Canonical Energy and Linearized-Gravitational Stability

Canonical energy measures a linearized gravitational disturbance by pairing it with its time translate in the symplectic form. A positive conserved energy can exclude growth, but only of the physical quantities it actually controls. Constraints, gauge, conserved charges, the hypersurface and its boundary terms are all part of that statement. Here we develop the criterion for asymptotically flat vacuum black holes with a nondegenerate bifurcate Killing horizon, concentrating on the axisymmetric sector. We distinguish an energy norm, a stronger boundedness estimate, an obstruction to relaxation, and a theorem establishing exponential growth.

Required background. PDEs and Green Functions for Gravitation supplies energy estimates and boundary flux identities.

Helpful background. Black-Hole Geometry and Mechanics fixes stationary horizons and charges; Formulations of General Relativity supplies constraints and gauge; Mathematical Relativity supplies the functional-analytic viewpoint; Black-Hole Perturbation Theory supplies the linearized solutions. The local quadratic-action and symplectic-current construction develops the variational starting point.

Canonical energy and its sign

Let $g$ be the stationary vacuum background, $t^a$ its Killing field normalized as a unit time translation at infinity, and $h_{ab}$ a real smooth solution of the linearized vacuum Einstein equation. Fix the Einstein–Hilbert potential, including its normalization $1/(16\pi G)$, and use the Hollands–Wald field-space sign:

$$\boldsymbol\omega(g;h_1,h_2) =\delta_1\boldsymbol\Theta(g;h_2) -\delta_2\boldsymbol\Theta(g;h_1), \qquad W_\Sigma(h_1,h_2)=\int_\Sigma\boldsymbol\omega(g;h_1,h_2).$$

This is the negative of the Iyer–Wald current convention used on Presymplectic Potentials and Currents. The sign and the order of arguments must be translated together. With a future-oriented spacelike slice, define

$$\mathcal E_\Sigma(h_1,h_2)=W_\Sigma(h_1,\mathcal L_t h_2), \qquad \mathcal E_\Sigma(h)=\mathcal E_\Sigma(h,h).$$

Because $t$ is a background symmetry, $\mathcal L_t h$ is another linearized solution. The current is closed on two such solutions. On complete exterior Cauchy slices extending from the same bifurcation surface $B$ to spatial infinity, the prescribed falloff and horizon conditions eliminate the side flux. Hence $\mathcal E_\Sigma$ is conserved. Its bilinear form is symmetric: the antisymmetric part is the integral of $\mathcal L_t\boldsymbol\omega=d(i_t\boldsymbol\omega)$, whose boundary integrals vanish. These are boundary-dependent statements, not consequences of antisymmetry alone Hollands and Wald 2013, arXiv v4, Eqs. (19), (24)–(28), pp. 13–14, and Eqs. (46)–(51), pp. 17–18, PDF.

A simple normalization check is an oscillator with $L=C(\dot q^2-\nu^2q^2)/2$, $C,\nu\gt0$. Its form is $W(q_1,q_2)=C(\dot q_1q_2-\dot q_2q_1)$, so

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

The last equality uses the equation of motion and the ordinary quadratic Hamiltonian $H=C(\dot q^2+\nu^2q^2)/2$. Canonical energy in this Hessian normalization is twice the coefficient of the quadratic physical energy. For a complex mode one must specify a Hermitian extension or work with its real and imaginary parts. Literally substituting $\mathcal L_t h=-i\omega h$ into the complex-bilinear expression $W(h,\mathcal L_t h)$ gives zero by antisymmetry; it does not measure a complex mode’s physical energy.

The physical space and its zero directions

Fix four spacetime dimensions for the explicit boundary formulas below. Assume a stationary asymptotically flat vacuum exterior with a compact bifurcation surface, surface gravity $\kappa\gt0$, and $K=t+\Omega_H\phi$ the future horizon Killing field. Here $\phi$ is the axial Killing field and $\Omega_H$ is the background horizon angular velocity. Axisymmetry means $\mathcal L_\phi h=0$; it allows $\mathcal L_t h$ to be replaced by $\mathcal L_K h$ in the horizon flux. The same construction uses all relevant axial symmetries in higher dimensions.

