Symplectic Flux and Boundary Conditions

The covariant presymplectic form is independent of the hypersurface only when the net symplectic flux through the intervening boundary vanishes or is accounted for by boundary degrees of freedom. This requirement is weaker than demanding that every perturbation vanish at the boundary, and stronger than requiring the background equations alone. Reflective, radiative, and mixed conditions differ by which boundary data are fixed, which responses may vary, and whether the resulting flux is zero, prescribed, or dissipative.

Required background. The Gibbons–Hawking–York Boundary Term supplies a differentiable boundary variational principle; Presymplectic Potentials and Currents supplies hypersurface currents and flux; and Killing Fields and Isometry Groups supplies symmetry generators.

Helpful background. The Einstein Equation supplies the linearized bulk equations; Hypersurface Normals and Tangential Projectors supplies boundary normals and orientations; and Formulations of General Relativity orients the chapter.

Flux between two hypersurfaces

Let a spacetime region $V$ be bounded by spacelike hypersurfaces $\Sigma_1$ and $\Sigma_2$ and a lateral boundary $\mathcal B$. For two tangent perturbations, $d\omega=0$, so outward-normal-first orientation gives

$$\Omega_{\Sigma_2}(\delta_1,\delta_2) -\Omega_{\Sigma_1}(\delta_1,\delta_2) =-\mathcal F_{\mathcal B}(\delta_1,\delta_2),$$

where

$$\mathcal F_{\mathcal B} =\int_{\mathcal B}\omega(\delta_1, \delta_2).$$

Consequently the same bulk phase-space two-form can be used on every slice if $\mathcal F_{\mathcal B}=0$ for all allowed pairs. Iyer and Wald explain this hypersurface-dependence criterion in Iyer and Wald 1994, §3, eqs. (44)–(47), pp. 9–10, PDF.

If the flux is nonzero, there are two coherent interpretations. The bulk region is an open system whose presymplectic form changes, or boundary fields contribute a form $\Omega_{\mathcal B}$ such that the combined bulk-plus-boundary structure is conserved. Simply declaring the bulk integral conserved would discard the exchange.

Source–response pairing at a boundary

After the action has been supplemented by suitable boundary terms, its on-shell variation often takes the schematic form

$$\delta S_{\mathrm{on\ shell}} =\int_{\mathcal B} P_A\,\delta Q^A.$$

Here $Q^A$ are boundary configuration data and $P_A$ their responses. The boundary pullback of the presymplectic current then has the canonical structure

$$\omega_{\mathcal B}(\delta_1,\delta_2) =\delta_2 P_A\,\delta_1 Q^A -\delta_1 P_A\,\delta_2 Q^A,$$

up to tangential exact forms and the site’s ordering convention. Fixing $Q^A$ makes the flux vanish because every allowed perturbation has $\delta Q^A=0$. Fixing $P_A$ is a Neumann-type alternative. A mixed condition $P_A=P_A(Q)$ is flux-free only if its linearization is symmetric in field space—equivalently, if it derives locally from a boundary functional.

Harlow and Wu derive the boundary-compatible variational and symplectic structure in Harlow and Wu 2020, §§2.1–2.4, pp. 6–23, PDF.

Reflective, radiative, and dissipative choices

These labels describe flux properties, not merely coordinate formulas:

Boundary type	Allowed exchange	Phase-space consequence
Conservative reflective	Zero symplectic flux for allowed perturbations	Bulk phase space can be conserved; energy conservation also needs a time symmetry and an appropriate boundary energy
Radiative	Incoming and/or outgoing characteristic data are prescribed	Bulk symplectic structure generally exchanges flux with the exterior
Dissipative	Energy leaves according to a sign-controlled boundary law	Evolution can have an energy estimate, but bulk evolution is not Hamiltonian in isolation
Mixed conservative	Source and response obey a variational relation	Flux may vanish after an allowed boundary functional is included

Vanishing symplectic flux and vanishing stress-energy flux are related but not identical statements. Symplectic flux is bilinear in perturbations and governs the phase-space two-form. Energy flux uses a chosen generator and is quadratic or nonlinear. One must check the condition appropriate to the claim.

For instance, the scalar mixed condition $\partial_n\phi=W'(\phi)$ is symplectically flux-free, but its bulk energy flux contains $\partial_t\phi\,\partial_n\phi=\partial_tW(\phi)$. The bulk energy can change by energy stored in the boundary potential. A conserved total energy includes that boundary term with the sign fixed by the action and outward-normal convention. Likewise, externally time-dependent Dirichlet data have $\delta Q=0$ for variations within that prescribed-source phase space, yet the prescribed source can do work. Zero symplectic flux does not establish a time-translation symmetry.

At a null radiative boundary, allowing news generally produces physical flux; at a timelike wall, reflective conditions can remove it. At asymptotic AdS infinity, common Dirichlet conditions are reflective for the gravitational phase space after the appropriate renormalized boundary terms are fixed. These examples require their own asymptotic analysis and are not universal templates.

Corners and time-local conditions

Tangential exact terms in $\omega_{\mathcal B}$ integrate to cuts $S_1=\partial\Sigma_1$ and $S_2=\partial\Sigma_2$. They may be absorbed into a corner presymplectic potential or interpreted as boundary state variables. A condition stated only on each time slice is insufficient if it leaves an uncancelled corner variation when slices meet the boundary.

For evolution problems, boundary conditions also need analytic admissibility: they must control incoming characteristic fields and satisfy corner compatibility with initial data. Flux-free does not by itself imply a well-posed initial-boundary value problem.

Common pitfalls

Bulk closure does not eliminate boundary flux. It turns the change between slices into a boundary integral.

Reflective does not mean every field vanishes. It means the allowed source–response subspace has zero relevant flux.

A dissipative estimate is not a conserved symplectic form. It describes an open bulk system unless the environment is included.

Exercise: mixed scalar boundary condition

Suppose a scalar field has boundary source $Q=\phi$ and response $P=n^a\nabla_a\phi$. Show that the local mixed condition $P=W'(Q)$ gives zero symplectic flux for allowed perturbations.

Solution

Allowed perturbations satisfy

$$\delta P=W''(Q)\,\delta Q.$$

Therefore the boundary current is

$$\omega_{\mathcal B} =W''(Q)\delta_2Q\,\delta_1Q -W''(Q)\delta_1Q\,\delta_2Q=0.$$

The cancellation works because the response is the derivative of one boundary functional. For several fields the Jacobian $\partial P_A/\partial Q^B$ must be symmetric; an arbitrary mixed relation need not define a flux-free subspace.

References

• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, 146 (2020). 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, no. 2 (1994): 846–864. DOI; Open PDF.