Boundedness and Decay of Linear Fields

Mode stability excludes a growing separated solution. Boundedness and decay ask more: what happens to an arbitrary wave packet, and which initial-data norm controls it for all future time? On every fixed subextremal Kerr exterior, a massless scalar has uniformly bounded nondegenerate energy, integrated local energy decay, and polynomial decay with additional weighted regularity. The norm, the measured region and the initial derivative count are different parts of that statement. This page makes those distinctions explicit, develops the estimate mechanism and its counterexamples, and separately explains what changes for gravitational perturbations of Schwarzschild. It does not reproduce the complete global Kerr PDE proof or infer nonlinear stability from a scalar estimate.

Required background. PDEs and Green Functions for Gravitation supplies energy methods, Morawetz estimates, and Sobolev decay.

Helpful background. Black-Hole Geometry and Mechanics fixes redshift, trapping, and asymptotic regions; Formulations of General Relativity distinguishes scalar and tensor systems; Mathematical Relativity supplies normed theorem statements; Black-Hole Perturbation Theory supplies mode and scattering inputs.

Scalar energy on the Kerr exterior

For this sustained estimate use geometrized units $G=c=1$: $M$ and $a=J/M$ are lengths. Fix $M\gt0$ and $|a|\lt M$, and consider a real massless scalar satisfying

$$\Box_{g_{M,a}}\psi=0,\qquad r\ge r_+=M+\sqrt{M^2-a^2}.$$

Complex solutions are treated by summing the energies of their real and imaginary parts. No axisymmetry or fixed angular multipole is imposed. Restoring physical mass replaces $M$ by $GM$; in particular the dimensionless late-time variable below is $\tau/(GM)$, not $\tau/M$ with a physical mass.

Choose a smooth spacelike slice $\Sigma_0$ crossing the future horizon, avoiding the past horizon, and agreeing with a Boyer–Lindquist $t$ slice far out. Its translates under the stationary Killing field $T$ are $\Sigma_\tau$. Thus the slice geometry and lapse are stationary. Initially take $\psi$ and $n_0\psi$ smooth and compactly supported, where $n_\tau$ is the future unit normal. The future horizon is an outflow boundary of this Cauchy evolution, not an additional reflecting or incoming boundary condition. Write

$$\mathcal D_{0,\tau}=\bigcup_{0\le s\le\tau}\Sigma_s, \qquad dV_g=\alpha\,d\tau\,d\Sigma_\tau, \qquad \alpha=(-g^{-1}(dt_*, dt_*))^{-1/2},$$

where $t_*=\tau$ labels the slices. The lapse $\alpha$ is bounded above and away from zero for this fixed foliation. It cannot simply be omitted when converting a spacetime integral into a time integral of slice norms.

Choose a smooth stationary future timelike redshift field $N$, equal to $T$ sufficiently far out, and uniformly comparable to $n_\tau$. For any real scalar $f$, define

$$\begin{aligned} Q_{\mu\nu}[f]&=\nabla_\mu f\nabla_\nu f -\frac12g_{\mu\nu}\nabla^\rho f\nabla_\rho f,\\ J^N_\mu[f]&=Q_{\mu\nu}[f]N^\nu,\qquad E_N[f](S)=\int_S J^N_\mu[f]n_S^\mu\,dS. \end{aligned}$$

On the spacelike slices this is a positive first-derivative energy. It is uniformly comparable to the integral of $|n_\tau f|^2+|\nabla_{\Sigma_\tau}f|^2$, including at the horizon. This is what nondegenerate means here. It is not a conserved energy: $N$ is generally not Killing. The stationary $T$ energy is conserved with its fluxes, but is not positive throughout Kerr’s ergoregion. The local current calculations are developed on Energy Estimates and Test-Field Stability.

Boundedness and the two escape channels

For each fixed integer $j\ge1$, set

$$\mathcal E_j(\tau)= \sum_{i=0}^{j-1}M^{2i}E_N[N^i\psi](\Sigma_\tau).$$

The factors of $M$ put every summand in the same units. The wave equation determines the higher normal derivatives from the Cauchy data, so these are initial-data norms, not extra freely prescribed data. In particular, $\mathcal E_1=E_N[\psi]$ controls one derivative; $\mathcal E_2$ controls two. Elliptic estimates identify the higher norms with the corresponding geometric Sobolev norms on these stationary, spatial-end slices.

