Quadratic Actions, Symplectic Currents, and Canonical Energy

The first variation of an action gives the background equation; its second variation gives the linear perturbation operator. Antisymmetrizing two variations of the boundary potential then produces a conserved current, while pairing a perturbation with its time translate produces canonical energy. These constructions organize dynamics, flux, and stability, but only after the allowed boundary conditions and gauge directions are fixed.

Helpful background. Black-Hole Geometry and Mechanics fixes the background and horizon data used in applications; PDEs and Green Functions for Gravitation supplies the differential-operator viewpoint; Formulations of General Relativity separates covariant equations, constraints, and evolution; Presymplectic Potentials and Currents prepares the variational language used on this page.

Quadratic actions

Let $\phi^A$ denote the metric and any matter fields, and let the Lagrangian $n$-form satisfy

$$\delta\mathbf L =\mathbf E_A\,\delta\phi^A+d\boldsymbol\Theta(\phi;\delta\phi).$$

For a background solution $\bar\phi$ and an admissible perturbation $\gamma$, define the quadratic action by the Hessian

$$S^{(2)}[\gamma] =\frac12\left. \frac{d^2}{d\varepsilon^2}S[\bar\phi+\varepsilon\gamma] \right|_{\varepsilon=0}.$$

The background equation and the boundary conditions make the first variation vanish. Varying $S^{(2)}$ with respect to $\gamma$ yields the linearized equation $\mathbf E_A^{(1)}[\gamma]=0$, together with its own boundary term. This is the clean reason that the linearized operator is formally self-adjoint: the two slots of a Hessian commute, up to the boundary terms created by integration by parts.

The qualifier admissible matters. The Einstein–Hilbert bulk term alone does not define every Dirichlet, null, asymptotic, or finite-region variational problem. Boundary and corner terms, falloff, and which fields are held fixed must be specified before the quadratic form is a differentiable action. A boundary term may leave the bulk equation unchanged while altering surface momenta or flux.

From the potential to symplectic flux

For two commuting linearized variations, use the Hollands–Wald field-space sign convention appropriate to the canonical energy below:

$$\boldsymbol\omega(\bar\phi;\gamma_1,\gamma_2) =\delta_1\boldsymbol\Theta(\bar\phi;\gamma_2) -\delta_2\boldsymbol\Theta(\bar\phi;\gamma_1).$$

The current is an $(n-1)$-form on spacetime and is antisymmetric in its two field-space arguments. Applying $\delta_1\delta_2-\delta_2\delta_1$ to the Lagrangian gives

$$d\boldsymbol\omega =\delta_2\mathbf E_A\,\gamma_1^A -\delta_1\mathbf E_A\,\gamma_2^A.$$

It is therefore closed when the background and both perturbations satisfy their respective equations. The current here is the negative of the Iyer–Wald convention used on Presymplectic Potentials and Currents. The variational construction and ambiguities are unchanged, but the current, its off-shell identity and the energy pairing must use one consistent sign. The present convention is Hollands and Wald 2013, arXiv v4, Eq. (19), p. 13, PDF.

For an oriented hypersurface $\Sigma$,

$$W_\Sigma(\gamma_1,\gamma_2) =\int_\Sigma\boldsymbol\omega(\bar\phi;\gamma_1,\gamma_2).$$

If two slices bound a region with side boundary $\mathcal B$, Stokes’ theorem and outward-normal-first orientation give their difference as the negative of the flux through $\mathcal B$ when $d\boldsymbol\omega=0$. Thus current conservation does not imply slice independence in a radiating black-hole exterior: flux can cross the future horizon and null infinity.

Gauge degeneracy and the meaning of norm

For a gauge vector $\xi$ that vanishes suitably at every boundary, $\gamma_\xi=\mathcal L_\xi\bar g$ is a degeneracy of $W_\Sigma$ on the linearized constraint surface. If $\xi$ has nonzero boundary action, the same calculation leaves a surface term that may be a charge. “Pure gauge” is therefore a property of a phase space and its boundaries, not of the local formula $\mathcal L_\xi\bar g$ alone.

Because $W_\Sigma$ is antisymmetric, it is not a positive norm. After complexifying a mode space, one may form a Hermitian product from $-iW_\Sigma(\gamma^*,\gamma)$, but positivity depends on the frequency sector, boundary conditions, and background. Quasinormal modes create additional difficulties because their usual spatial profiles are not square-integrable on ordinary Cauchy slices. A statement that the symplectic product “normalizes a mode” must name the domain, regularization, and surface.

There are also potential ambiguities: $\boldsymbol\Theta\mapsto\boldsymbol\Theta+d\mathbf Y$ changes $\boldsymbol\omega$ by an exact spacetime form. Closed-surface integrals may be unchanged, while a hypersurface with boundary acquires a corner term. Fixing the potential convention and boundary counterterms is therefore part of defining any numerical canonical-energy value.

