Horizon Area, Surface Gravity, Angular Velocity, and Potentials

Horizon area is intrinsic to a chosen cross-section, but surface gravity, angular velocity, electromagnetic potential, temperature, and energy depend on a normalized generator, boundary frame, or gauge reference. This record fixes a stationary Einstein–Maxwell baseline, then states which entries survive for isolated or dynamical horizons. The canonical derivations belong to Conserved Charges, Smarr Relations, and First Laws.

Helpful background. Killing Fields and Isometry Groups supplies stationary and axial generators; Hypersurface Normals and Tangential Projectors supplies cross-section geometry; Black-Hole Geometry and Mechanics supplies the horizon taxonomy; The Gibbons–Hawking–York Boundary Term supplies Hamiltonian boundary terms; and Quantum Fields in Curved Spacetime supplies the stationary Hawking effect.

The stationary Einstein–Maxwell package

Work in four-dimensional, asymptotically flat Einstein–Maxwell theory with $c=\hbar=k_B=1$, explicit $G$, signature $(-,+,+,+)$, and rationalized Maxwell action density $-F_{ab}F^{ab}/4$. Normalize the stationary Killing field $t^a$ to unit time translation at the selected infinity and the axial field $\phi^a$ to have $2\pi$-periodic orbits. For physical mass $M\gt0$, angular momentum $J$, and electric charge

$$Q=\int_{S^2_\infty}\star F,$$

define

$$a=\frac{J}{M}, \qquad q^2=\frac{GQ^2}{4\pi}, \qquad r_\pm=GM\pm\sqrt{G^2M^2-a^2-q^2}.$$

The quantities $r_\pm$ are the algebraic roots of the Kerr–Newman radial polynomial. When $0\lt a^2+q^2\lt G^2M^2$, they are the outer and inner horizon radii. At $a=q=0$, only $r_+=2GM$ is a horizon and $r_-=0$ is the central curvature singularity; at equality the roots merge into one degenerate horizon.

The future outer horizon generator fixes surface gravity by

$$\chi^a=t^a+\Omega_H\phi^a, \qquad \chi^b\nabla_b\chi^a=\kappa\chi^a, \qquad \kappa\equiv\kappa_H.$$

For Kerr–Newman, its principal stationary quantities are

$$\begin{aligned} \Omega_H&=\frac{a}{r_+^2+a^2}, &A_H&=4\pi(r_+^2+a^2),\\ \kappa_H&=\frac{r_+-r_-}{2(r_+^2+a^2)}. \end{aligned}$$

Here $A_H$ is the intrinsic area of a horizon cross-section; $r_+$ is a Boyer–Lindquist coordinate value, not an areal radius. The generator, rotation, area, surface gravity, and mechanics normalization are given in Bardeen, Carter, and Hawking 1973, pp. 162–166. The exact charged normalization and limiting checks are collected in Kerr–Newman Horizon Data.

Choose the electromagnetic gauge relative to the same boundary clock. The gauge-qualified potential difference is

$$\Phi_H =-\chi^aA_a\big|_{\mathcal H} +\chi^aA_a\big|_\infty.$$

In the gauge that vanishes at infinity,

$$\Phi_H=\frac{Qr_+}{4\pi(r_+^2+a^2)}.$$

The factors of $4\pi$ in $q^2$, $Q$, and $\Phi_H$ are one convention package; mixing them with a Gaussian Maxwell action breaks the first law. Use the exact Gaussian-to-rationalized Maxwell map when importing a charged exact-spacetime formula.

Geometry, mechanics, and thermodynamics

The area defines the four-dimensional irreducible mass

$$M_{\mathrm{irr}}=\sqrt{\frac{A_H}{16\pi G^2}}.$$

It equals $M$ for Schwarzschild and tracks the area component that cannot decrease under the hypotheses of the classical area law. It is not a local matter rest mass or, by itself, a claim about an extraction process. See Irreducible Mass and Rotational Energy.

For a regular stationary nonextremal state, the Hawking and leading Einstein entropy identifications are

$$T_H=\frac{\kappa_H}{2\pi}, \qquad S_H=\frac{A_H}{4G}.$$

The temperature coefficient comes from the stationary particle-creation calculation in Hawking 1975, §§ 2–3, printed pp. 202–214. The covariant stationary entropy and horizon term are developed in Iyer and Wald 1994, §§ 2–4 and § VI. Their exact scope is explained under Surface Gravity and Hawking Temperature and Area Entropy and the First Law.

Quantity	What fixes it	Invariant content	Boundary of use
$A_H$	A specified spacelike horizon cross-section	Intrinsic area; independent of the scaling of a null normal	Different horizon notions or foliations can select different sections
$\kappa_H$	Horizon generator, time orientation, and normalization	Nonaffinity of that normalized generator	A generic marginal surface has no preferred Killing normalization
$\Omega_H$	Normalized $t^a$, oriented $2\pi$ axial field, and asymptotic frame	Coefficient making $\chi^a$ null on the stationary horizon	No unique value without stationarity and axial symmetry
$\Phi_H$	$\chi^a$, Maxwell normalization, and potential reference at two boundaries	Gauge-qualified horizon-to-boundary potential difference	A bare value of $-\chi\mathbin{\cdot}A$ is gauge dependent
$M_{\rm irr}$	Four-dimensional horizon area	Area mass used in classical mechanics	Does not determine total mass when spin, charge, or exterior fields are present
$S_H$	Stationary Einstein action and renormalized $G$ at leading order	Leading area entropy	Higher-curvature theories use a Noether-charge entropy; generalized entropy also includes the exterior state
$T_H$	Normalized stationary generator and a regular nonextremal quantum state	Temperature relative to the same boundary time	No exact thermal identification for a generic evolving marginal horizon