The boundedness theorem gives

$$\mathcal E_j(\tau)\le C_j\mathcal E_j(0),\qquad \tau\ge0.$$

There is no derivative loss in this boundedness statement: the same order appears on both sides. Constants depend on the fixed background, foliation, multiplier and $j$, but not on time or the solution. No uniform constant as $|a|\to M$ is asserted. Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Theorems 3.1–3.2, Eqs. (23)–(29), pp. 15–16, PDF.

The same first-order theorem bounds flux through the whole future horizon and whole future null infinity separately. On the horizon let $K=T+\Omega_H\partial_\phi$, with $\Omega_H=a/(r_+^2+a^2)$, and normalize its parameter $v_H$ by $K v_H=1$. Let $dA_H$ be the area measure of a horizon section. At infinity let $u$ be normalized retarded time and $\mathcal R\psi=\lim_{r\to\infty}r\psi$ the radiation field, with the limit interpreted in the radiation-energy sense when needed. Then

$$\begin{aligned} \mathcal F_H^N&=\int_{\mathcal H^+\cap\mathcal D_{0,\infty}} Q[\psi](N,K)\,dv_H\,dA_H,\\ \mathcal F_\infty^T&=\int_{\mathscr I^+\cap\overline{\mathcal D_{0,\infty}}} |\partial_u\mathcal R\psi|^2\,du\,d\Omega,\\ \mathcal F_H^N+\mathcal F_\infty^T&\le C\mathcal E_1(0). \end{aligned}$$

Here $d\Omega$ is unit-sphere area. The $N$ horizon flux is nonnegative; it is not the possibly negative superradiant $T$ horizon flux. These bounds follow by adding the two inequalities with an enlarged constant. Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Theorem 3.1, Eqs. (21)–(22), p. 15, PDF.

A finite slab between $\Sigma_0$ and $\Sigma_\tau$ has no finite segment of null infinity: both slices end at spatial infinity. A flux between finite null-infinity cuts requires a different foliation. This will matter when we ask the energy itself to decay.

Integrated decay and trapping loss

Boundedness permits energy to remain in a fixed region forever. Integrated local energy decay, abbreviated ILED, bounds how long it can remain there. The derivatives and weight in that integral must be specified.

For this fixed Kerr background choose an interval $[r_A,r_B]$, outside the horizon and containing the radial projection of the trapped null geodesics. It may be enlarged slightly, and is chosen to include $3M$. Define

$$\zeta(r)=\left(1-\frac{3M}{r}\right)^2 \mathbf1_{\{r\notin[r_A,r_B]\}}.$$

For Schwarzschild take $[r_A,r_B]=\{3M\}$, so $\zeta=(1-3M/r)^2$. For Kerr the interval represents the photon region, not one photon sphere. This deliberately coarse physical-space weight vanishes over a region; the frequency-localized theorem resolves trapping more precisely.

Let $Z=\partial_r$ at fixed Boyer–Lindquist time and angles, and let $Z_*$ be the regular radial coordinate vector in horizon-crossing Kerr-star coordinates. Choose a smooth radial cutoff $\chi$ equal to one near and outside the trapping interval, and zero near the horizon. Put $\widetilde Z=\chi Z+(1-\chi)Z_*$. The angular derivative $\not\nabla$ uses the positive induced metric of the constant-$t_*,r$ two-spheres. These definitions follow Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, §§ 2.2.3–2.2.6, pp. 12–13, PDF.

For each fixed finite $R\gt r_B$, a consequence of the theorem is

$$\begin{aligned} \mathcal M_R[\psi] &=\int_{\mathcal D_{0,\infty}\cap\{r\le R\}} \left[|\widetilde Z\psi|^2+ \zeta\bigl(|T\psi|^2+|\not\nabla\psi|^2\bigr) +M^{-2}|\psi|^2\right]dV_g,\\ \mathcal M_R[\psi]&\le C_R M\mathcal E_1(0). \end{aligned}$$

This follows from Eq. (20): fix any positive radial-weight exponent there, restrict to $r_+\le r\le R$, and bound its positive powers of $r$ above and below. Restoring the length scale supplies the factor $M$ on the right. $C_R$ depends on $R/M$; the display is not uniform as $R\to\infty$. The first-order radial derivative remains controlled at trapping, while the time and angular terms degenerate. Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Theorem 3.1, Eq. (20), p. 15, PDF.