Canonical energy on a stationary background

Let $t^a$ be the stationary Killing field normalized by the chosen asymptotic time. The canonical energy of a real linearized solution is

$$\mathcal E_\Sigma(\gamma) =W_\Sigma(\gamma,\mathcal L_t\gamma).$$

For a real oscillator with $L=C(\dot q^2-\nu^2q^2)/2$, $C,\nu\gt0$, the potential is $\Theta=C\dot q\,\delta q$. Our sign gives

$$W(q_1,q_2)=C(\dot q_1q_2-\dot q_2q_1),\qquad \mathcal E(q)=C(\dot q^2-q\ddot q) =C(\dot q^2+\nu^2q^2)=2H.$$

The equation of motion gives the second equality. $H$ is the ordinary quadratic Hamiltonian, whereas $\mathcal E$ uses second-variation normalization. For $q=Ae^{-i\nu t}$, the positive-frequency Hermitian pairing is $-iW(q^*,q)=2C\nu|A|^2$. Reversing the current without reversing the associated pairing would reverse both signs. For a field on a finite interval, integration by parts can additionally leave a boundary term; the canonical-energy stability page derives the corresponding master-field identity.

It is conserved only when the net symplectic flux through the boundaries vanishes; otherwise a balance law relates its change to horizon and infinity fluxes. It is a global quadratic functional, not a local gravitational stress–energy tensor. Parameter variations toward nearby stationary black holes and admissible gauge perturbations can lie in its degeneracy space.

Hollands and Wald formulate an energy criterion for axisymmetric perturbations of stationary vacuum black holes with a bifurcate Killing horizon. On the subspace satisfying the linearized constraints, suitable horizon gauge conditions, and vanishing first-order ADM charges, positivity and nondegeneracy of canonical energy modulo gauge and stationary directions give a conserved energy norm. Controlling a specified stronger metric norm requires coercivity and well-posed evolution in a suitable gauge. Negative canonical energy obstructs sufficiently strong stationary relaxation; the positive-flux argument alone is not a proof of exponential growth Hollands and Wald 2013, arXiv v4, § 4.2, pp. 34–35, PDF. The stability criterion and its spectral extension state the distinct conserved-slice, future-cut and exponential-growth hypotheses.

The qualifications prevent common overreach. Positivity in a restricted axisymmetric sector is not a proof for arbitrary nonaxisymmetric data. Thermodynamic instability of a black-hole family need not supply an allowed negative-energy perturbation with the required charge conditions. Conversely, a negative value computed with an unremoved gauge or corner contribution is not physical evidence of an instability.

Checks, boundaries, and limitations

Three checks should accompany the calculation. Exchange the two perturbations and verify antisymmetry of $\boldsymbol\omega$. Differentiate it and recover the two linearized equations with the displayed sign. Then evaluate a compactly supported gauge variation and confirm that only constraints and surface terms remain. Dimensional analysis gives $\mathcal E$ the dimension of energy once the Killing time and $G$ normalization are restored.

Exact background construction remains in Exact Solutions of Einstein’s Equations, including the stationary Killing normalization and horizon geometry. Full finite-amplitude evolution remains in Black-Hole Dynamics. Canonical energy is a powerful linear stability diagnostic, but applying its sign outside the proven phase space does not establish a nonlinear endpoint.

Exercise: when is a gauge direction degenerate?

Let $\gamma$ satisfy the linearized equations and constraints, and let $\gamma_\xi=\mathcal L_\xi\bar g$. Explain why $W_\Sigma(\gamma,\gamma_\xi)$ vanishes for compactly supported $\xi$, and why the same conclusion can fail if $\xi$ approaches a nonzero asymptotic symmetry.

Solution

Diffeomorphism covariance converts the symplectic current with one gauge slot into a term proportional to the linearized constraints plus an exact spacetime derivative. The constraint term vanishes for $\gamma$, and the integral of the exact term becomes an integral over $\partial\Sigma$. If $\xi$ has compact support away from that boundary, the surface integral is zero, so the gauge direction lies in the kernel of the presymplectic form.

If $\xi$ tends to an asymptotic time translation, rotation, or another allowed boundary symmetry, the surface term need not vanish; it is the variation of the corresponding charge under appropriate integrability conditions. Such a transformation is not discarded as a redundant gauge direction on that phase space. The boundary behavior, not just the local Lie-derivative form, decides the answer.

References

• Hollands, Stefan, and Robert M. Wald. “Stability of Black Holes and Black Branes.” Communications in Mathematical Physics 321 (2013): 629–680. DOI. Open PDF, arXiv v4. Page locators above refer to this arXiv edition.
• 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.