechanics develops event, isolated, and dynamical horizons; Black-Hole Perturbation Theory supplies near-equilibrium fluxes; Formulations of General Relativity explains constraints and asymptotic charges; and Computational Relativity develops horizon tracking and waveform extraction.

Horizon and asymptotic balance laws

For a smooth segment of axisymmetric dynamical horizon between leaves $S_1,S_2$, a canonical quasi-local balance has the form

$$M_H(S_2)-M_H(S_1) =F_H^{\rm matter}+F_H^{\rm grav},$$

where $M_H$ is determined by the leaf’s area and spin, and the fluxes are integrals of stress-energy, shear, normal connection, and—when rotation is included—the appropriate angular-momentum term. Every quantity refers to the same spacelike tube and evolution vector. The identity requires a specified branch, foliation, symmetry prescription, and Einstein dynamics; energy conditions enter positivity claims.

The future event horizon

$$\mathcal H^+=\partial J^-(\mathscr I^+)$$

is a null boundary determined by the complete future. Under global hypotheses and the null energy condition, its cross-sectional area is nondecreasing toward the future. Yet an event horizon can expand in a region where $T_{ab}\ell^a\ell^b$ and local shear vanish, because later infall changes which null generators escape. Ashtekar and Krishnan explain this obstruction to a local event-horizon flux account in Ashtekar and Krishnan 2004, §1, pp. 1–9, PDF.

Near equilibrium, the physical-process first law relates a small perturbative event-horizon area change to absorbed energy and angular momentum. That linearized result is not the exact finite dynamical-horizon balance and should not be applied to a violent merger without its quasistationary assumptions.

A global mass and angular-momentum closure

Choose the initial center-of-momentum frame and a consistent asymptotic rotation frame. For asymptotically flat collapse with a settled final black hole, a schematic late-time closure is

$$\begin{aligned} M_{\rm ADM} ={}&E_{\rm BH}^{\rm final} +E_{\mathscr I^+}^{\rm rad} +E_{\rm exterior} +\delta M,\\ J_{\rm ADM} ={}&J_{\rm BH}^{\rm frame} +J_{\mathscr I^+}^{\rm rad} +J_{\rm exterior} +\delta J. \end{aligned}$$

The horizon mass $M_H$ is the remnant’s rest mass. If it recoils with momentum $\boldsymbol P_f$, its energy in the chosen frame is

$$E_{\rm BH}^{\rm final}=\sqrt{(M_H^{\rm final})^2+\lvert\boldsymbol P_f\rvert^2}.$$

The corresponding angular momentum $J_{\rm BH}^{\rm frame}$ includes the remnant’s motion about the selected origin; it equals intrinsic horizon spin only after the appropriate center and rest-frame alignment. The exterior terms include remaining matter and field energy with a consistent binding-energy prescription. A local matter-density integral alone is not a general decomposition of gravitational energy. In the clean late vacuum limit the exterior remainder vanishes.

The residuals collect extraction, boundary, and numerical error; they are not physical flux categories. Massive matter remaining outside is not determined by the flux at null infinity. Horizon absorption is already included in the final remnant energy and must not be added again. If accretion continues, “final” is premature.

At future null infinity, the Bondi mass decreases by outgoing matter and gravitational radiation. The dynamical-horizon balance records net energy exchange with the hole. Comparing the two provides a powerful closure, but equality of their instantaneous rates is not expected: they use different time parameters, propagation delays, and regions of the spacetime.

Retrospective event-horizon comparison

An event horizon is normally found by evolving a late null surface backward through stored spacetime data. Vary the late seed time and shape, generator density, time interpolation, and treatment of caustics. Thornburg describes this method and its convergence requirements in Thornburg 2007, §2, pp. 11–24, PDF.

Compare event and quasi-local horizons using invariant cross-sectional area, intrinsic multipoles, and causal nesting, not coordinate radius. During positive accretion the event horizon generally lies outside the future outer marginal tube; they approach one another as flux decays. A difference between their areas at equal coordinate time is not missing energy.

The event horizon is indispensable for the global statement that signals cannot reach $\mathscr I^+$. The dynamical horizon is better suited to asking how much matter and gravitational energy entered a black hole between two quasi-local states. Neither replaces the asymptotic radiation budget.

Common budget errors

Do not integrate flux across an apparent-horizon jump as if the old and new outermost surfaces formed one smooth tube. Do not identify coordinate-time synchronization between horizon and extraction sphere with retarded-time synchronization. Do not subtract irreducible mass from ADM mass and call the remainder radiation when spin and exterior matter are present. And do not interpret an event-horizon area increase before infall as negative or acausal energy: it is a consequence of the boundary’s global definition.

The isolated limit supplies the joint check. When all fluxes vanish and the exterior settles, the outer dynamical horizon becomes null, its charges become constant, and its late cross-sections coincide with the stationary event horizon. The global closure should then be time independent within errors.

Check your understanding

A simulation in the initial rest frame has $M_{\rm ADM}=1$, final remnant energy $0.94$ (including recoil), radiated energy $0.04$, and a consistently evaluated exterior remainder $0.015$. What is the mass residual, and may it be assigned immediately to horizon absorption?

Solution

The closure gives

$$\delta M=1-0.94-0.04-0.015=0.005.$$

It is a $0.5\%$ residual. It cannot be assigned to horizon absorption without checking horizon flux, finite-radius extrapolation, matter still on the grid, outer-boundary loss, and numerical convergence. Horizon absorption is already represented in the measured final horizon mass and its quasi-local flux history.

References

• Ashtekar, Abhay, and Badri Krishnan. “Dynamical Horizons and Their Properties.” Physical Review D 68 (2003): 104030. doi:10.1103/PhysRevD.68.104030.
• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. doi:10.12942/lrr-2004-10.
• Thornburg, Jonathan. “Event and Apparent Horizon Finders for 3+1 Numerical Relativity.” Living Reviews in Relativity 10 (2007): 3. doi:10.12942/lrr-2007-3.