Conserved Charges, Smarr Relations, and First Laws

Black-hole mechanics begins only after mass, angular momentum, charge, and horizon energy have precise definitions. This chapter connects four ways of organizing those quantities: asymptotic and Komar surface charges, bulk identities from the field equations, Euler scaling, and covariant Hamiltonian variations. Their agreement produces the stationary Einstein–Maxwell first law under controlled hypotheses. Their differences expose matter sources, nonintegrable flux, multiple boundaries, and variable couplings rather than being erased by a universal formula.

Helpful background. Exact Solutions of Einstein’s Equations supplies the stationary families used for checks; Causality and Global Structure supplies asymptotic ends and boundary orientation; Einstein Equations fixes curvature and matter normalizations; Gravitational Actions and Boundary Terms supplies the variational setting.

Enter through the charge question

Start by asking where a charge is defined and what symmetry generates it. ADM mass and angular momentum use asymptotic initial data. Komar integrals use normalized Killing fields. Maxwell charge is a flux of $\star F$. Horizon and covariant phase-space charges are boundary terms tied to a variational problem. None is simply “the integral of local gravitational energy density,” and equality between two boundaries must be derived.

In stationary, asymptotically flat vacuum gravity, moving a Komar surface from infinity to a horizon crosses no Ricci source. For $\chi=t+\Omega_H\phi$, the two boundary evaluations give

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

The normalization of $t^a$, the $2\pi$ periodicity of $\phi^a$, and the relative orientation of the two boundaries fix every sign and factor. Matter or $\Lambda$ makes the intervening bulk term nonzero. Bardeen, Carter, and Hawking develop the stationary horizon mass formula in Bardeen, Carter, and Hawking 1973, Open PDF, pp. 163–166.

Three relations that should not be conflated

The Komar identity is a spacetime integral identity: Stokes’ theorem turns the difference of boundary integrals into a Ricci-source volume integral. It is valuable for stationary solutions and makes bulk matter visible.

The Smarr relation is algebraic at one solution. In four-dimensional Einstein–Maxwell theory, homogeneity under length rescaling assigns weights two to $A_H$ and $J$, one to $Q$ and $M$, and yields

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

Dimension or a new length scale changes the Euler weights. A cosmological pressure varied in an extended phase space contributes $-2PV$ to the four-dimensional Smarr formula; see Kubizňák, Mann, and Teo 2017, §2.2, Eqs. (2.19)–(2.21), Open PDF, pp. 8–9.

The first law is a one-form on solution space,

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

It compares nearby stationary solutions satisfying the linearized equations and boundary conditions. It is neither the finite difference of the Smarr formula nor the result of replacing every variation by a numerical change during a dynamical process.

The covariant bridge

For a covariant Lagrangian four-form,

$$\delta L=E(\phi)\delta\phi+d\theta(\phi;\delta\phi).$$

The diffeomorphism Noether current leads, on shell, to a surface charge $Q_\xi$. On one selected closed boundary component $S$, define

$$\delta H_\xi[S] =\int_S k_\xi, \qquad k_\xi=\delta Q_\xi-\xi\mathbin{\cdot}\theta.$$

For a background symmetry and an allowed linearized solution, $dk_\xi=0$. Thus, if $\partial\Sigma=S_\infty\cup(-B)$,

$$0=\int_{\partial\Sigma}k_\xi =\int_{S_\infty}k_\xi-\int_B k_\xi,$$

so the same charge variation is evaluated on either component. At infinity it gives the variation of asymptotic charges. At a bifurcation surface the horizon generator vanishes, removing its contraction with $\theta$ and leaving the gravitational area term. Maxwell theory requires one consistent representative at both boundaries. If $A_a$ and $\delta A_a$ are smooth through the bifurcation surface, the explicit inner Maxwell term vanishes and $-\Phi_H\delta Q$ is carried at infinity. If instead they vanish asymptotically, the usual charged potential is generally singular at the bifurcation surface; one evaluates a suitable future-horizon section, where $+\Phi_H\delta Q$ appears. The two allocations give the same first law and cannot be mixed. Iyer and Wald develop the symplectic and Noether-charge objects in Iyer and Wald 1994, §§III–IV and §VI, equations (73)–(82) and (92)–(102), Open PDF, pp. 14–18; Gao gives the two Einstein–Maxwell boundary allocations in Gao 2003, §I, equations (2)–(8), and §III, equations (27)–(50), Open PDF.

