Angular-Momentum and Energy Flux Laws

Rotation extends the dynamical-horizon area law into coupled angular-momentum and energy balances. The angular law follows directly from the momentum constraint and is meaningful for any declared tangential vector, but only an axial symmetry or a documented approximate-symmetry prescription makes it “the” black-hole spin. A canonical mass law then matches each cross-section to the Kerr mass with the same area and angular momentum.

Required background. Hypersurface normals and tangential projectors supplies the extrinsic curvature and surface normals entering the charges.

Helpful background. Black-Hole Geometry and Mechanics develops quasi-local spin and first laws; Black-Hole Perturbation Theory supplies radiative fluxes; Formulations of General Relativity provides the momentum constraint; and Computational Relativity develops approximate axial vectors and budget checks.

Quasi-local angular momentum

Let a spacelike dynamical horizon $H$ have induced metric $q_{ab}=g_{ab}+\hat\tau_a\hat\tau_b$, future unit normal $\hat\tau^a$, and leaves $S$ with outward unit normal $\hat r^a$ within $H$. Adopt

$$K_{ab}=q_a{}^c q_b{}^d\nabla_c\hat\tau_d$$

for the extrinsic curvature of $H$. This is the opposite sign from the Cauchy-slice convention used in Angular Momentum in Collapse; the compensating minus sign in the surface charge below gives the same physical angular momentum. Let $\varphi^a$ be tangent to every leaf, divergence free there, and normalized so its flow parameter has period $2\pi$ around closed orbits. This normalization fixes the vector’s scale, not the orbits’ proper circumference. Define

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

If $\varphi^a$ is an axial Killing field, this is the intrinsic horizon angular momentum and agrees with the appropriate Komar charge in the stationary case. If no exact symmetry exists, different approximate Killing-vector constructions give different $J$; their spread is a prescription uncertainty.

Applying the momentum constraint on the portion $\Delta H$ between $S_1$ and $S_2$ gives

$$J_{S_2}-J_{S_1} =\mathcal J_{\rm matter}+\mathcal J_{\rm grav},$$

with

$$\begin{aligned} \mathcal J_{\rm matter} &=-\int_{\Delta H} T_{ab}\hat\tau^a\varphi^b\,d^3V,\\ \mathcal J_{\rm grav} &=-\frac{1}{16\pi G} \int_{\Delta H}P^{ab}\mathcal L_\varphi q_{ab}\,d^3V, \end{aligned}$$

where $P^{ab}=K^{ab}-Kq^{ab}$. The signs follow the displayed definition and outward orientation. Ashtekar and Krishnan derive these relations in Ashtekar and Krishnan 2004, §4.2.1, eqs. (42)–(45), pp. 32–33, PDF.

If $\varphi$ is a Killing field of the full horizon metric, $\mathcal L_\varphi q_{ab}=0$ and the gravitational angular-momentum flux vanishes in this representation. On a dynamical distorted horizon, a leafwise axial field need not be a symmetry of $q_{ab}$ along $H$, so the gravitational term can be nonzero.

Canonical horizon energy

Write $A=4\pi R^2$ and suppose the leaves admit an axial vector defining $J$. Associate to each leaf the Kerr mass with the same $R,J$:

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

This relation is not an assertion that the distorted leaf is isometric to Kerr. It selects a canonical energy function of the two charges. The corresponding effective angular velocity and surface gravity are the Kerr functions

$$\Omega_H=\frac{2GJ} {R\sqrt{R^4+4G^2J^2}}, \qquad \kappa_H=\frac{R^4-4G^2J^2} {2R^3\sqrt{R^4+4G^2J^2}}.$$

Direct differentiation yields

$$dM_H=\frac{\kappa_H}{8\pi G}\,dA +\Omega_H\,dJ.$$

Integrated along $H$, the corresponding evolution vector is $t^a=N\ell^a-\Omega_H\varphi^a$, with the lapse fixed by the chosen Kerr energy. The minus sign agrees with the negative sign in $J_S[\varphi]$: in a stationary spacetime the horizon generator is proportional to $t+\Omega_H\varphi$. Matter flux, the shear and normal-connection contribution, and angular-momentum work then equal $M_{H,2}-M_{H,1}$. Angular-momentum work has no universal sign, so positivity of the area-radius flux does not imply positivity of every rotational-energy contribution. The construction and canonical flux formula are in Ashtekar and Krishnan 2004, §§4.2.2–4.2.3, eqs. (48)–(57), pp. 33–36, PDF.

Physical checks and limitations

For $J=0$, $M_H=R/(2G)$, $\Omega_H=0$, and $\kappa_H=1/(2R)$, recovering the spherical area balance. In the isolated limit all matter and gravitational fluxes vanish, so $A,J,M_H$ are constant. Reversing the axial orientation sends $J\to-J$ and $\Omega_H\to-\Omega_H$ while leaving $M_H$ and $\Omega_HdJ$ unchanged.

The outer-Kerr-horizon condition $2G\lvert J\rvert\leq R^2$ makes $\kappa_H\geq0$, with strict inequality for a subextremal hole. This domain matters: outside it the algebraic mass function still exists, but its $R$ would correspond to an inner Kerr horizon and $\kappa_H\lt0$. Area–angular-momentum inequalities require energy, symmetry, and stability hypotheses; the formula alone does not prove them. If a noisy approximate spin violates the condition, inspect normalization, surface resolution, and the approximate axial vector before interpreting it physically.

These laws are quasi-local and depend on the foliation and branch of the dynamical horizon. They do not automatically equal an event-horizon balance or the angular momentum radiated to infinity. A closed global budget must also include exterior matter, null-infinity flux, and boundary error.

Check your understanding

A nonrotating leaf has areal radius $R$. A small angular momentum $J$ is added at fixed $R$. Expand the horizon mass through order $J^2$.

Solution

Using $\sqrt{1+x}=1+x/2+O(x^2)$,

$$M_H=\frac{R}{2G} \sqrt{1+\frac{4G^2J^2}{R^4}} =\frac{R}{2G} +\frac{GJ^2}{R^3}+O(J^4).$$

There is no term linear in $J$ because reversing the axial orientation cannot change the mass.

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.