Isolated-Horizon First Laws

An isolated-horizon first law is not obtained by naming a surface gravity and integrating it. It is the condition that evolution preserving an internal weakly isolated boundary admit a Hamiltonian on the covariant phase space. The horizon contribution to the Hamiltonian variation is a one-form built from area, angular momentum, and charge. A horizon energy exists exactly when that one-form is integrable. Matching its coefficients to a stationary family selects a useful normalization without requiring the exterior itself to be stationary.

Required background. Hypersurface Normals and Tangential Projectors supplies the null inner boundary. Helpful background. Exact Spacetimes supplies the Kerr–Newman normalization family; Causality and Global Structure supplies boundary components; Einstein Equations fixes the theory; Black-Hole Geometry and Mechanics supplies weakly isolated-horizon data; Variational and Symplectic Geometry supplies Hamiltonian integrability.

Phase space with an isolated inner boundary

Consider four-dimensional Einstein–Maxwell solutions in a region bounded internally by a weakly isolated horizon (WIH) $\Delta$ and externally by an asymptotic end. Allowed variations preserve the WIH boundary conditions, the axial structure when rotation is included, and the asymptotic falloff. The symplectic form contains the bulk integral plus the boundary term required to make it independent of the Cauchy slice.

Choose an evolution vector $t^a$ that approaches a fixed time translation at infinity. On the horizon let it take the symmetry form

$$t^a\widehat{=}B_{(t)}\ell^a-\Omega_{(t)}\varphi^a,$$

where $\widehat{=}$ denotes equality on $\Delta$, $B_{(t)}$ is constant on each solution’s horizon, and $\varphi^a$ is the normalized axial field. This choice assigns

$$\kappa_{(t)}=B_{(t)}\kappa_{(\ell)}$$

and an electromagnetic potential $\Phi_{(t)}$ defined relative to the outer boundary. The coefficients may be functions of horizon charges across phase space, but on each WIH they are constant over the boundary.

The horizon Hamiltonian one-form

Contract the phase-space symplectic form with the variation generated by $t^a$. The asymptotic contribution is the variation of the chosen total energy. The inner-boundary contribution is

$$X_t^\Delta(\delta) =\frac{\kappa_{(t)}}{8\pi G}\delta A_\Delta +\Omega_{(t)}\delta J_\Delta +\Phi_{(t)}\delta Q_\Delta.$$

Evolution is Hamiltonian only if there is a function $E_\Delta^{(t)}$ such that

$$\delta E_\Delta^{(t)}=X_t^\Delta(\delta).$$

This equality is the isolated-horizon first law. It is simultaneously the condition for a horizon energy and the differentiability condition on the evolution generator. Ashtekar and Krishnan derive the Hamiltonian criterion and the gravitational-rotational law in Ashtekar and Krishnan 2004, §4.1.3, Eqs. (35)–(37). The full rotating Einstein–Maxwell boundary one-form, its closure conditions, and the generalized law are given in Ashtekar, Beetle, and Lewandowski 2001, §IV.D, Eqs. (4.9)–(4.12).

The boundary conditions alone do not supply a unique $t^a$. Different normalizations can generate different, internally consistent Hamiltonians. This freedom reflects the absence of a preferred unit time translation at a finite null boundary, not an ambiguity in the area or flux itself.

Integrability conditions

Use $(A,J,Q)$ as local coordinates on the horizon state space. On a simply connected coordinate patch, the one-form is exact precisely when its mixed derivatives agree:

$$\frac{\partial}{\partial J} \left(\frac{\kappa_{(t)}}{8\pi G}\right) =\frac{\partial\Omega_{(t)}}{\partial A},$$ $$\frac{\partial}{\partial Q} \left(\frac{\kappa_{(t)}}{8\pi G}\right) =\frac{\partial\Phi_{(t)}}{\partial A}, \qquad \frac{\partial\Omega_{(t)}}{\partial Q} =\frac{\partial\Phi_{(t)}}{\partial J}.$$

These relations are not additional black-hole field equations. They restrict which phase-space-dependent choices of horizon evolution define a path-independent energy. If they fail, integrating $X_t^\Delta$ from one solution to another depends on the path, so the proposed $E_\Delta^{(t)}$ is not a function on the full phase space.

If the relevant state-space region is not simply connected, these local conditions are not by themselves enough for a single-valued global energy: the integral around every noncontractible closed cycle must also vanish.

If symplectic flux crosses the boundary, the surface is not isolated and the Hamiltonian-conservation argument changes into a balance law. One cannot fix that failure by adding an arbitrary energy function: the boundary class and phase space must first be changed.

Selecting the Kerr–Newman normalization

A natural prescription identifies the coefficient functions with those of the Kerr–Newman outer horizon having the same $(A_\Delta,J_\Delta,Q_\Delta)$. Use the rationalized Maxwell action $S_{\mathrm{EM}}=-\tfrac14\int F_{ab}F^{ab}\sqrt{-g}\,d^4x$ and flux charge $Q_\Delta=\int_S\star F$. To display the prescription in this convention, work in the purely electric, single-potential sector; magnetic charge requires a duality rotation or a patched or dual-potential treatment. Define

$$R_\Delta=\sqrt{\frac{A_\Delta}{4\pi}}, \qquad q_\Delta^2=\frac{GQ_\Delta^2}{4\pi},$$

and

$$\mathcal N_\Delta =\sqrt{\left(R_\Delta^2+q_\Delta^2\right)^2 +4G^2J_\Delta^2}.$$

