Area Balance on Dynamical Horizons

The dynamical-horizon area law is an exact finite balance equation in the strong-field region. It equates the increase of areal radius to matter energy crossing the horizon plus a positive geometric flux built from outgoing shear and the normal connection. Unlike the event-horizon area theorem, it is quasi-local and attributes growth to flux through the portion of horizon under consideration.

Required background. Hypersurface normals and tangential projectors supplies the geometry of a two-surface inside a spacelike horizon.

Helpful background. Black-Hole Geometry and Mechanics develops dynamical horizons and Hawking mass; Black-Hole Perturbation Theory supplies shear flux; Formulations of General Relativity provides the constraint equations; and Computational Relativity develops quasi-local flux diagnostics.

Geometry of a spacelike dynamical horizon

Let $H$ be a smooth spacelike dynamical horizon foliated by closed surfaces $S$ of spherical topology. Let $\hat\tau^a$ be its future unit timelike normal and $\hat r^a$ the outward unit spacelike normal to $S$ tangent to $H$. Fix

$$\ell^a=\hat\tau^a+\hat r^a,\qquad n^a=\hat\tau^a-\hat r^a,$$

so $\ell\cdot n=-2$, $\theta_{(\ell)}=0$, and $\theta_{(n)}\lt 0$. Write each leaf’s area as $A=4\pi R^2$, define the lapse $N_R=\lvert D R\rvert$ intrinsic to $H$, and set $\xi_{(R)}^a=N_R\ell^a$.

The relevant geometric fields on a leaf are

$$\sigma_{ab} =\widetilde q_a{}^c\widetilde q_b{}^d\nabla_c\ell_d, \qquad \zeta^a =\widetilde q^{ab}\hat r^c\nabla_c\ell_b,$$

where $\sigma_{ab}$ is trace free because $\theta_{(\ell)}=0$. Their values depend on the fixed null normalization, while the complete balance law is geometrically consistent with that choice.

The finite area balance

For zero cosmological constant and leaves $S_1,S_2$ bounding $\Delta H$, the scalar and momentum constraints on spacelike $H$ imply

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

The first term is matter energy flux associated with $\xi_{(R)}^a$. The second is interpreted as ingoing gravitational energy flux. Ashtekar and Krishnan derive this equation from the constraints in Ashtekar and Krishnan 2004, §3.2, eq. (25), pp. 21–24, PDF and discuss the geometric flux in §3.3, eq. (26), pp. 24–27, PDF.

If matter satisfies the dominant energy condition, $T_{ab}\hat\tau^a\ell^b\geq0$. The shear and $\zeta$ terms are manifestly nonnegative, so $R_2\geq R_1$. Dimensional consistency is explicit: $T\,d^3V$ has dimension $G^{-1}L$, and the geometric squares contribute the same after multiplication by $G^{-1}$.

Checks in spherical and isolated limits

In spherical symmetry, $\sigma_{ab}=0$ and $\zeta_a=0$, because no gravitational radiation or angular structure crosses the round horizon. The law reduces to change in Hawking mass $R/(2G)$ equaling matter flux. For ingoing Vaidya with marginal tube $R=2Gm(v)$, it becomes $m_2-m_1$ equal to the integrated null-dust energy.

In vacuum but without spherical symmetry, matter flux vanishes while shear and $\zeta$ can grow the area. This is the strong-field gravitational contribution; it does not require a background metric or perturbative wave amplitude.

Approaching equilibrium requires a regular null scaling. The unit vectors $\hat\tau,\hat r$ cease to exist on a null hypersurface, and their associated $\ell$ can diverge while $N_R$ tends to zero. The limit is taken in complete flux expressions and regular rescaled fields, not by setting every boost-dependent field, especially $\zeta$, to zero. A rotating isolated horizon retains a nontrivial normal connection even though its area and charges are constant. This transition is treated explicitly in Ashtekar and Krishnan 2003, §VI.B–C, Eqs. (72)–(83).

The $\zeta$ term is essential. On a spacelike horizon it encodes normal-bundle and rotational information not captured by outgoing shear alone. Dropping it would spoil the exact constraint identity and can underestimate growth in rotating situations.

Scope and failure boundaries

The displayed form assumes a spacelike dynamical horizon, spherical leaf topology, four-dimensional Einstein gravity, $\Lambda=0$, and the chosen lapse. A more general Gauss–Bonnet factor enters for other topology, and $\Lambda$ contributes to the matter-like side. Negative cosmological constant removes term-by-term positivity of that flux expression without additional conditions. It does not reverse the geometric area increase on a future spacelike dynamical horizon: $\mathcal L_{\hat r}dA=-\theta_{(n)}dA/2\gt0$ still follows from its defining expansion signs.

The law does not apply unchanged to an arbitrary timelike or mixed-signature marginal tube, nor does it assign a local energy density to gravitational waves on every spacetime hypersurface. Its flux is special to the dynamical-horizon geometry and chosen evolution vector. It also does not say that every individual MOTS branch has monotone area; the outer/future/spacelike assumptions must be checked.

Check your understanding

Across a vacuum dynamical-horizon segment, the shear integral contributes $0.012M$ and the $\zeta$ integral contributes $0.003M$ after their prefactors are included, where $M$ is a physical mass scale. What is $R_2-R_1$, and what happens in spherical vacuum?

Solution

The total geometric flux is $0.015M$, so

$$\frac{R_2-R_1}{2G}=0.015M, \qquad R_2-R_1=0.030GM.$$

In spherical vacuum both geometric terms and the matter flux vanish. A nontrivial spacelike dynamical-horizon segment is then impossible: its geometric definition would require increasing area. The corresponding stationary marginal tube is null and has constant areal radius.

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.