Dynamical Horizons and Flux Laws

Dynamical horizons make black-hole growth quantitatively local. Their marginal leaves carry area, spin, mass, and multipoles, while the Einstein constraints turn changes in those quantities into finite matter and gravitational fluxes across the horizon. This chapter derives those laws, explains their symmetry and branch assumptions, and shows how they complement rather than duplicate event-horizon and null-infinity budgets.

Helpful background. Black-Hole Geometry and Mechanics supplies quasi-local and global horizon definitions, and Hypersurface Normals and Tangential Projectors supplies the leaf, tube, and normal geometry entering the balances.

Enter this chapter

The default regime is four-dimensional Einstein gravity with a smooth marginally trapped tube, declared future/outward null orientation, and specified matter energy condition. The simplest positive balance laws assume a spacelike future outer branch with spherical leaf topology and zero cosmological constant. Rotation additionally requires an axial vector or a documented approximate-symmetry prescription. The preceding chapter, Trapped Surfaces and Horizon Formation, supplies the expansion, stability, branch, and surface-finding tests needed before these balances can be used.

Read the pages in this order:

1. Marginal Tubes and Their Signature derives spacelike, null, and timelike character from preserved marginality.
2. Area Balance on Dynamical Horizons obtains the exact matter-plus-geometry flux law.
3. Angular-Momentum and Energy Flux Laws adds quasi-local spin and canonical Kerr-matched mass.
4. Horizon Multipoles in Evolution tracks invariant distortion and rotation toward equilibrium.
5. Transitions, Jumps, and Multiple Horizons separates smooth tubes from outermost-branch changes.
6. Quasi-Local and Event-Horizon Budgets closes horizon, exterior, and radiation accounting.

Signature before flux

For a marginal tube with $\theta_{(\ell)}=0$, $\theta_{(k)}\lt 0$, choose future null normals normalized by $\ell\cdot k=-1$ and set

$$\mathcal V^a=\ell^a-Ck^a, \qquad \mathcal V^2=2C.$$

Preserving marginality gives an elliptic equation for $C$, including angular derivatives on a distorted leaf. Under the null energy condition and strict outer stability, matter flux or outgoing shear anywhere on a connected leaf makes $C\gt0$ everywhere: the tube is spacelike and its area grows. When both vanish over the whole leaf, the tube is null. Only in a leafwise-constant reduction, such as spherical symmetry, does the equation become a quotient of two expansion variations. If the outer condition, ingoing-expansion sign, or energy condition fails, timelike and mixed-signature branches are possible.

This hypothesis check comes before applying a positive balance. Inner merger horizons can violate the future or outer condition and change signature, so their areas need not obey the simple monotone reading. Ashtekar and Krishnan set out the definitions and signature relations in Ashtekar and Krishnan 2004, §§2.2–2.3, pp. 13–20, PDF.

Area, spin, and mass balances

For the finite area law, let $H$ be the spacelike horizon, $\hat\tau^a$ its future unit timelike normal, and $\hat r^a$ the outward unit spacelike normal to a leaf within $H$. The normalization used here is $\ell^a=\hat\tau^a+\hat r^a$ and $n^a=\hat\tau^a-\hat r^a$, so $\ell\cdot n=-2$. If $R=\sqrt{A/(4\pi)}$, then $N_R=\lvert D R\rvert_H$ and $\xi_{(R)}^a=N_R\ell^a$. With $\widetilde q_{ab}$ the leaf metric, the two geometric flux fields are the outgoing shear $\sigma_{ab}=\widetilde q_a{}^c\widetilde q_b{}^d\nabla_c\ell_d$ (its trace vanishes because $\theta_{(\ell)}=0$) and $\zeta^a=\widetilde q^{ab}\hat r^c\nabla_c\ell_b$.

On a spacelike dynamical horizon with areal radius $R$, the exact area-radius law is

$$\begin{aligned} \frac{R_2-R_1}{2G} ={}&\int_{\Delta H} T_{ab}\hat\tau^a\xi_{(R)}^b\,d^3V\\ &+\frac{1}{16\pi G}\int_{\Delta H}N_R \left(\lvert\sigma\rvert^2+2\lvert\zeta\rvert^2\right)d^3V. \end{aligned}$$

Spherical symmetry removes both geometric terms; the isolated limit removes every flux. Positivity follows from the dominant energy condition and the squared geometric fields.