The Kerr–Newman outer-horizon parameter domain has $R_\Delta\gt0$ and

$$R_\Delta^4\ge q_\Delta^4+4G^2J_\Delta^2.$$

Equality is the extremal boundary; strict inequality is the subextremal outer-horizon branch. A nonzero constant rescaling cannot change whether a WIH normal has zero surface gravity. The Kerr–Newman calibration is therefore defined on a phase space restricted by

$$\kappa_{(\ell)}=0 \quad\Longleftrightarrow\quad R_\Delta^4=q_\Delta^4+4G^2J_\Delta^2, \qquad B_{(t)}\ne0,$$

where $B_{(t)}\ne0$ excludes a degenerate choice with no genuine horizon-time evolution. On this restricted domain the Kerr–Newman-calibrated energy is

$$E_\Delta=\frac{\mathcal N_\Delta}{2GR_\Delta}.$$

This is not merely an integrable expression whose coefficients are asserted afterward. Since $A_\Delta=4\pi R_\Delta^2$, differentiation at fixed $(J_\Delta,Q_\Delta)$ gives

$$8\pi G\left.\frac{\partial E_\Delta}{\partial A_\Delta} \right|_{J_\Delta,Q_\Delta} =\frac{R_\Delta^4-q_\Delta^4-4G^2J_\Delta^2} {2R_\Delta^3\mathcal N_\Delta} =\kappa_{(t)}.$$

The other two derivatives are

$$\left.\frac{\partial E_\Delta}{\partial J_\Delta} \right|_{A_\Delta,Q_\Delta} =\frac{2GJ_\Delta}{R_\Delta\mathcal N_\Delta} =\Omega_{(t)},$$

and, using $\partial q_\Delta^2/\partial Q_\Delta=GQ_\Delta/(2\pi)$,

$$\left.\frac{\partial E_\Delta}{\partial Q_\Delta} \right|_{A_\Delta,J_\Delta} =\frac{Q_\Delta\left(R_\Delta^2+q_\Delta^2\right)} {4\pi R_\Delta\mathcal N_\Delta} =\Phi_{(t)}.$$

Thus the three coefficients arise from one scalar potential, so their mixed derivatives agree and the first law is exact. The extremal equality makes the displayed $\kappa_{(t)}$ vanish, as it must. These formulas are the rationalized-charge translation of Ashtekar, Beetle, and Lewandowski 2001, §IV.E, Eqs. (4.13)–(4.16). The additive constant is fixed by $E_\Delta\to0$ at the uncharged, nonrotating, zero-area endpoint; a zero-area limit at fixed nonzero $J_\Delta$ or $Q_\Delta$ is outside the stated outer-horizon domain.

This does not assert that the exterior geometry is Kerr–Newman. It uses a stationary family to calibrate the time normalization and energy function on the finite boundary. Radiation may remain outside while the horizon itself satisfies WIH conditions.

For a nonrotating uncharged horizon, matching Schwarzschild gives

$$E_\Delta=\sqrt{\frac{A_\Delta}{16\pi G^2}}, \qquad \kappa_{(t)}=\sqrt{\frac{\pi}{A_\Delta}}.$$

Indeed,

$$dE_\Delta =\frac{1}{8\sqrt{\pi}G\sqrt{A_\Delta}}\,dA_\Delta =\frac{\kappa_{(t)}}{8\pi G}\,dA_\Delta.$$

This elementary limit checks the normalization and dimensions.

Relation to the stationary and physical-process laws

The stationary first law compares globally stationary solutions and evaluates one Hamiltonian variation at infinity and a Killing horizon; the equilibrium-law theorem page separates its stationary-family and arbitrary-linearized domains. The isolated-horizon law needs equilibrium only at the internal boundary, but it adds WIH boundary data and a choice of horizon evolution.

A physical-process law instead treats weak flux across a perturbed event or Killing horizon and relates the resulting area change to energy and angular momentum flux after appropriate settling. A dynamical horizon has exact flux balances for its spacelike geometry. These statements can agree in shared limits without having identical hypotheses.

The quasi-local law also does not choose a universal quasi-local energy for arbitrary two-surfaces. It defines the Hamiltonian associated with a particular isolated-boundary phase space and evolution vector. General quasi-local mass proposals answer a wider and different problem.

Diagnose an integrability failure

On a two-parameter horizon state space $(A,J)$, suppose a proposed evolution has

$$\frac{\kappa_{(t)}}{8\pi G}=A+J, \qquad \Omega_{(t)}=2A.$$

Does a horizon energy exist for arbitrary variations? If not, give one simple restriction under which the pullback becomes integrable.

Solution

The mixed derivatives are

$$\partial_J(A+J)=1, \qquad \partial_A(2A)=2.$$

They disagree, so the one-form $(A+J)\,dA+2A\,dJ$ is not exact on the full two-dimensional state space. Its integral depends on path, and there is no global $E(A,J)$ with these coefficients. On the one-dimensional subspace $J=0$ with allowed variations $dJ=0$, the pullback is $A\,dA=d(A^2/2)$ and is integrable. That restricted energy does not repair the full rotating phase space; it defines a smaller boundary problem.

Horizon Multipoles and Distortion extracts more detailed invariant shape and rotation information from horizon cross-sections.

References

• Ashtekar, Abhay; Beetle, Christopher; Lewandowski, Jerzy. “Mechanics of Rotating Isolated Horizons.” Physical Review D 64 (2001): 044016. DOI. Open PDF.
• Ashtekar, Abhay; Krishnan, Badri. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): article 10. DOI.