Compact support fixes the otherwise invisible asymptotic constant. For finite derivative-energy data with a nonzero asymptotic constant $\psi_\infty$, replace the zeroth-order term by $M^{-2}|\psi-\psi_\infty|^2$. A constant solution has zero stress-energy; it immediately disproves an estimate of $|\psi|^2$ by derivative energy alone without this qualification.

One can instead control a nondegenerate local integral at the cost of an extra derivative of the data. For example,

$$\begin{aligned} \mathcal L_R[\psi] &=\int_{\mathcal D_{0,\infty}\cap\{r\le R\}} \left[|n_\tau\psi|^2+|\nabla_{\Sigma_\tau}\psi|^2 +M^{-2}|\psi|^2\right]dV_g,\\ \mathcal L_R[\psi]&\le C_R M\mathcal E_2(0). \end{aligned}$$

To see the derivative count, take $j=2$ in Eq. (25). Its unweighted first-derivative terms, together with Eq. (20), control the displayed integrand on the compact radial region; its right-hand side is the energy of $\psi$ and $N\psi$. Uniform frame comparison and the bounded lapse give the stated norm. This is a sufficient one-derivative loss, not a claim that one full derivative is the optimal loss. Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Theorem 3.2, Eq. (25), p. 16, PDF.

Why is some change necessary? A high-frequency packet can follow a trapped ray for any prescribed finite observation time if its frequency is chosen high enough. The third exercise turns that fact into a contradiction to a uniform nondegenerate bound using only $\mathcal E_1(0)$. It does not claim that one exact wave remains localized forever.

Which energy decays?

An outgoing wave can leave every bounded region without losing its total energy on a complete spatial slice. In flat space, $f(x-t)$ with $f$ smooth and compactly supported carries the same total one-dimensional wave energy at every time, even though its local energy eventually vanishes on each fixed interval. The outgoing part of a black-hole solution raises the same distinction.

For quantitative decay choose a fixed sufficiently large interface radius $R_0$, beyond the initial support. Replace the outer part of $\Sigma_\tau$ by an outgoing null hypersurface $\mathcal C_\tau$ from its boundary two-sphere. Write the resulting spacelike/null surface as

$$\Gamma_\tau=(\Sigma_\tau\cap\{r\le R_0\})\cup\mathcal C_\tau.$$

Fix a smooth, caustic-free stationary outgoing optical family extending from that interface to null infinity, with its asymptotically flat normalization. Its optical function satisfies $g^{-1}(du, du)=0$ and $Tu=1$, and $\mathcal C_\tau$ is $u=\tau$. Smoothness and absence of caustics are part of this foliation choice, not consequences of $Tu=1$ alone. On rotating Kerr, do not set $u=t-r_*(r)$ using only the Boyer–Lindquist tortoise coordinate: those surfaces are not null away from the axis.

On a null branch there is no unit normal. Use its future generator $L=d/d\lambda$ together with its parameter and area measure, so that

$$\mathscr E_N[f](\Gamma_\tau)= \int_{\Sigma_\tau\cap\{r\le R_0\}}Q[f](N,n_\tau)\,d\Sigma_\tau +\int_{\mathcal C_\tau}Q[f](N,L)\,d\lambda\,dA.$$

Rescaling $L$ and its parameter together leaves the flux unchanged. The surfaces now end on different null-infinity cuts. In particular, energy that crossed an earlier cut is not counted again on a later $\Gamma_\tau$.

Surface	Outer end	Meaning of the energy
$\Sigma_\tau$	Spatial infinity	Includes the outgoing wave still present at large radius
$\Gamma_\tau$	A cut of future null infinity	Measures energy remaining after radiation through earlier cuts
Smooth hyperboloidal $\widetilde\Sigma_\tau$	A cut of future null infinity	Uses its own unit normal and weighted flux comparison

The stationary spatial Sobolev equivalence used above must not be copied unchanged to hyperboloidal slices. The normal becomes asymptotically null relative to stationary observers. The foliation distinction is explicit in Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, § 3.3, p. 17, PDF; the spacelike/null construction and weighted-energy argument are in Dafermos and Rodnianski 2009, § 4, pp. 8–10, PDF.