For a concrete smooth class, take constraint-satisfying initial data that extend smoothly across $B$ and vanish near spatial infinity. Fix the first-order charges,

$$\delta M=\delta P_i=\delta J=0.$$

Use a horizon-adapted Gaussian null gauge in which the background horizon remains null to first order, and impose $\delta\theta|_B=0$ and $\delta\varepsilon_B=0$. The first condition fixes the perturbed outgoing expansion; the second fixes the area element pointwise. The first law gives $\delta A=0$ from the charge conditions, allowing the area-element gauge. Linearized Raychaudhuri then propagates $\delta\theta=0$ along the horizon. For noncompact data, the four-dimensional asymptotic conditions also include the appropriate falloff and Regge–Teitelboim parity; smoothness by itself does not ensure convergence of $W_\Sigma$ Hollands and Wald 2013, arXiv v4, § 2.1, pp. 8–11, PDF.

Hollands and Wald place these data in a weighted initial-data Hilbert space and use a dense smooth domain $\mathcal T$ satisfying the constraints, zero-charge conditions and horizon gauge. The concrete class above lies within that domain; extensions to its completion require the source’s functional analysis. There are two distinct reductions:

• An admissible gauge variation $h=\mathcal L_\xi g$ has zero symplectic pairing with physical data when its constraint and boundary-charge terms vanish. Compact support away from the boundaries is a sufficient example. A nonzero boundary action must be checked, not discarded merely because the variation is a Lie derivative.
• A perturbation toward a stationary black hole can be stationary only up to a gauge transformation relative to the fixed background $t$. Such a direction is $\mathcal E$-orthogonal to $\mathcal T$. It is a degeneracy of the restricted form only when it also belongs to $\mathcal T$.

Thus let $\mathcal T'$ be $\mathcal T$ modulo its stationary degeneracies, including admissible pure-gauge directions. The form is well defined and nondegenerate on $\mathcal T'$. Nondegeneracy means that $\mathcal E(h,k)=0$ for every $k$ forces $[h]=0$; it does not mean that an indefinite form has no nonzero vectors with $\mathcal E(h,h)=0$ Hollands and Wald 2013, arXiv v4, Proposition 4, pp. 20–21, and § 4.1, p. 33, PDF.

A stationary family can have negative energy

For a smooth axisymmetric vacuum family $g(\lambda)$ in the prescribed horizon gauge, the second-variation identity is

$$\mathcal E_\Sigma(h) =\delta^2M-\Omega_H\delta^2J -\frac{\kappa}{8\pi G}\delta^2A.$$

Here $h=\partial_\lambda g|_0$, $\delta^2=\partial_\lambda^2|_0$, $A$ is the area of the fixed coordinate surface $B$, and $\kappa,\Omega_H$ are held at their background values. The individual second variations depend on the second-order extension; their combination depends only on $h$, because the difference of two extensions satisfies the linearized first law. Without axisymmetry there is also the term $-\Omega_HW_\Sigma(h,\mathcal L_\phi h)$ Hollands and Wald 2013, arXiv v4, Proposition 1, Eqs. (44)–(53), pp. 17–18, PDF.

For Schwarzschild with physical ADM mass $M(\lambda)=M_0+\alpha\lambda$, the area and background surface gravity are $A(\lambda)=16\pi G^2M(\lambda)^2$ and $\kappa_0=1/(4GM_0)$. Therefore

$$\delta^2M=0,\qquad \delta^2A=32\pi G^2\alpha^2,\qquad \mathcal E_\Sigma=-\frac{\alpha^2}{M_0}.$$

This has the dimension of energy and is negative when $\alpha\ne0$, but $\delta M=\alpha\ne0$ excludes it from $\mathcal T$. It is not an instability of Schwarzschild. Although every member of the family is stationary, its normalized Killing field varies in the required horizon gauge: $\mathcal L_t h=-\mathcal L_{\delta t}g$, not necessarily zero. Thus the negative value does not contradict the definition of $\mathcal E$ Hollands and Wald 2013, arXiv v4, Eqs. (56)–(64), pp. 19–20, PDF.

Future cuts and modified canonical energy

A complete slice retains radiation that has moved far outward. A slice $S_\tau$ ending instead on a finite cut $C_\tau$ of future null infinity and a cut $B_\tau$ of the future horizon excludes radiation already beyond those cuts. Let $\mathscr I^+_{12}$ and $\mathcal H^+_{12}$ denote only the boundary segments between cuts 1 and 2. Define $\Delta X=X_2-X_1$.

