Equilibrium First Law

The equilibrium first law is a variational identity on the space of solutions, not an ordinary time-evolution equation. For a four-dimensional, asymptotically flat Einstein–Maxwell black hole it relates the first-order changes of canonical charges and horizon area. Each coefficient belongs to the unperturbed stationary black hole and depends on fixed asymptotic, rotational, electromagnetic, and generator conventions. The formula is classical mechanics. Its thermodynamic interpretation is a separate semiclassical result.

Required background. The Einstein Equation supplies the constraints and linearized field equation.

Helpful background. Exact Solutions of Einstein’s Equations supplies Kerr–Newman checks; Causality and Global Structure supplies asymptotic boundaries; Flat-Background Hilbert Stress-Energy from Metric Variation supplies matter variations; and Black-Hole Geometry and Mechanics supplies horizon generators and charges.

The Einstein–Maxwell variational statement

In the conventions used here, the law is

$$\delta M =\frac{\kappa}{8\pi G}\,\delta A +\Omega_H\,\delta J +\Phi_H\,\delta Q.$$

Assume a smooth stationary, axisymmetric, asymptotically flat background with a connected Killing horizon. Normalize the stationary and axial Killing fields at infinity and set

$$\chi^a=t^a+\Omega_H\phi^a.$$

Let $M$ and $J$ be the corresponding ADM charges. Let $Q$ be the electric charge in a fixed Maxwell normalization, and let $\Phi_H$ be the gauge-invariant potential difference between the horizon and infinity. Magnetic charge or multiple gauge fields require their own potential–charge pairs.

Two perturbation domains must not be blended:

• For the stationary-family route, $\delta\Psi=(\delta g_{ab},\delta A_a)$ is tangent to a smooth family of stationary solutions, obeys the linearized field equations and asymptotic falloffs, and preserves the declared boundary data. In the gauge that vanishes at infinity, only the pullbacks of $A_a$ and $\delta A_a$ need be smooth on the chosen future-horizon section; the one-forms may be singular at a bifurcation surface.
• For the strengthened arbitrary-linearized theorem, the nondegenerate background admits a bifurcation surface and every perturbed field is smooth there. The perturbation need not be stationary, but it satisfies the linearized constraints and the stated asymptotic and boundary conditions. In the theorem’s single-potential formulation, the stationary Maxwell potential is globally defined and smooth from the exterior through the bifurcation surface, as are its allowed perturbations. This excludes magnetic charge unless one first applies electromagnetic duality or supplies a patchwise extension. A nonzero asymptotic electromagnetic reference is then allowed and carries the electric work term.

Historically, the equilibrium law compares neighboring stationary black-hole solutions. Bardeen, Carter, and Hawking derive the mass variation and state the first law in Bardeen, Carter, and Hawking 1973, pp. 166–168, equations (3.1)–(3.3), Open PDF. A stronger Hamiltonian theorem allows the second class of arbitrary nonsingular perturbations of the stationary background, not necessarily ones tangent to a stationary family, provided they satisfy the linearized constraints and stated boundary conditions Wald 1993, §1, equation (15), pp. 358–362, Open manuscript.

This strengthening does not turn the formula into a finite dynamical law. It is still a first-order identity evaluated about a stationary background.

Covariant phase-space proof architecture

The Covariant Phase Space and Horizon Hamiltonians page constructs the presymplectic potential, Noether current and charge, and integrability test. For the theorem, package them into

$$k_\chi(\delta\Psi) =\delta\mathbf Q_\chi -\chi\mathbin{\cdot}\boldsymbol\theta(\Psi,\delta\Psi).$$

On the stationary background $\mathcal L_\chi\Psi=0$. For an admissible linearized solution—or, in Hamiltonian form, a perturbation satisfying the needed linearized constraints—the bulk variation reduces to the boundary identity

$$\int_{S_\infty} \!k_\chi = \int_{C} \!k_\chi .$$

The two standard evaluations differ in domain, not in the resulting law:

Route	Perturbation and regularity	Infinity term	Inner term
Future-horizon stationary family	Tangent to a stationary family; $A_a,\delta A_a=O(r^{-1})$ at infinity; smooth pullbacks on a suitable $S_H$ strictly to the future of $B$ when $B$ exists	$\delta M-\Omega_H\delta J$	$\kappa\delta A/(8\pi G)+\Phi_H\delta Q$
Smooth-bifurcation theorem	Arbitrary admissible linearized perturbation; nondegenerate background; all perturbed fields smooth through $B$; nonzero asymptotic potential reference permitted	$\delta M-\Omega_H\delta J-\Phi_H\delta Q$	$\kappa\delta A/(8\pi G)$

On $B$, $\chi^a=0$, so a smooth Maxwell potential has no explicit inner electric work term. Its derivative remains nonzero, $\nabla_a\chi_b=\kappa\epsilon_{ab}$, and the Einstein–Hilbert Noether charge produces the area term. In the future-section route the asymptotically vanishing charged potential may be singular at $B$ but has a smooth pullback to $S_H$, where the work term appears. Combining the infinity expression of one row with the inner expression of the other uses no single gauge representative and is invalid.

