First Law for Stationary Einstein–Maxwell Black Holes

The stationary first law is a differential identity on a specified family of black-hole solutions. In four-dimensional asymptotically flat Einstein–Maxwell theory it relates the mass variation to changes of horizon area, angular momentum, and electric charge. Its coefficients are fixed by the normalized horizon generator and a referenced electromagnetic potential. The law does not apply to arbitrary finite differences or to perturbations that violate the linearized constraints and boundary conditions.

Required background. Variational and Symplectic Geometry supplies Hamiltonian charge variations. Helpful background. Exact Spacetimes supplies the Kerr–Newman family; Causality and Global Structure fixes the asymptotic end; Einstein Equations fixes electrovacuum dynamics; Gravitational Actions and Boundary Terms fixes the variational boundary data; Black-Hole Geometry and Mechanics fixes horizon conventions.

Statement and hypotheses

Consider a smooth one-parameter family of four-dimensional, asymptotically flat, stationary and axisymmetric Einstein–Maxwell black holes. Use one asymptotic end, normalize $t^a$ to unit time translation there, and normalize the axial Killing field $\phi^a$ to have $2\pi$-periodic orbits. Assume a regular connected Killing horizon with generator

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

The perturbation is tangent to the solution family, satisfies the linearized field equations and asymptotic conditions, and uses a gauge in which the potential difference

$$\Phi_H=-\chi^aA_a\big|_H+\chi^aA_a\big|_\infty$$

is well defined. With the rationalized electric charge $Q=\int_{S_\infty}\star F$, the first law is

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

Bardeen, Carter, and Hawking give the original stationary derivation in Bardeen, Carter, and Hawking 1973, pp. 166–168, equations (3.1)–(3.3), Open PDF.

The exact-spacetime chapters instead use Gaussian flux charge. Their variables describe the same solution after the complete Gaussian-to-rationalized Maxwell map, for which $Q_{\rm R}=\sqrt{4\pi}\,Q_{\rm G}$, $\Phi_{\rm R}=\Phi_{\rm G}/\sqrt{4\pi}$, and therefore $\Phi_{\rm R}\delta Q_{\rm R}=\Phi_{\rm G}\delta Q_{\rm G}$.

The coefficients are evaluated on the background solution. The potential difference is invariant under admissible symmetry-compatible gauge transformations, although a transformation that is nonzero at a boundary can move the separate Maxwell contribution between the horizon and infinity. A rescaling of $t^a$ changes $M$, $\chi^a$, $\kappa$, $\Omega_H$, and $\Phi_H$ together; asymptotic unit normalization removes that freedom.

Equality of two Hamiltonian variations

The preceding Covariant Phase Space and Horizon Hamiltonians page constructs the boundary form and explains integrability and exact-form ambiguities. Here the formalism is applied to one concrete stationary Einstein–Maxwell family, with the two gauge representatives followed all the way through the horizon and infinity terms.

Fix the background generator $\chi=t+\Omega_H\phi$ while evaluating a tangent variation. The covariant phase-space identity gives equal boundary integrals on the two ends of a spacelike hypersurface. The placement of the Maxwell work term depends on which gauge representative is regular at the chosen inner boundary.

First impose $A_a,\delta A_a=O(r^{-1})$ at infinity. For a charged black hole, the usual stationary representative need not be smooth at the bifurcation surface even though its pullback to the future horizon is smooth. End the hypersurface on a suitable future-horizon cross-section $S_H$; when a bifurcation surface $B$ exists, choose $S_H$ strictly to its future. Write $k_\chi=\delta Q_\chi-\chi\mathbin{\cdot}\theta$. Then

$$\int_{S_\infty} k_\chi =\delta M-\Omega_H\delta J.$$

On $S_H$, the gravitational and Maxwell pieces give

$$\int_{S_H}k_\chi =\frac{\kappa}{8\pi G}\delta A_H +\Phi_H\delta Q.$$

Equating the integrals immediately yields the first law. For example, the Reissner–Nordström representative $A=-Q\,dt/(4\pi r)$ in the declared rationalized convention belongs to this route: it vanishes at infinity and its pullback to the future horizon is smooth, but the one-form is singular at the bifurcation surface.

There is an equally valid second route. Choose $A_a$ and $\delta A_a$ smooth through the bifurcation surface $B$. Because $\chi^a|_B=0$, the Maxwell Noether charge and contraction term vanish there, and stationarity makes the horizon value $-\chi\mathbin{\cdot}A$ zero. The potential difference is then represented by the nonzero asymptotic value $\chi\mathbin{\cdot}A|_\infty=\Phi_H$, so the two boundary expressions are

$$\int_{S_\infty}k_\chi =\delta M-\Omega_H\delta J-\Phi_H\delta Q, \qquad \int_B k_\chi =\frac{\kappa}{8\pi G}\delta A_H.$$

This route gives the same law after rearrangement. It is inconsistent to take the vanishing-at-infinity expression from the first route and the smooth bifurcation-surface expression from the second: that combination uses no single gauge representative and spuriously deletes $\Phi_H\delta Q$. Iyer and Wald derive the closed boundary form in equations (73)–(77) and prove the smooth-field first law in Iyer and Wald 1994, §VI, Theorem 6.1 and equations (92)–(102), Open PDF, pp. 17–18. Gao describes the smooth- bifurcation gauge and its infinity allocation in Gao 2003, §I, equations (2)–(8), Open PDF, then gives the explicit Einstein–Maxwell future-horizon calculation in Gao 2003, §III, equations (27)–(50), Open PDF.