Let $q^{(H)}_{ab}=g_{ab}+\hat\tau_a\hat\tau_b$ and adopt the horizon extrinsic-curvature sign

$$K_{ab}=q^{(H)}_a{}^c q^{(H)}_b{}^d\nabla_c\hat\tau_d.$$

With an axial field $\varphi^a$, the momentum constraint gives change in

$$J_S=-\frac{1}{8\pi G}\oint_S K_{ab}\varphi^a\hat r^b\,dA$$

as matter plus gravitational angular-momentum flux. Area and spin select a canonical energy

$$M_H(R,J)=\frac{\sqrt{R^4+4G^2J^2}}{2GR},$$

whose differential is $dM_H=(\kappa_H/8\pi G)dA+\Omega_HdJ$. These are exact quasi-local relations on the specified branch, not claims that a distorted leaf is already Kerr.

The full constraint derivations and finite first law appear in Ashtekar and Krishnan 2004, §§3–4.2, pp. 21–36, PDF.

Multipoles and approach to equilibrium

Area and spin do not characterize a dynamical leaf. Intrinsic scalar curvature and the curl of its rotational one-form contain higher distortion and rotation. Projecting them against geometrically selected weighting functions gives source multipoles that are independent of coordinates on the leaf; the equilibrium construction and its Kerr normalization are given by Ashtekar et al. 2004, §§ II–IV, Eqs. (2.7)–(4.5).

In exact axisymmetry the axial field supplies a canonical latitude. In a generic evolution, weighting fields must be transported consistently from leaf to leaf, and approximate-symmetry residuals become part of the uncertainty. Higher multipoles should approach the Kerr relations as the common outer horizon settles, but source multipoles at the horizon need not equal field multipoles at infinity. Ashtekar, Campiglia, and Shah develop the dynamical construction and multipole evolution in Ashtekar, Campiglia, and Shah 2013, §§II–IV, pp. 2–22, PDF.

Branches and complete budgets

A marginal equation can create an inner/outer pair when its stability operator develops a zero eigenvalue. In the head-on merger studied by Pook-Kolb and collaborators, the common pair is born with a zero principal eigenvalue and then separates into outer and inner branches Pook-Kolb et al. 2021, § II.B, pp. 4–5, and Figs. 1–2. Selecting only the outermost leaf makes the apparent horizon jump. Flux laws require a smooth tube segment; subtracting areas across a branch switch creates a bookkeeping discontinuity, not an instantaneous physical flux.

The quasi-local mass increase should be compared with ADM initial data, Bondi radiation, and exterior matter:

$$M_{\rm ADM} =E_{\rm BH}^{\rm late} +E_{\mathscr I^+}^{\rm rad} +E_{\rm exterior} +\delta M.$$

Here $E_{\rm BH}^{\rm late}=\sqrt{(M_H^{\rm late})^2+\lvert\boldsymbol P_f\rvert^2}$ includes recoil in the initial rest frame, and the exterior remainder requires consistent matter, field, and binding-energy accounting. See the budget page for the matching of frames and angular momentum. An event horizon supplies the invariant global boundary but can grow before local influx because its definition knows the future. Its area difference at equal coordinate time is not a missing quasi-local flux. Only in the isolated late limit do the outer dynamical horizon, stationary event horizon, and constant quasi-local charges meet cleanly.

Review the chapter

A vacuum marginal tube has $\theta_{(k)}\lt 0$, satisfies the outer condition, and has nonzero shear. What are its signature and area behavior? Which terms remain in its area law, and what additional condition is needed to define an unambiguous spin?

Solution

Raychaudhuri makes the source of the elliptic signature equation positive somewhere; strict outer stability and the maximum principle imply $C\gt0$ throughout the connected leaf, so the tube is spacelike. Since $\theta_{(k)}\lt 0$, its area increases outward. The matter term vanishes, but the positive shear contribution and possibly the $\zeta$ contribution remain. An unambiguous spin requires a normalized axial symmetry; without exact symmetry one must declare and test an approximate axial-vector prescription.

References

• 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.
• Ashtekar, Abhay, Jonathan Engle, Tomasz Pawlowski, and Chris Van Den Broeck. “Multipole Moments of Isolated Horizons.” Classical and Quantum Gravity 21 (2004): 2549–2570. doi:10.1088/0264-9381/21/11/003.
• Ashtekar, Abhay, Miguel Campiglia, and Samir Shah. “Dynamical Black Holes: Approach to the Final State.” Physical Review D 88 (2013): 064045. doi:10.1103/PhysRevD.88.064045.
• Pook-Kolb, Daniel, Ofek Birnholtz, José Luis Jaramillo, Badri Krishnan, and Erik Schnetter. “Horizons in a Binary Black Hole Merger I: Geometry and Area Increase.” Physical Review D 103 (2021): 026005. doi:10.1103/PhysRevD.103.026005.