Slice and functional	Boundary endpoints	Balance
Raw $\mathcal E_\Sigma$	The same $B$ and spatial infinity	Conserved
Raw $\mathcal E_{S_\tau}$	Moving future cuts $B_\tau,C_\tau$	Symplectic flux; not generally monotone
Modified $\overline{\mathcal E}_{S_\tau}$	The same moving cuts, with their correction terms	Decreases by nonnegative radiative flux

Boundary fields, generators and measures

On $\mathcal H^+$ use an affine parameter $u$ with $u=0$ at $B$, $n=\partial_u$, and $K=\kappa u\,n$. Let $\mu_{AB}$ be the background cut metric and $h^H_{AB}$ the pullback of the metric perturbation. Indices on horizon tensors are raised with $\mu^{AB}$; the measure is $du\,dA_\mu$. The zero-expansion gauge gives the linearized shear

$$\delta\sigma_{AB}=\frac12\partial_u h^H_{AB}.$$

At $\mathscr I^+$ take $\widetilde g=\Omega^2g$ in a Bondi conformal frame with unit-sphere cuts, affine generator $\widetilde n=\partial_{\widetilde u}$ and $\widetilde t=a_\infty\widetilde n$, where $a_\infty\gt0$ is constant. The four-dimensional rescaled perturbation is $\widetilde h_{ab}=\Omega h_{ab}$, not $\Omega^2h_{ab}$. Assume it extends smoothly to $\mathscr I^+$ with the source’s trace and transversality conditions $\widetilde g^{ab}\widetilde h_{ab}=O(\Omega)$ and $\widetilde h_{ab}\widetilde n^a=O(\Omega)$. Its cut pullback is trace free. In this convention the perturbed news is

$$\delta\widetilde N_{AB} =-\partial_{\widetilde u}\widetilde h_{AB}.$$

The minus sign differs from the convention $N_{AB}=\partial_u C_{AB}$ on the radiation page; the squared flux agrees. Infinity indices use the unit-sphere metric and the measure is $d\widetilde u\,d\widetilde A$. These hypotheses are stronger than merely requiring a finite coordinate perturbation Hollands and Wald 2013, arXiv v4, Eqs. (88)–(96), pp. 25–27, PDF.

With these measures understood explicitly, write the integrated positive fluxes as

$$\begin{aligned} F_{\mathscr I,12} &=\frac1{16\pi G}\int_{\mathscr I^+_{12}} a_\infty\,\delta\widetilde N_{AB}\delta\widetilde N^{AB} \,d\widetilde u\,d\widetilde A,\\ F_{\mathcal H,12} &=\frac1{4\pi G}\int_{\mathcal H^+_{12}} \kappa u\,\delta\sigma_{AB}\delta\sigma^{AB}\,du\,dA_\mu. \end{aligned}$$

Both are nonnegative: the contractions are of real tensors with positive cut metrics, and the generator weights are positive to the future of $B$. Neither integral is an unspecified whole-future flux.

Why the cut terms are subtracted

The endpoint functionals are

$$\begin{aligned} \mathcal C_\tau[h] &=\frac1{32\pi G}\int_{C_\tau} a_\infty\,\widetilde h^{AB} \partial_{\widetilde u}\widetilde h_{AB}\,d\widetilde A,\\ \mathcal B_\tau[h] &=\frac1{32\pi G}\int_{B_\tau} \kappa u\,(h^H)^{AB}\partial_u h^H_{AB}\,dA_\mu. \end{aligned}$$

Their signs are not fixed. They arise from integrating by parts, not from an extra physical source. For example, the horizon current contains, with the cut contraction implicit,

$$\kappa u\,h'h'-h(\kappa h'+\kappa u\,h'') =2\kappa u\,h'h'-\partial_u(\kappa u\,hh').$$

