ADM, Komar, and Horizon Charges

A stationary black hole carries several charges that look different because they are defined on different boundaries and from different structures. ADM mass and angular momentum use asymptotic initial data; Komar charges use Killing fields; electric charge uses the Maxwell flux; horizon charges use a cross-section of the horizon. They agree only after the field equations, orientations, asymptotic normalization, topology, and absence or inclusion of bulk sources are handled consistently. This page fixes those choices and derives the vacuum horizon check behind the Smarr relation.

Required background. Variational and Symplectic Geometry supplies surface charges and Hamiltonian variations. Helpful background. Exact Solutions of Einstein’s Equations supplies Schwarzschild and Kerr checks; Causality and Global Structure supplies asymptotic ends; Einstein Equations relates curvature to bulk sources; Gravitational Actions and Boundary Terms fixes the variational problem; Black-Hole Geometry and Mechanics fixes the horizon normalization used here.

Charges at spatial infinity

Work first in four-dimensional, asymptotically flat general relativity. On an asymptotically Euclidean slice, write the spatial metric in asymptotically Cartesian coordinates as $h_{ij}=\delta_{ij}+s_{ij}$. With the usual falloff and parity conditions, the ADM energy is

$$M_{\mathrm{ADM}} =\frac{1}{16\pi G}\lim_{r\to\infty} \int_{S_r}(\partial_js_{ij}-\partial_is_{jj})N^i\,dS,$$

where $N^i$ is the outward unit normal within the slice. This expression is not a coordinate scalar under arbitrary transformations; its invariance uses the allowed asymptotic coordinate class. ADM angular momentum likewise uses the asymptotic rotational generator and the canonical momentum of the slice.

For a stationary spacetime, normalize the Killing field $t^a$ to unit time translation at the chosen asymptotic end. If $S$ is an oriented closed two-surface homologous to a sphere at infinity, define the Komar mass by

$$M_K[S]=-\frac{1}{8\pi G}\int_S {\star dt},$$

where $t=t_a dx^a$ and the surface orientation is induced by the outward-normal-first convention. For an axial Killing field $\phi^a$ whose orbits have period $2\pi$,

$$J_K[S]=\frac{1}{16\pi G}\int_S {\star d\phi}.$$

The different prefactors are essential. They encode the familiar Komar factor of two between time translations and rotations. At spatial infinity in a stationary asymptotically flat vacuum solution, $M_K=M_{\mathrm{ADM}}$ and $J_K=J_{\mathrm{ADM}}$. Without stationarity there need not be a Killing form to insert in a Komar integral, while ADM charges remain meaningful under their own asymptotic hypotheses.

Maxwell flux and gauge potential

Use the Einstein–Maxwell action with matter term $-\tfrac14\int F_{ab}F^{ab}\sqrt{-g}\,d^4x$. The electric charge enclosed by $S$ is

$$Q[S]=\int_S {\star F},$$

with this rationalized normalization. Maxwell’s equation $d{\star F}=0$ makes the charge independent of the homologous surface when no electric current crosses the intervening region. This convention gives the Reissner–Nordström metric term $GQ^2/(4\pi r^2)$ and the horizon discriminant $G^2M^2-GQ^2/(4\pi)$.

The charge is gauge invariant, but the electrostatic potential is not until a reference boundary is chosen. For horizon generator $\chi^a=t^a+\Omega_H\phi^a$, define

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

A constant gauge shift then cancels between the two boundaries. This potential difference, not an unreferenced value of $A_t$, multiplies $\delta Q$ in the first law.

Moving a Komar surface to the horizon

Let $\Sigma$ run from a horizon cross-section $B$ to spatial infinity, and assume for the moment a stationary vacuum region between them. For any Killing field $\xi^a$,

$$\nabla_a\nabla^a\xi^b=-R^b{}_a\xi^a.$$

Vacuum sets the right-hand side to zero, so Stokes’ theorem makes the Komar integral independent of the homologous surface. Apply this to $\chi=t+\Omega_H\phi$. At infinity, linearity and the definitions above give

$$-\frac{1}{8\pi G}\int_{S_\infty}{\star d\chi} =M-2\Omega_HJ.$$

On a nondegenerate Killing horizon, choose the binormal orientation so that $\nabla_a\chi_b=\kappa\epsilon_{ab}$ on a bifurcation surface. Then

$$-\frac{1}{8\pi G}\int_B{\star d\chi} =\frac{\kappa A_H}{4\pi G}.$$

Equality of the two boundary integrals yields

$$M-2\Omega_HJ=\frac{\kappa A_H}{4\pi G},$$

the vacuum four-dimensional Smarr relation. Bardeen, Carter, and Hawking give the stationary horizon identities and mass formula in Bardeen, Carter, and Hawking 1973, Open PDF, pp. 163–166.

With electromagnetic fields, matter, or a cosmological constant present, the Komar form is not generally closed across the bulk. The boundary difference then equals an integral of the Ricci source; after the Einstein and Maxwell equations are used, this produces the electromagnetic work term or other bulk contributions. It is incorrect to declare horizon and infinity charges equal before doing that step.

Schwarzschild and Kerr normalization checks

For Schwarzschild, $t^a$ is unit normalized at infinity, $A_H=16\pi G^2M^2$, and $\kappa=1/(4GM)$. Hence

$$\frac{\kappa A_H}{4\pi G}=M,$$

which checks the sign, $G$, and numerical factor in the horizon integral.

For Kerr, take $a=J/M$ and $\chi=t+\Omega_H\phi$. The horizon identity gives $M-2\Omega_HJ=\kappa A_H/(4\pi G)$. Rescaling $t^a$ would rescale the Komar mass, $\chi^a$, and $\kappa$, so asymptotic unit normalization is part of the physical definition. Reversing the surface orientation flips every Komar integral; mixing orientations at the two ends creates a false sign failure.

The covariant phase-space charge later reproduces the same stationary variations while making integrability and boundary ambiguities explicit. The agreement is a theorem within one phase space, not evidence that every quasi-local energy proposal must equal ADM mass. The Noether-charge derivation and its stationary horizon evaluation are given in Iyer and Wald 1994, §§III–IV and §VI, Open PDF, printed pp. 848–862.

Check the factors on Schwarzschild

Using $A_H=16\pi G^2M^2$ and $\kappa=1/(4GM)$, evaluate the horizon Komar charge. Then suppose one mistakenly defines the Komar mass with $-1/(16\pi G)$ rather than $-1/(8\pi G)$. What inconsistency results?

Solution

The correct horizon value is

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

The incorrect prefactor gives $M/2$ on both a horizon and a sphere at infinity when applied to $t^a$. It therefore fails to agree with the ADM mass of Schwarzschild. The rotational Komar charge legitimately uses $1/(16\pi G)$; transferring that prefactor to the time-translation charge is the factor-of-two error.

Komar Identities and Bulk Contributions now derives the intervening source term rather than assuming it vanishes.

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.
• 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.