Iyer and Wald define the symplectic and Noether-charge objects in §§III–IV, derive the closed boundary form and Hamiltonian construction in equations (73)–(82), and prove the smooth-field result in Iyer and Wald 1994, §VI, Theorem 6.1 and equations (92)–(102), Open PDF, pp. 14–18. Gao develops the future-horizon Einstein–Maxwell route in Gao 2003, §III, equations (27)–(50), Open PDF. The full stationary-family boundary calculation checks both representatives on Reissner–Nordström and differentiates Kerr–Newman. An extremal horizon has no ordinary bifurcation surface, so it uses that future-section route, a limiting family, or another degenerate-horizon argument under separately stated hypotheses.

What is held fixed

Several apparent paradoxes come from varying the boundary problem while writing the fixed-boundary formula.

The asymptotic time translation and $2\pi$ normalization of the axial field are fixed. The coefficients $\Omega_H$, $\kappa$, and $\Phi_H$ in the displayed law are background values; terms such as $J\,\delta\Omega_H$ do not appear because the identity varies the charges while holding the generator used in the Hamiltonian variation fixed. Electromagnetic gauge transformations are restricted so the potential difference is well defined. Newton’s constant, the cosmological constant, and other couplings are fixed; varying them creates additional work terms and a different phase space.

The perturbation must be tangent to the constraint surface. An arbitrary variation of metric components that violates the linearized Hamiltonian or momentum constraint is not a counterexample to the first law—it is not a variation between admissible solutions. Likewise, changing the normalization of $t^a$ rescales both the energy and the intensive horizon quantities and changes the boundary Hamiltonian.

Schwarzschild and Kerr–Newman checks

For Schwarzschild,

$$A=16\pi G^2M^2, \qquad \kappa=\frac{1}{4GM}.$$

Therefore

$$\frac{\kappa}{8\pi G}\,\delta A =\frac{1}{32\pi G^2M} \left(32\pi G^2M\,\delta M\right) =\delta M,$$

with $J=Q=0$. This confirms the factor of $8\pi G$ and restores dimensions: $\kappa\delta A/G$ has the dimension of mass.

For Kerr–Newman, set

$$R=\sqrt{\frac{A}{4\pi}}, \qquad q^2=\frac{GQ^2}{4\pi}, \qquad \mathcal N=\sqrt{(R^2+q^2)^2+4G^2J^2}.$$

The exact mass relation is

$$M(A,J,Q)=\frac{\mathcal N}{2GR}.$$

Direct differentiation gives

$$8\pi G\frac{\partial M}{\partial A}=\kappa, \qquad \frac{\partial M}{\partial J}=\Omega_H, \qquad \frac{\partial M}{\partial Q}=\Phi_H,$$

with the derivatives taken at fixed values of the other two charges. This is the solution-family check, not the proof of the arbitrary-linearized theorem; the cancellations and gauge normalization are worked through on First Law for Stationary Einstein–Maxwell Black Holes. The mass relation is the rationalized-charge form of Christodoulou and Ruffini 1971, pp. 3552–3555.

Static $J\to0$, uncharged $Q\to0$, and Schwarzschild limits must agree. On the extremal boundary $R^4=q^4+4G^2J^2$, $\kappa=0$, so tangent variations that remain on the extremal family have no area term in the first-order identity; this does not say that arbitrary perturbations preserve extremality.

Check your understanding

A student varies the Schwarzschild relation but writes the horizon term as $\kappa\,\delta A/(4\pi G)$. Use the exact area and surface gravity to diagnose the error.

Solution

Substitution gives $\kappa\delta A/(4\pi G)=2\delta M$, so the coefficient is twice too large. The Einstein–Hilbert Noether-charge normalization produces $\kappa\delta A/(8\pi G)$, which equals $\delta M$. The check is independent of thermodynamic interpretation.

Next, Hawking Area Theorem develops the classical nondecrease law for event-horizon cross-sections and exposes its global hypotheses.

References

• Bardeen, J. M., B. Carter, and S. W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Christodoulou, Demetrios, and Remo Ruffini. “Reversible Transformations of a Charged Black Hole.” Physical Review D 4 (1971): 3552–3555. DOI.
• Gao, Sijie. “First Law of Black Hole Mechanics in Einstein–Maxwell and Einstein–Yang–Mills Theories.” Physical Review D 68 (2003), 044016. DOI. Open PDF, arXiv:gr-qc/0304094v2.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994): 846–864. DOI.
• Wald, Robert M. “The First Law of Black Hole Mechanics.” In Directions in General Relativity, Volume 1: Papers in Honor of Charles Misner, edited by B. L. Hu, M. P. Ryan Jr., and C. V. Vishveshwara, 358–366. Cambridge University Press, 1993. DOI. Open manuscript.