A sufficient weighted initial norm

Here is one concrete, nonminimal norm giving useful rates. Keep the initial data smooth and supported strictly inside $r\lt R_0$. Finite propagation then makes the field and all its commuted jets zero on the initial outgoing branch $\mathcal C_0$.

Let $\mathcal Z$ consist of $MT$, $MN$ and three smooth stationary cutoff angular rotations. The latter vanish in the interior and equal the standard infinitesimal rotations of the asymptotic spheres sufficiently far out. $\mathcal Z^\alpha$ denotes any ordered word of length $|\alpha|$, with the empty word the identity. Define

$$\mathcal W[\psi]= \sum_{i=0}^{2}\ \sum_{|\alpha|\le2} \int_{\Gamma_0}\left(\frac rM\right)^2 J^N_\mu[\mathcal Z^\alpha(MT)^i\psi]\,d\Sigma^\mu.$$

The oriented measure here means $n_0^\mu d\Sigma_0$ on the spacelike part and $L^\mu d\lambda dA$ on the null part. Every summand is nonnegative. This norm uses at most five total differential orders once the derivative inside the current is counted. It is stronger than the minimal collection needed for energy decay; no minimal-regularity claim is being made. All cutoff fields, the interface and the optical geometry are fixed before comparing solutions.

Set $s=\tau/M$. With constants independent of the solution and $\tau$, but allowed to depend on those fixed choices and $a/M$, one obtains

$$\begin{aligned} \mathscr E_N[\psi](\Gamma_\tau) &\le C\mathcal W[\psi](1+s)^{-2},\\ \sup_{\Gamma_\tau}\sqrt{\frac rM}\,|\psi| &\le C\sqrt{\frac{\mathcal W[\psi]}M}\,(1+s)^{-1},\\ \sup_{\Gamma_\tau}\frac rM\,|\psi| &\le C\sqrt{\frac{\mathcal W[\psi]}M}\,(1+s)^{-1/2}. \end{aligned}$$

These follow from the weighted norm and commutation scheme in Dafermos and Rodnianski 2009, Eqs. (10)–(11), §§ 5–6, pp. 10–12, PDF, with the full-subextremal boundedness and ILED inputs supplied by the later DRSR theorem. The 2009 Kerr discussion was conditional on those then-forthcoming inputs. Summing all words gives a conservative sufficient norm; scaling by $M$ restores consistent units. Boundedness on an initial finite time interval replaces late-time powers of $s$ by powers of $1+s$.

These rates are not sharp Price laws. On a fixed radial region the second inequality gives $|\psi|=O((1+s)^{-1})$; the third gives the different weighted control needed near null infinity. DRSR’s Corollary 3.1 also gives the stronger local rates $|\psi|=O((1+s)^{-3/2+\delta})$ and regular first derivatives $O((1+s)^{-2+\delta})$, for smooth compact data and fixed $0\lt\delta\lt1/2$, on translated hyperboloidal slices. Its local energy rate $(1+s)^{-4+2\delta}$ applies to $N\psi$, not to the energy of $\psi$. In these stronger statements the constants depend on additional higher-order weighted data norms specified only schematically in that corollary; they are not asserted here to be controlled by the particular $\mathcal W$ above. Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Corollary 3.1, p. 17, PDF.

How the estimates cooperate

The global proof combines several mechanisms. None is a replacement for the others.

Horizon control. Positive surface gravity permits a redshift multiplier with a favorable bulk term near the horizon. Extending it outward creates cutoff errors, which other estimates must absorb. This restores regular transverse derivative control without pretending that $T$ becomes timelike throughout the ergoregion.

Trapping and superradiance. Carter separation organizes real frequencies and angular eigenfunctions. Frequency-dependent currents control the radial equation, degenerating appropriately at trapping. Quantitative mode stability handles the bounded-frequency obstruction; the qualitative contradiction on Whiting Mode Stability of Kerr alone does not supply a uniform resolvent bound. Plancherel recombines frequencies, and a continuity argument removes an initially imposed future-integrability assumption. It would be circular simply to assume that decay already holds to justify the transform. The sequence is described in Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, § 3.4, pp. 17–18, PDF.

