Quasi-Local and Isolated-Horizon Mechanics

Quasi-local horizon mechanics describes equilibrium and change at a finite black-hole boundary without assuming that the exterior is stationary. The construction begins with a null hypersurface of fixed intrinsic area, adds controlled time-independence of its connection, constructs charges from surface fields, and asks when their variation defines a Hamiltonian energy. Intrinsic curvature and rotation multipoles then resolve distortion, while a final comparison separates invariant data on a chosen boundary from the slicing and branch choices that select that boundary.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary checks; Causality and Global Structure distinguishes finite boundaries from event horizons; Einstein Equations turns nonexpansion into flux restrictions; Variational and Symplectic Geometry supplies the horizon Hamiltonian.

Enter through boundary equilibrium

An event horizon answers a global escape question. A quasi-local horizon asks what the geometry and fields are doing at one finite boundary. The first equilibrium structure is a nonexpanding horizon $\Delta$: a null hypersurface whose generator $\ell^a$ has

$$\theta_{(\ell)}=0$$

everywhere, together with the field equations, compact sections, and a causal energy-flux condition. Raychaudhuri’s equation then forces the generator shear and $T_{ab}\ell^a\ell^b$ to vanish. Consequently the cross-sectional metric and area form are Lie dragged along $\ell^a$.

This result is intrinsic. It prohibits matter or shear flux through the boundary under the stated conditions, but it does not require a Killing vector in a neighborhood. Radiation can remain in the exterior, and the boundary’s global event-horizon status must be established separately.

From metric equilibrium to connection equilibrium

The induced horizon derivative satisfies

$$D_a\ell^b=\omega_a\ell^b, \qquad \kappa_{(\ell)}=\ell^a\omega_a.$$

A weakly isolated horizon selects an equivalence class $[\ell]$ under positive constant rescalings and imposes $\mathcal L_\ell\omega_a=0$. Cartan’s identity then gives $d\kappa_{(\ell)}=0$, the quasi-local zeroth law. The value of surface gravity still rescales with $\ell^a$, so a mechanics prescription must select a normalization.

A fully isolated horizon requires the whole intrinsic derivative operator to be time independent. As specified structures, the three levels obey

$$\mathrm{IH}\Longrightarrow\mathrm{WIH} \Longrightarrow\mathrm{NEH}.$$

Every NEH admits suitable, generally nonunique WIH structures, but the bare NEH definition does not specify which WIH class is included as part of the horizon structure; not every WIH admits an IH structure. Ashtekar and Krishnan state the hierarchy and its mechanics in Ashtekar and Krishnan 2004, §§2.1 and 4.1.3.

Charges and Hamiltonian energy

On a cross-section $S$, use the rationalized Maxwell normalization $S_{\mathrm{EM}}=-\tfrac14\int F_{ab}F^{ab}\sqrt{-g}\,d^4x$. The electric charge is then the flux

$$Q_S=\int_S\star F.$$

If $S$ is axisymmetric, its normalized axial field pairs with the rotation one-form to give gravitational angular momentum; in Einstein–Maxwell theory a gauge-field surface term completes the total charge. Closedness of $S$, together with the axial identity $D_a\varphi^a=0$, makes the gravitational integral invariant under null boosts, while gauge invariance requires a consistent symmetry-preserving potential or patching prescription.

For an allowed horizon evolution $t^a$, the internal-boundary contribution to the symplectic form is

$$X_t^\Delta =\frac{\kappa_{(t)}}{8\pi G}\,dA_\Delta +\Omega_{(t)}\,dJ_\Delta +\Phi_{(t)}\,dQ_\Delta.$$

A horizon energy exists precisely when this one-form is exact on the chosen phase space. The equality $dE_\Delta^{(t)}=X_t^\Delta$ is the isolated-horizon first law. Matching the coefficient functions to Kerr–Newman with the same horizon charges selects a useful canonical normalization without claiming the exterior is Kerr–Newman.

Intrinsic distortion and rotation

For a smooth axisymmetric nonextremal isolated horizon, scalar curvature and the curl of $\omega_a$ define harmonic moments $I_n$ and $L_n$. Together with area, the full sequences reconstruct the intrinsic horizon geometry up to diffeomorphism within their stated class. Physical rescalings give mass and current source moments with $M_0=M_\Delta$ and $J_1=J_\Delta$.

These are finite-boundary source multipoles, not automatically the asymptotic field multipoles. Exterior matter and gravitational fields contribute to moments defined at infinity. The invariant construction and reconstruction theorem appear in Ashtekar et al. 2004, §III.A–C, pp. 8–16.

Routes through the chapter

For foundational geometry, read the first two pages in order. They prove the NEH consequences and show exactly what WIH and IH add.

For mechanics, continue through charges and the isolated-horizon first law. This route emphasizes axial and gauge normalization, integrability, and the distinction between boundary energy and general quasi-local mass.

For numerical interpretation, study multipoles and the slicing comparison. Intrinsic spectra can track relaxation on a selected horizon, but branch selection, approximate symmetry, stability, and resolution must accompany the data. Production horizon-finding algorithms belong to Computational Relativity rather than this geometric treatment.

Contents in construction order

1. Nonexpanding Horizons derives shear, flux, and intrinsic-metric constancy from the boundary conditions.
2. Weakly Isolated and Isolated Horizons freezes increasing amounts of connection data and proves the zeroth law.
3. Quasi-Local Horizon Angular Momentum and Charge constructs the surface charges and checks their boost and gauge behavior.
4. Isolated-Horizon First Laws identifies the first law with Hamiltonian integrability.
5. Horizon Multipoles and Distortion extracts invariant curvature and rotation moments.
6. Slicing, Foliation, and Quasi-Local Ambiguity distinguishes fixed-boundary invariants, slice dependence, tube multiplicity, and foliation uniqueness through the 2026-08-31 evidence cutoff. This page begins where the earlier horizon taxonomy ends: the horizon class is no longer the only question, because the surface, slice, or tube held fixed must also be named. The modern merger and transition questions are surveyed in Ashtekar and Krishnan 2025, §4.2, especially Open Issue 7.

Review the chapter

A simulation finds a null marginal tube with constant cross-sectional area and reports that it is a weakly isolated horizon with Kerr mass and spin. What must be checked beyond area constancy, and what conclusions would still remain unjustified after those checks pass?

Solution

One must verify the NEH topology and regularity, zero expansion on the whole null tube, the field equations and energy-flux condition, and hence vanishing shear and null flux. For a WIH, one must select a constant-rescaling class $[\ell]$ and show $\mathcal L_\ell\omega_a=0$. To report spin, an axial field with $2\pi$ orbits and a consistent Maxwell gauge term are needed. To report a horizon energy, the chosen evolution coefficients must make the area–spin– charge one-form integrable; Kerr–Newman matching is one normalization prescription. Even then, the exterior need not be stationary or Kerr, and the boundary is not proven to be the global event horizon. Those are separate spacetime claims.

Ergoregions, Penrose Processes, and Energy Extraction next uses stationary symmetry and horizon flux to quantify rotational-energy extraction.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): article 10. DOI.
• Ashtekar, Abhay, and Badri Krishnan. “Quasi-Local Black Hole Horizons: Recent Advances.” Living Reviews in Relativity 28 (2025): article 8. DOI.
• 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.