Here primes mean affine derivatives. At infinity the corresponding identity is $\dot h\dot h-h\ddot h=2\dot h\dot h-\partial_{\widetilde u}(h\dot h)$. Consequently the outward symplectic fluxes are $W_{\mathscr I}=F_{\mathscr I,12}-\Delta\mathcal C$ and $W_{\mathcal H}=F_{\mathcal H,12}-\Delta\mathcal B$. Stokes’ theorem then gives

$$\begin{aligned} \Delta\mathcal E_S &=-F_{\mathscr I,12}-F_{\mathcal H,12} +\Delta\mathcal C+\Delta\mathcal B,\\ \overline{\mathcal E}_{S_\tau} &:=\mathcal E_{S_\tau}-\mathcal C_\tau-\mathcal B_\tau,\\ \Delta\overline{\mathcal E}_S &=-F_{\mathscr I,12}-F_{\mathcal H,12}\leq0. \end{aligned}$$

This is the modified-energy law of Hollands and Wald 2013, arXiv v4, Theorem 1, Eqs. (105)–(106), p. 28, PDF. There is a sign inconsistency in that version’s intermediate Eqs. (97) and (103): their printed plus endpoint differences do not follow from Eqs. (94) and (102) or agree with the final law. The integrations above give the consistent minus endpoint differences.

The horizon correction vanishes at $B$ because $K=0$ there. The two corrections also vanish when their respective shear or news vanish. To identify an initial modified energy with $\mathcal E_\Sigma$, one must take the limit $S_\tau\to\Sigma$ and verify these endpoint terms vanish; the compact-spatial-infinity data class supplies the infinity limit.

Schwarzschild master fields fix the normalization

For one real $\ell\geq2$ harmonic of the complete CPM or Zerilli–Moncrief field $\Psi(t,x)$, use $x=r_*$ and the corresponding Schwarzschild potential $V(x)$. The reduced action and its normalization give

$$S^{(2)}=\frac{C_\ell}{2}\int (\Psi_t^2-\Psi_x^2-V\Psi^2)\,dt\,dx, \qquad C_\ell=\frac{N_\ell}{64\pi G},\qquad N_\ell=\frac{(\ell+2)!}{(\ell-2)!}.$$

Thus $\Psi_{tt}=\Psi_{xx}-V\Psi$ and $W=C_\ell\int(\Psi_{1,t}\Psi_2-\Psi_{2,t}\Psi_1)\,dx$. On a finite fixed interval $[x_a,x_b]$, integration by parts yields

$$\begin{aligned} H_{[a,b]}&=\frac{C_\ell}{2}\int_{x_a}^{x_b} (\Psi_t^2+\Psi_x^2+V\Psi^2)\,dx,\\ \mathcal E^{\rm master}_{[a,b]} &=2H_{[a,b]}-C_\ell[\Psi\Psi_x]_{x_a}^{x_b}. \end{aligned}$$

This is the energy obtained from the stated reduced symplectic potential. Equality to a gravitational energy on a finite region requires the corresponding gravitational boundary convention; identical bulk equations alone do not establish it. On a domain where the bracket vanishes, the Hollands–Wald normalization is $2H$, not $H$.

The same factor follows directly from the gravitational radiation. For a real tensor harmonic $T_{AB}$ with $\int T_{AB}T^{AB}d\Omega=N_\ell/2$, the outgoing cut field is $\widetilde h_{AB}=\psi_\infty T_{AB}$. At the horizon $r_H=2GM$, the ingoing metric is $h^H_{AB}=\pm r_H\psi_H T_{AB}$, with plus for even and minus for odd parity. Use $\mu^{AB}=r_H^{-2}\Omega^{AB}$ and $dA_\mu=r_H^2d\Omega$. In Killing time $v$, $K=\partial_v=\kappa u\partial_u$, so $du=\kappa u\,dv$ and $\delta\sigma^{(K)}=\kappa u\,\delta\sigma^{(n)}$. Both the weight and the measure are needed to obtain

$$\frac{dF_{\mathscr I}}{du_\infty} =2C_\ell(\partial_{u_\infty}\psi_\infty)^2,\qquad \frac{dF_{\mathcal H}}{dv} =2C_\ell(\partial_v\psi_H)^2.$$

Here $a_\infty=1$ and $u_\infty$ is the normalized retarded time. These are twice the physical quadratic Bondi and shear fluxes given by the metric-reconstruction and radiation dictionary. The radius powers cancel in the horizon integral. For either cut the correction reduces to $C_\ell\psi\dot\psi$ in its normalized Killing time. This checks the sign, angular norm and factor two without identifying a finite-cut horizon shear flux with an instantaneous horizon mass derivative.

