Covariant Phase Space and Horizon Hamiltonians

Covariant phase space turns a spacetime symmetry into a charge variation without choosing a time coordinate or canonical decomposition. The key object is not a charge density by itself but the boundary form $\delta Q_\xi-\xi\mathbin{\cdot}\theta$. Its integral is the variation of a Hamiltonian only when the boundary conditions make it finite, conserved, and integrable on the chosen solution space. At a bifurcation surface the horizon generator vanishes, exposing the gravitational area term that enters the stationary first law. For Maxwell fields, the boundary on which the electric work term appears depends on the gauge representative and its regularity.

Required background. Variational and Symplectic Geometry supplies field-space variations, symplectic currents, and gauge degeneracies. Helpful background. Exact Solutions of Einstein’s Equations supplies stationary test families; Causality and Global Structure supplies the boundaries of a hypersurface; Einstein Equations fixes the equations of motion; Gravitational Actions and Boundary Terms fixes the action variation; Black-Hole Geometry and Mechanics supplies the normalized horizon generator.

From a Lagrangian to the symplectic current

Let $\Psi$ denote all dynamical fields and let $L[\Psi]$ be the covariant four-form Lagrangian. Its first variation can be written

$$\delta L=E(\Psi)\,\delta\Psi+d\theta(\Psi;\delta\Psi),$$

where $E=0$ denotes the field equations and $\theta$ is the presymplectic potential three-form. For two commuting variations, define

$$\omega(\Psi;\delta_1\Psi,\delta_2\Psi) =\delta_1\theta(\Psi;\delta_2\Psi) -\delta_2\theta(\Psi;\delta_1\Psi).$$

Its integral over a Cauchy hypersurface is the presymplectic form. “Pre” is important: diffeomorphisms that vanish appropriately at the boundary are degenerate gauge directions, while transformations with nontrivial boundary action can carry charges.

For the Einstein–Hilbert Lagrangian, the metric potential can be represented in components by

$$\theta_{abc}(g;\delta g) =\frac{1}{16\pi G}\epsilon_{dabc} \left(\nabla^d\delta g-\nabla_e\delta g^{de}\right),$$

where $\delta g=g_{ab}\delta g^{ab}$ because the independent variation here is $\delta g^{ab}$. If instead $h_{ab}=\delta g_{ab}$ is the independent variation, then $\delta g^{ab}=-h^{ab}$ and the same potential reads $\epsilon_{dabc}(\nabla_eh^{de}-\nabla^dh)/(16\pi G)$. The two familiar formulas therefore agree after translating the varied metric variable.

Noether current and surface charge

For a vector field $\xi^a$, define the diffeomorphism Noether current

$$J_\xi=\theta(\Psi;\mathcal L_\xi\Psi)-\xi\mathbin{\cdot}L.$$

Covariance implies that, on shell, $J_\xi$ is closed and can locally be written $J_\xi=dQ_\xi$. For Einstein gravity the Noether charge two-form is

$$Q_\xi=-\frac{1}{16\pi G}\star d\xi,$$

which is one half of the time-translation Komar normalization. The missing half is not an error: the Hamiltonian variation also contains $-\xi\mathbin{\cdot}\theta$, and the relation between Noether and ADM charges uses both terms.

When the background solves the equations, the perturbation solves the linearized equations, and $\xi$ is field independent during the variation, set

$$k_\xi(\delta\Psi) =\delta Q_\xi-\xi\mathbin{\cdot}\theta(\Psi;\delta\Psi).$$

The fundamental identity is

$$\Omega_\Sigma(\delta\Psi,\mathcal L_\xi\Psi) =\int_{\partial\Sigma}k_\xi.$$

On a selected closed boundary component $S$, the candidate charge one-form is

$$\delta H_\xi[S]=\int_S k_\xi.$$

If $\xi$ is a symmetry of the background, the left side of the full-boundary identity vanishes. For $\partial\Sigma=S_\infty\cup(-B)$, the orientations therefore give

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

so the same charge variation can be evaluated at infinity or at the horizon. Iyer and Wald define the symplectic current and Noether charge in §§III–IV, derive the closed boundary form in Iyer and Wald 1994, §VI, equations (73)–(77), Open PDF, p. 14, and construct the Hamiltonian in Iyer and Wald 1994, §VI, equations (78)–(82), Open PDF, p. 15.

Integrability is an additional condition

The displayed boundary integral is initially a one-form on solution space. It equals the variation of a function $H_\xi$ only if its integral around closed loops vanishes. For field-independent $\xi$, the local obstruction is controlled by the symplectic flux:

$$\int_S \xi\mathbin{\cdot}\omega (\Psi;\delta_1\Psi,\delta_2\Psi)=0$$

for all allowed tangent variations on the selected boundary component $S$, up to the boundary prescription used. Nonzero radiation flux, varying sources, or insufficient falloff can prevent integrability or conservation. One may then obtain a flux-balance law rather than a time-independent Hamiltonian charge.