One should not insert a term $J\delta\Omega_H$ in this calculation. The Hamiltonian variation is evaluated for the fixed background vector field $\chi$; after the identity is obtained, the coefficients vary from one background point to another across the family. If a field-dependent symmetry prescription is used instead, its additional variation must be treated within that prescription.

Direct differentiation of Kerr–Newman

The exact family supplies an independent analytic check. Define

$$a=\frac{J}{M}, \qquad q^2=\frac{GQ^2}{4\pi}, \qquad \Delta(r)=r^2-2GMr+a^2+q^2.$$

For a subextremal black hole, $\Delta=(r-r_+)(r-r_-)$ and

$$A_H=4\pi(r_+^2+a^2), \quad \Omega_H=\frac{a}{r_+^2+a^2}, \quad \kappa=\frac{r_+-r_-}{2(r_+^2+a^2)}, \quad \Phi_H=\frac{Qr_+}{4\pi(r_+^2+a^2)}.$$

Differentiate the horizon equation $\Delta(r_+)=0$ while temporarily using $(M,a,q)$ as independent coordinates:

$$(r_+-r_-)\delta r_+ -2Gr_+\delta M+2a\delta a+2q\delta q=0.$$

Also,

$$\delta A_H=8\pi(r_+\delta r_++a\delta a), \qquad \delta J=a\delta M+M\delta a,$$

and $Q\delta Q/(4\pi)=q\delta q/G$. Substitute these relations into the right-hand side of the first law. The area term becomes

$$\frac{\kappa}{8\pi G}\delta A_H =\frac{r_+^2}{r_+^2+a^2}\delta M -\frac{aM}{r_+^2+a^2}\delta a -\frac{r_+q}{G(r_+^2+a^2)}\delta q.$$

The rotational term supplies $a^2\delta M/(r_+^2+a^2)+aM\delta a/(r_+^2+a^2)$, and the electric term supplies $r_+q\delta q/[G(r_+^2+a^2)]$. The unwanted variations cancel and the coefficient of $\delta M$ sums to one. This verifies every factor, including $4\pi$ in the rationalized charge.

Static, uncharged, and extremal checks

For Schwarzschild, $J=Q=0$, $A_H=16\pi G^2M^2$, and $\kappa=1/(4GM)$. Therefore

$$\frac{\kappa}{8\pi G}\delta A_H=\delta M.$$

For Kerr, set $Q=0$ while retaining the rotational term. For Reissner–Nordström, set $J=0$ while retaining the referenced electric potential. These limits detect the two most common normalization mistakes: omitting the rotational coefficient through an inconsistent generator, or mixing rationalized and Gaussian electric charge.

On an extremal family, $r_+=r_-$ and $\kappa=0$. The first law restricted to tangent variations of the extremal solution surface becomes

$$\delta M=\Omega_H\delta J+\Phi_H\delta Q.$$

This does not permit arbitrary independent variations of $M$, $J$, and $Q$; they must remain tangent to the extremality constraint. Nor does the bifurcation-surface proof apply directly, because an extremal horizon has no bifurcation surface in the original spacetime. The future-horizon derivation above does not require one and applies directly to a stationary extremal family with $\kappa=0$, subject to its gauge, smooth-pullback, and boundary hypotheses Gao 2003, §V, p. 044016-6, Open PDF. A limiting or other degenerate-horizon argument is an alternative.

What the law does not assert

The stationary law compares equilibrium solutions. A physical process in which matter crosses a horizon uses a different perturbative argument and requires settling assumptions. A fully dynamical horizon obeys flux-balance relations instead. A finite change obtained by replacing every $\delta$ with $\Delta$ neglects the variation of $\kappa$, $\Omega_H$, and $\Phi_H$ along the path.

The formula also depends on the phase space. Allowing $\Lambda$, a higher-curvature coupling, a conical deficit, magnetic charge, or additional matter boundary data to vary can add work terms. Those terms are justified by one consistent variational problem, not by dimensional analogy alone.

Check an extremal tangent

For extremal Reissner–Nordström in the present convention, $GM=\sqrt{G/(4\pi)}\,|Q|$. Take $Q\gt0$. Differentiate the extremality relation and verify the extremal first law using $\Phi_H=Q/(4\pi r_+)$ and $r_+=GM$.

Solution

Extremality gives $M=Q/\sqrt{4\pi G}$, so a tangent variation satisfies $\delta M=\delta Q/\sqrt{4\pi G}$. At $r_+=GM=\sqrt{G/(4\pi)}Q$,

$$\Phi_H=\frac{Q}{4\pi r_+}=\frac{1}{\sqrt{4\pi G}}.$$

Because $\kappa=0$ and $J=0$, the first law becomes $\delta M=\Phi_H\delta Q$, exactly the differentiated extremality relation. An arbitrary variation with fixed $Q$ and nonzero $\delta M$ would leave the extremal surface and is not tested by this restricted equation.

Equilibrium First Law next recasts this calculation as a theorem with distinct stationary-family and arbitrary-linearized perturbation domains.

Multiple Horizons and Variable Couplings examines how the phase space changes when there is no single asymptotic/horizon pair or when a coupling itself is varied.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. 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.