What positivity proves

If $\mathcal E$ is positive on $\mathcal T'$, then $\|[h]\|_{\mathcal E}^2:=\mathcal E_\Sigma(h)$ is a conserved norm. It excludes growth measured by that norm on this domain. It does not automatically control every component or derivative of a gauge-fixed metric. To prove a bound in some specified phase-space norm $\mathcal X$, one needs a uniform lower bound on the evolving, reduced data:

$$c\|[h](\Sigma_\tau)\|_{\mathcal X}^2 \leq\mathcal E_{\Sigma_\tau}(h) =\mathcal E_{\Sigma_0}(h),\qquad c\gt0.$$

The constant must be independent of $\tau$. An upper estimate on the initial energy in the chosen data norm then relates the bound to the initial size. Constraint propagation, an admissible gauge and well-posed evolution must keep the solution in the declared space. Positivity on a dense smooth domain does not alone provide continuity of evolution in a different completion.

For future-cut slices the corresponding implication is instead

$$c\|[h](S_\tau)\|_{\mathcal X}^2 \leq\overline{\mathcal E}_{S_\tau}(h) \leq\overline{\mathcal E}_{S_0}(h).$$

This requires an independent coercivity estimate for the modified functional, or a proved comparison controlling its cut terms. The raw complete-slice bound cannot simply be substituted here. Neither inequality alone proves decay, an integrated local energy estimate, or nonlinear stability Hollands and Wald 2013, arXiv v4, § 4.2, pp. 34–35, PDF.

Negative energy and growth

Suppose $h\in\mathcal T$ has $\mathcal E_\Sigma(h)\lt0$ and the initial-cut limit just described holds. Then every later modified energy is at most this strictly negative value. If $h$ were to approach a stationary perturbation in $\mathcal T$ strongly enough that the raw energy converged and both cut corrections tended to zero, the limiting modified energy would be zero, a contradiction. This excludes that kind of relaxation. Pointwise news decay alone need not control the product $\widetilde h\,\partial_{\widetilde u}\widetilde h$.

This argument does not prove unbounded growth or produce an exponential mode. The original source explicitly identifies that gap Hollands and Wald 2013, arXiv v4, § 4.2, pp. 34–35, PDF. A negative-energy oscillator with Lagrangian $-(\dot q^2-\nu^2q^2)/2$ still satisfies $\ddot q+\nu^2q=0$. It is not a model of vacuum gravity; it shows why the sign of an arbitrarily chosen conserved quadratic form is not a growth theorem.

The additional spectral argument

Prabhu and Wald give a stronger result under additional structure. Assume the asymptotically flat stationary-axisymmetric vacuum background admits a $t$- or $(t,\phi)$-reflection isometry: it reverses the stationary direction and, in the second case, the axial direction. Choose a reflection-invariant maximal slice, whose extrinsic-curvature trace is zero, ending smoothly on compact $B$. Use their completely gauge-fixed, axisymmetric smooth-data space $\mathcal V^\infty$: the linearized constraints and zero ADM charges hold, both perturbed null expansions and the area-element variation vanish at $B$.

Here is the meaning of the smooth-data requirement. Let $s_{ij}$ be the induced background metric, $q_{ij}=\delta s_{ij}$, and $p_{ij}$ the perturbation of the ADM momentum density divided by the background volume density, with indices lowered using $s$. Fix reference units to express the data and coordinates dimensionlessly. Let $D$ and $d\mu_s$ be the spatial covariant derivative and volume measure, and let $\rho$ be a smooth positive weight equal to 1 near $B$ and asymptotic to the radial coordinate. For $\delta X=(p,q)$, define

$$\|\delta X\|_{W^k_\rho}^2 =\sum_{j=0}^k\int_\Sigma\rho^{2j} \bigl(|D^jp|_s^2+|D^jq|_s^2\bigr)\,d\mu_s.$$

For each $k$, take the closure of smooth data that vanish near spatial infinity in this norm; require membership in all these spaces. Fixed positive normalization factors do not change this domain. It is a mathematical data norm, not the canonical energy. Finally, in the direct-sum $L^2$ inner product of $(p,q)$, project the constrained data orthogonally to the remaining allowed gauge directions. Their generators preserve the horizon conditions and approach translations or rotations at infinity. The precise boundary conditions and commuting constraint/gauge projections are constructed in Prabhu and Wald 2015, arXiv v2, §§ 2–3, Eqs. (3.3)–(3.4), (3.14)–(3.16) and (3.24)–(3.30), pp. 11–21, PDF. Thus $\mathcal V^\infty$ is not the space of arbitrary smooth coordinate perturbations; the elliptic projection construction is outside this page’s scope.