The boundary one-form must be integrable before it is called $\delta H_\xi$. Radiative symplectic flux, varying sources, or an inconsistent field-dependent generator can obstruct path independence. In that case the correct result may be a balance law rather than a conserved Hamiltonian.

Routes through the chapter

For a geometric route, read the first three pages. You will move Komar integrals between boundaries, identify source terms, and reproduce Smarr by scaling. Schwarzschild, Reissner–Nordström, and Kerr limits serve as independent normalization tests.

For a variational route, start with covariant phase space and continue to the stationary first law. This route explains why the bifurcation surface is so effective, why charge variations require boundary conditions, and why extremal horizons need a limiting or adapted argument.

The chapter’s printed order places scaling before the variational construction. A first-time reader may instead read covariant phase space and the stationary first law before returning to the Smarr page, whose Euler argument takes the differential law as an input.

For AdS, de Sitter, or modified-coupling work, read the scaling page and then the final comparison. Keep fixed-coupling, extended-pressure, separate- horizon, and globally constrained variations in distinct phase spaces.

Chapter sequence

1. ADM, Komar, and Horizon Charges fixes asymptotic, Killing, electromagnetic, and horizon normalizations.
2. Komar Identities and Bulk Contributions derives the source term between infinity and the horizon.
3. Smarr Relations from Scaling applies Euler homogeneity and changes dimension or cosmological scale.
4. Covariant Phase Space and Horizon Hamiltonians constructs the charge variation and states its integrability condition.
5. First Law for Stationary Einstein–Maxwell Black Holes equates the infinity and bifurcation-surface Hamiltonians and differentiates Kerr–Newman as a check.
6. Multiple Horizons and Variable Couplings compares competing phase-space choices through the 2026-08-12 evidence cutoff.

Recurring normalization checks

The Schwarzschild relation $A_H=16\pi G^2M^2$, $\kappa=1/(4GM)$ must give both $M=\kappa A_H/(4\pi G)$ and $\delta M=\kappa\delta A_H/(8\pi G)$. Failure signals an orientation, $G$, or Komar factor-of-two error.

For rationalized electric charge, the horizon equation contains $GQ^2/(4\pi)$. The potential must then contain the matching $4\pi$. The Reissner–Nordström mass formula detects a mixed convention immediately. Use the exact Gaussian-to-rationalized Maxwell map before importing a charged metric, potential, flux, or work term from the Gaussian-normalized exact-spacetime chapters.

Finally, a proposed extended first law must reduce to its fixed-coupling form when those variations vanish, and its Smarr relation must include the scaling weight of every varied dimensionful coupling.

Review the chapter

A calculation in four-dimensional Kerr–AdS proposes

$$\delta M=\frac{\kappa}{8\pi G}\delta A_H +\Omega_H\delta J+V\delta P,$$

but defines $M$ relative to a rotating frame at infinity, uses $\Omega_H$ relative to a nonrotating frame, and says $\Lambda$ is fixed. Identify the three consistency checks that fail and how to repair the statement.

Solution

First, energy and angular velocity must refer to the same asymptotic time and rotational frame; mixing rotating-frame $M$ with nonrotating-frame $\Omega_H$ spoils the Hamiltonian pairing. Second, $V\delta P$ belongs only to a phase space in which $P=-\Lambda/(8\pi G)$ varies. If $\Lambda$ is fixed, $\delta P=0$ and the term is absent from the differential law. Third, the result must be integrable under one set of boundary conditions and must pass the associated Smarr scaling test, including the $-2PV$ term when pressure is varied. The repair is to declare one frame, one fixed- or variable-$\Lambda$ phase space, and charges derived consistently from it.

The Laws of Black-Hole Mechanics now uses these charges to state the zeroth, equilibrium-first, area, physical-process, and third laws with their distinct hypotheses.

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.
• Kubizňák, David, Robert B. Mann, and Mae Teo. “Black Hole Chemistry: Thermodynamics with Lambda.” Classical and Quantum Gravity 34 (2017): 063001. DOI.