Escape toward infinity. The $r^p$ method trades an initial radial weight for a spacetime estimate. Its sign mechanism can be seen without Kerr notation. In flat spacetime set $u=t-r$, $v=t+r$, and $\Psi=r\psi$. Then

$$-4\partial_u\partial_v\Psi+r^{-2}\Delta_{S^2}\Psi=0, \qquad r=(v-u)/2.$$

Define the angular integrals $A=\int_{S^2}|\partial_v\Psi|^2d\Omega$ and $B=\int_{S^2}|\nabla_{S^2}\Psi|^2d\Omega$. Multiplication by $r^p\partial_v\Psi$ and integration by parts on the sphere give the exact identity

$$2\partial_u(r^p A)+\frac12\partial_v(r^{p-2}B) +p r^{p-1}A+\frac{2-p}{4}r^{p-3}B=0.$$

For $0\le p\le2$ both bulk coefficients are nonnegative. At $p=2$ the angular bulk term vanishes; at $p=1$ both derivative types occur. The boundary terms must be kept with their outward signs when integrating over an exterior region. Applying the $p=2$ and $p=1$ estimates in sequence, choosing good times in dyadic intervals and using boundedness between those times yields polynomial energy decay. The necessary compact region terms are supplied by ILED, including its derivative cost.

The displayed identity is a flat-space calculation, not the exact Kerr identity. Curvature and rotation add terms; sufficiently far out their errors are controlled by the corresponding weighted estimates and the interior bounds. The Schwarzschild calculation and Kerr application are explained in Dafermos and Rodnianski 2009, §§ 4–6, pp. 8–12, PDF.

Pointwise control. Energy bounds are integral bounds. Commutation with $T$, the redshift field and cutoff angular rotations, followed by elliptic and Sobolev estimates, supplies the spatial regularity needed for pointwise decay. The non-Killing commutators have errors; they do not commute with the wave operator merely because they appear in a norm. A hidden-symmetry operator can be useful in other approaches, but is not a substitute for checking those estimates in this one. The broader PDE treatment continues in Scalar-Wave Boundedness and Decay.

Scalar and gravitational stability

The scalar theorem concerns one test field on a fixed metric. For a metric perturbation, the linearized constraints, gauge and stationary family directions must also be controlled.

The Schwarzschild gravitational theorem of Dafermos, Holzegel and Rodnianski uses smooth characteristic seed data on two intersecting null hypersurfaces, with the finite weighted curvature and connection energy in their Eq. (246). An initial-data gauge normalization gives boundedness. A second, horizon-determined gauge normalization gives decay after subtracting the reference linearized Kerr solution. Thus the decaying object is the normalized radiative remainder, not arbitrary raw metric components. Dafermos, Holzegel, and Rodnianski 2019, §§ 10.3.2–10.4, Theorems 3–4 and Corollaries 10.2–10.3, pp. 111–116.

The stationary remainder includes a mass change and three angular-momentum directions. Such perturbations should not decay to zero: they describe nearby stationary black holes. The theorem controls the gauge correction and gives decay of the appropriate normalized metric quantities to their linearized Kerr values. Its metric rates must not be assigned indiscriminately to all curvature components. The complete tensorial norms and gauge construction belong to Linear Stability of Schwarzschild.

Neither result is a theorem for arbitrary gravitational perturbations of rotating Kerr or for the quadratic errors of the full Einstein equation. Compare Canonical Energy and Linearized-Gravitational Stability and From Linear Decay to Nonlinear Stability. The extremal limit is also separate: vanishing surface gravity removes the redshift mechanism used here, as explained on Extremal Horizons and Aretakis Instability.

Exercises

The following examples separate three different obstructions. The nonnormal evolution example on the Whiting page supplies a complementary warning: stable eigenvalues alone do not give a uniform evolution-norm bound.

Example	What is fixed?	What it disproves
Reflecting cavity	One exact wave for all times	Boundedness implies local decay
Narrow smooth spikes	One integrable function for all times	Time integrability alone implies a pointwise limit
Trapped packets	A different wave may be chosen for each finite duration	A uniform nondegenerate ILED bound using only initial first-derivative energy

A stable reflecting cavity

In dimensionless coordinates consider $u_{tt}-u_{xx}=0$ on $0\lt x\lt\pi$ with $u(t,0)=u(t,\pi)=0$. Verify that $u=\sin x\cos t$ has constant energy, zero boundary flux, but no local energy decay even on $J=[\pi/4,3\pi/4]$. Which stability statements survive?