Reflection splits data into odd $P$ and even $Q$ parts and removes cross terms from $\mathcal E=\mathscr K(P,P)+\mathscr U(Q,Q)$. This reflection parity is not spherical-harmonic parity. Their Theorem 1 proves the kinetic form $\mathscr K$ positive definite on $\mathcal V^\infty_{\rm odd}$. In the fixed gauge write $\dot Q=\mathbf K P$ and $\dot P=-\mathbf UQ$. Complete the range of $\mathbf K$ in

$$\langle\mathbf K P_1,\mathbf K P_2\rangle_{\mathscr H} =\mathscr K(P_1,P_2).$$

Then $\ddot Q=-\mathbf A Q$, with $\mathbf A=\mathbf K\mathbf U$ symmetric on the dense domain $\operatorname{dom}\mathbf A=\mathbf K[\mathcal V^\infty_{\rm odd}]$ and $\langle Q,\mathbf A Q\rangle_{\mathscr H}=\mathscr U(Q,Q)$. Not every even perturbation belongs to $\mathscr H$: the range restriction is essential. The source uses a self-adjoint extension and spectral projections; it does not assume without proof that the original operator is essentially self-adjoint or bounded below Prabhu and Wald 2015, arXiv v2, Theorem 1 and §§ 5–6, pp. 25–26 and 31–33, PDF.

For $P_0=0$ and $Q_0=\mathbf K P'_0$ with smooth admissible odd $P'_0$, the sufficient condition $\mathscr U(Q_0,Q_0)\lt0$ gives exponential growth in $\mathscr H$. Indeed, the negative quadratic expectation forces nonzero spectral weight on some bounded interval $[-b^2,-a^2]$, $b\gt a\gt0$. Since $P_0=0$ gives $\dot Q(0)=0$, on that subspace evolution multiplies the data by $\cosh(t\sqrt{-\lambda})$, with $\lambda$ the negative spectral parameter. If $\Pi$ is the projection onto this interval, then

$$\|Q(t)\|_{\mathscr H} \geq\frac12 e^{at}\|\Pi Q_0\|_{\mathscr H},\qquad t\geq0.$$

The kinetic energy and minus the potential energy have exponential lower growth bounds as well:

$$\|\dot Q\|_{\mathscr H}^2=\mathscr K(P,P),\qquad -\mathscr U(Q,Q)=\mathscr K(P,P)-\mathcal E.$$

The derivative of the negative-spectrum evolution grows like a hyperbolic sine, and $\mathcal E$ is conserved. These quadratic forms are gauge invariant, so the growth is not a gauge artifact. These data are the time derivative of data $(P'_0,0)$; this explains the time-derivative qualification in the theorem. The source also gives a Rayleigh–Ritz bound on the growth rate Prabhu and Wald 2015, arXiv v2, Proposition 6.2 and Theorem 2, pp. 35–36, PDF. The estimate above retains the spectral restriction and the factor $1/2$; the intermediate Eq. (6.18) in that version has inconsistent integration limits, spectral signs and exponential prefactors.

This is a spectral growth result, not automatically a discrete normal-mode eigenfunction. It neither exhibits a negative direction for every black hole nor removes the restriction to the admitted symmetry and data sector.

Relation to linearized Einstein stability

Canonical energy organizes constraints, charges and fluxes; it does not replace the estimates that control trapping, recover a metric from curvature, or establish late-time decay. For Schwarzschild, the linearized-Einstein theorem proves boundedness and decay toward linearized Kerr using its specified characteristic data, weighted norms, gauge normalizations and reconstruction Dafermos, Holzegel, and Rodnianski 2019, Theorems 3–4 and Corollary 10.3, pp. 111–116. The boundedness and decay page separates those hypotheses from scalar-wave and mode-stability statements.