The finite-inner-boundary version of this question is developed on Isolated-Horizon First Laws, where exactness of the horizon one-form selects the permitted evolution fields and energies.

The vector field may itself depend on the solution, as happens when $\chi=t+\Omega_H\phi$ is compared across a family with changing $\Omega_H$. Then one must specify whether $\xi$ or its coefficients are held fixed in the variation and include the corresponding correction. Silently varying the generator double-counts work terms.

There are also standard ambiguities: $L\mapsto L+d\mu$, $\theta\mapsto\theta+\delta\mu+dY$, and a corresponding shift of $Q_\xi$. Boundary conditions, counterterms, reference solutions, and the desired Hamiltonian select a finite representative. Exact-form ambiguity does not license arbitrary physical charges; consistent changes must be made at every boundary and tested against the variational identity.

Why the bifurcation surface is powerful

Take a stationary black hole with horizon generator $\chi=t+\Omega_H\phi$ and a smooth bifurcation surface $B$. The defining property $\chi^a|_B=0$ eliminates $\chi\mathbin{\cdot}\theta$ there. Moreover,

$$\nabla_a\chi_b=\kappa\epsilon_{ab} \quad\text{on }B,$$

so the Einstein Noether charge gives, under variations between nearby stationary solutions with the comparison generator handled consistently,

$$\int_B\delta Q_\chi =\frac{\kappa}{8\pi G}\delta A_H.$$

At infinity, the same Hamiltonian variation is $\delta M-\Omega_H\delta J$ in vacuum. Equality of the two boundary evaluations yields the vacuum stationary first law. The full smooth-field bifurcation-surface argument and its assumptions are given in Iyer and Wald 1994, §VI, Theorem 6.1 and equations (92)–(102), Open PDF, pp. 17–18.

For Einstein–Maxwell theory, define the potential difference, which is invariant under admissible symmetry-compatible gauge transformations,

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

There are two consistent ways to allocate its work term between the boundaries.

Maxwell representative	Permitted inner boundary	Infinity boundary form	Inner boundary form
$A_a,\delta A_a=O(r^{-1})$ at infinity; only their pullbacks are smooth on the future horizon	A suitable future-horizon cross-section $S_H$; when $B$ exists, choose $S_H$ strictly to its future	$\delta M-\Omega_H\delta J$	$\kappa\delta A_H/(8\pi G)+\Phi_H\delta Q$
$A_a$ and $\delta A_a$ smooth through $B$, with stationary gauge data	The bifurcation surface $B$	$\delta M-\Omega_H\delta J-\Phi_H\delta Q$	$\kappa\delta A_H/(8\pi G)$

The first row and second row are two complete boundary prescriptions. Mixing the vanishing-at-infinity outer expression from the first with the smooth-$B$ inner expression from the second would incorrectly lose the electric work term. Gao contrasts the smooth-bifurcation allocation with the future-horizon route in Gao 2003, §I, equations (2)–(8), and §III, equations (27)–(50), Open PDF.

The detailed Reissner–Nordström gauge check, the full boundary evaluation, and the extremal alternative are carried out on First Law for Stationary Einstein–Maxwell Black Holes. The structural point here is that boundary regularity and gauge reference determine where a work term is displayed, while the complete Hamiltonian identity is unchanged.

Boundary checks for a valid Hamiltonian

A proposed horizon Hamiltonian should pass four tests. Its variation must be finite under the declared falloff. Symplectic flux through excluded boundary segments must vanish or be included as flux. Mixed second variations must commute on the permitted solution family. Finally, the infinity and horizon expressions must use the same $t^a$, axial periodicity, electromagnetic gauge representative and reference, admissible inner cross-section, and orientation.

For Schwarzschild, $\delta H_t=\delta M$ at infinity and $\delta H_t=\kappa\delta A_H/(8\pi G)$ at the bifurcation sphere. Since $A_H=16\pi G^2M^2$ and $\kappa=1/(4GM)$, the horizon expression equals $\delta M$. This check tests the Noether prefactor and the absence of an unaccounted boundary term.

Test integrability of a model charge

Suppose a proposed two-parameter boundary charge has variation

$$\delta H=(2x+y)\,\delta x+x\,\delta y.$$

Is it integrable? Find $H$ up to a constant. What would change if the coefficient of $\delta y$ were $2x$?

Solution

For the displayed one-form, $\partial_y(2x+y)=1$ and $\partial_x(x)=1$, so mixed derivatives agree. An integral is

$$H=x^2+xy+C.$$

Indeed,

$$dH=(2x+y)\,dx+x\,dy,$$

which reproduces both coefficients of the proposed variation.

If the second coefficient were $2x$, the mixed derivatives would be $1$ and $2$. The one-form would not be closed, so no path-independent Hamiltonian would exist on the full $(x,y)$ space. One would need to restrict the allowed variations, add justified boundary data, or interpret the mismatch as flux; calling the original expression $\delta H$ would not make it integrable.

First Law for Stationary Einstein–Maxwell Black Holes now evaluates the two boundaries and checks the result directly on Kerr–Newman.

References

• 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.