Generator rescaling, gauge, and the first law

Surface gravity is not attached to an unparameterized null surface. If $\chi'^a=c\chi^a$ with constant $c\gt0$, then

$$\kappa'_H=c\kappa_H, \qquad \Phi'_H=c\Phi_H, \qquad H_{\chi'}=cH_\chi,$$

while $A_H$, $M_{\mathrm{irr}}$, and $S_H$ do not change. The same asymptotic normalization must propagate through $\kappa_H$, $\Omega_H$, $\Phi_H$, and the Hamiltonian energy. If the rescaled generator is decomposed as $\chi'=t'+\Omega'_H\phi$ with $t'=ct$ and the same $2\pi$-normalized $\phi$, then $\Omega'_H=c\Omega_H$. With the boundary time $t$ held fixed and normalized, however, this rescaling is not an allowed renormalization of the stationary package. Surface Gravity and Normalization develops this distinction.

Under $A\mapsto A+d\lambda$, the separate endpoint contractions can change, but the displayed potential difference is invariant for a stationary, single-valued gauge transformation with the same boundary reference. A constant shift cancels between the horizon and infinity. Large or patchwise transformations require transition data; magnetic charge may prevent a single global potential.

For nearby stationary Einstein–Maxwell black holes with all quantities normalized at the same boundary,

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

Rescaling only one work coefficient breaks this identity. The Hamiltonian statement, integrability conditions, and direct Kerr–Newman check belong to The First Law for Stationary Einstein–Maxwell Black Holes.

Which horizon is meant

Horizon notion	Available data	What is not automatic
Event horizon	Global null boundary and intrinsic area of a selected cross-section	A Killing generator, local detectability, preferred foliation, or equilibrium temperature
Killing horizon	Normalized $\chi^a$, $\kappa_H$, and, with axial symmetry, $\Omega_H$	Constancy of $\kappa_H$ still uses the zeroth-law hypotheses
Weakly isolated or isolated horizon	Quasi-local null equilibrium, null-normal equivalence class, area, and suitable horizon charges	A stationary exterior, unique normalization from infinity, or dynamical flux through the horizon
Dynamical or trapping horizon	Foliated marginal sections, area change, and matter/gravitational flux laws	A foliation-independent $\kappa_H$, exact $\Omega_H$, or the equilibrium identity $T_H=\kappa_H/(2\pi)$

The nonexpanding, weakly isolated, isolated, and dynamical conditions and their rescaling classes are compared in Ashtekar and Krishnan 2004, §§ 2–4. The canonical Weakly Isolated and Isolated Horizons page gives the quasi-local hierarchy. An area can therefore be meaningful on a nonstationary section even when no unique temperature, angular velocity, or potential belongs to it.

Limiting and normalization checks

Schwarzschild. Set $a=Q=0$. Then

$$\begin{gathered} r_+=2GM, \qquad r_-=0, \qquad A_H=16\pi G^2M^2,\\ \Omega_H=\Phi_H=0, \qquad \kappa_H=\frac{1}{4GM}, \qquad M_{\mathrm{irr}}=M,\\ S_H=4\pi GM^2, \qquad T_H=\frac{1}{8\pi GM}. \end{gathered}$$

Here the displayed $r_-=0$ is only the lower algebraic endpoint; Schwarzschild has no inner horizon there.

These values satisfy both the first law and the stationary Smarr check

$$M=\frac{\kappa_HA_H}{4\pi G} +2\Omega_HJ+\Phi_HQ.$$

Kerr. Set $Q=0$. The algebraic roots reduce to $r_\pm=GM\pm\sqrt{G^2M^2-a^2}$; reversing $J$ reverses $\Omega_H$ but leaves $A_H$ and $\kappa_H$ unchanged. The roots, area, and angular velocity are independently tabulated in Visser 2008, § 9, printed PDF pp. 24–29, especially Eqs. (104), (112), and (118).

Extremal limit. When $G^2M^2=a^2+q^2$, the roots coincide and $\kappa_H\to0$. The nonextremal stationary formula therefore has $T_H\to0$. This limiting statement does not by itself establish the state or Euclidean periodicity of an exactly extremal geometry. For extremal Kerr, $A_H=8\pi G^2M^2$ and $M_{\mathrm{irr}}=M/\sqrt2$.

Gauge shift. A constant potential shift leaves $\Phi_H$ unchanged because both endpoints move together. Reversing the electric charge reverses $\Phi_H$ but not the geometry; reversing the axial orientation reverses both $J$ and $\Omega_H$. Reversing the future orientation of $\chi^a$ also reverses $\kappa_H$ and is not an allowed positive-temperature normalization.

This page is a definition-and-normalization card. Numerical SI conversions, the uncharged Kerr scale table, light-crossing conventions, and ringdown rescaling remain in Black-Hole Horizon Scales.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. DOI.
• 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. Open PDF.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. DOI. Open PDF.
• 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. Open PDF.
• Visser, Matt. “The Kerr Spacetime: A Brief Introduction.” arXiv:0706.0622v3 (2008). Open PDF.