For nonaxisymmetric Kerr perturbations, replacing $t$ by the horizon generator improves the horizon sign but spoils the analogous infinity sign; keeping $t$ allows superradiant horizon flux. One cannot extend the axisymmetric positive-flux argument unchanged. Extremal horizons have no regular nondegenerate bifurcation surface of the kind assumed here; their Aretakis behavior needs a different analysis. Full nonlinear evolution remains in Black-Hole Dynamics, with the linear-to-nonlinear interface explaining the additional analytic requirements.

Exercises

Does the second-order path change the energy?

Replace the Schwarzschild family by $M(\lambda)=M_0+\alpha\lambda+\beta\lambda^2/2$. Compute $\delta^2A$ and $\mathcal E_\Sigma$. Why does $\beta$ cancel, and when is this a zero-charge perturbation?

Solution

With physical mass and $A=16\pi G^2M^2$,

$$\delta^2A=32\pi G^2(\alpha^2+M_0\beta),\qquad \mathcal E_\Sigma =\beta-\frac{\alpha^2+M_0\beta}{M_0} =-\frac{\alpha^2}{M_0}.$$

The background $\kappa_0=1/(4GM_0)$ is fixed in the identity. Changing $\beta$ changes the second-order extension, not the first-order field. Its contribution cancels by the first law. Here $\delta M=\alpha$, so the charge condition requires $\alpha=0$; this mass-family direction then has no nonzero first-order mass perturbation.

Positive but not coercive

On $\ell^2$, define $Q(a)=\sum_{n=1}^\infty |a_n|^2/n^2$. Is $Q$ positive definite? Can $c\|a\|_{\ell^2}^2\leq Q(a)$ hold with one $c\gt0$ for all $a$? Does failure of this estimate prove unstable evolution?

Solution

Every nonzero sequence has a nonzero entry with positive weight, so $Q(a)\gt0$. But the unit sequence $e_n$ has $\|e_n\|_{\ell^2}=1$ and $Q(e_n)=n^{-2}\to0$. No uniform $c\gt0$ exists. Equivalently, $a=ne_n$ has $Q(a)=1$ and $\|a\|_{\ell^2}=n$. A conserved positive $Q$ therefore does not by itself bound this stronger norm. It does not prove instability either: the stationary evolution $\dot a=0$ preserves both norms. An evolution equation and a suitable estimate are additional information.

A raw energy that increases while radiation leaves

For the flat master equation on $x\leq0$, take the real outgoing pulse $\Psi(t,x)=F(t-x)$ with $F(s)=A/[1+(s/T)^2]$, $A\ne0$, $T\gt0$, and action coefficient $C\gt0$. It has finite energy on this half-line and no boundary term at $x=-\infty$. Show that $\mathcal E_{\rm raw}=2H+CF(t)F'(t)$ and $\overline{\mathcal E}=\mathcal E_{\rm raw}-CF(t)F'(t)=2H$. Compare their derivatives at $t=2T$.

Solution

Here $\Psi_x=-F'$ and $H=C\int_t^\infty F'(s)^2\,ds$, so $\dot H=-CF'(t)^2$. The finite-interval boundary identity gives the stated plus corner term in $\mathcal E_{\rm raw}$. Thus

$$\dot{\mathcal E}_{\rm raw} =C(FF''-F'^2),\qquad \dot{\overline{\mathcal E}}=-2CF'^2.$$

At $2T$, $F=A/5$, $F'=-4A/(25T)$ and $F''=22A/(125T^2)$. Therefore

$$\dot{\mathcal E}_{\rm raw} =\frac{6CA^2}{625T^2}\gt0,\qquad \dot{\overline{\mathcal E}} =-\frac{32CA^2}{625T^2}\lt0.$$

The changing corner contribution permits raw-energy increase despite positive outgoing radiation. This exactly solvable finite-boundary model tests the integration-by-parts logic; its boundary is not a black-hole horizon or null infinity.

References

• Dafermos, Mihalis, Gustav Holzegel, and Igor Rodnianski. “The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations.” Acta Mathematica 222 (2019): 1–214. DOI.
• 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.
• Prabhu, Kartik, and Robert M. Wald. “Black Hole Instabilities and Exponential Growth.” Communications in Mathematical Physics 340 (2015): 253–290. DOI. Open PDF, arXiv v2. Page locators above refer to this arXiv edition.