Solution

Both the equation and boundary conditions hold directly. With density $e=(u_t^2+u_x^2)/2$ and flux $j=-u_tu_x$, one has $\partial_t e+\partial_xj=0$ and $j=0$ at the walls. Integration gives

$$E(t)=\frac\pi4,\qquad E_J(t)=\frac\pi8-\frac14\cos(2t),\qquad \int_0^T E_J(t)dt=\frac{\pi T}{8}-\frac18\sin(2T).$$

The last expression grows linearly. The cavity has real separated frequencies $\omega=\pm n$, $n=1,2,\ldots$, and bounded energy, but neither ILED nor pointwise decay. This is one persistent solution, not a transient packet chosen anew as the observation time grows.

When a finite time integral is insufficient

Let $\eta(z)=\exp(1-1/(1-z^2))$ for $|z|\lt1$ and zero otherwise. Show that the nonnegative smooth function

$$b(t)=\sum_{n=1}^{\infty}\eta\bigl(2^n(t-n)\bigr),\qquad t\ge0,$$

is integrable but does not tend to zero. Then show that a nonnegative integrable function with a global Lipschitz bound $|b'|\le K$, $K\gt0$, satisfies $b(t)^2\le2K\int_t^\infty b(s)ds$. What extra ingredient is needed for a polynomial rate?

Solution

The bump supports are disjoint and locally finite. The function is smooth, $0\le b\le1$, and

$$\int_0^\infty b(t)dt= \sum_{n=1}^{\infty}2^{-n}\int_{-1}^{1}\eta(z)dz\lt\infty.$$

But $b(n)=1$ and $b(n+1/2)=0$. Smoothness is not a uniform bound on how fast the spikes can narrow. This is an abstract function counterexample, not a claim that it is the local energy of a Kerr solution.

Under the Lipschitz assumption, $b(t+h)\ge b(t)-Kh$ for $0\le h\le b(t)/K$. Therefore

$$\int_t^\infty b(s)ds\ge \int_0^{b(t)/K}(b(t)-Kh)dh=\frac{b(t)^2}{2K}.$$

The tail integral tends to zero, so $b(t)\to0$. A polynomial rate still requires a quantitative bound on that tail. For waves, passage from local energy decay to pointwise field decay additionally needs higher-order spatial control, supplied by commutation and Sobolev estimates.

A trapped packet and the order of quantifiers

On Schwarzschild choose the stationary slices above and a timelike $N$ equal to $T$ near $r=3M$. In Schwarzschild coordinates check that

$$\gamma(\lambda)= \left(3\lambda,3M,\frac\pi2,\frac{\lambda}{\sqrt3M}\right)$$

is an affinely parametrized null geodesic with $-g(N,\dot\gamma)=1$. The Gaussian-beam approximation theorem gives, for each finite duration $T_0$, a sufficiently high-frequency smooth exact real wave $v_{T_0}$ with initial $N$ energy one and local $N$ energy greater than $1/2$ throughout $0\le\tau\le T_0$, in a fixed neighborhood of the orbit. Explain why no constant independent of the data and time can bound the nondegenerate time-integrated local energy by initial first-derivative energy alone. Sbierski 2015, Theorem 5.1, pp. 1396–1397, Theorem 5.7, p. 1398, and § 6A, pp. 1400–1401, PDF.

Solution

At $r=3M$ the Schwarzschild factor is $1/3$. Thus

$$g(\dot\gamma,\dot\gamma) =-\frac13(3)^2+(3M)^2\left(\frac1{\sqrt3M}\right)^2=0, \qquad -g(T,\dot\gamma)=1.$$

The radial geodesic equation also vanishes at the extremum of $(1-2M/r)/r^2$, and the remaining equations hold with the constant velocities shown. Conservation of the Killing energy keeps the ray energy equal to one. The Gaussian-beam theorem approximates that energy for every fixed finite time interval by an exact wave, with an error chosen below $1/2$. Consequently

$$\int_0^{T_0}E_{\mathrm{local}}[v_{T_0}](\tau)d\tau \gt\frac{T_0}{2}.$$

An alleged bound $CM E_N[v_{T_0}](0)$ fails when $T_0\gt2CM$. The bounded lapse also transfers this obstruction to the spacetime-volume form of a nondegenerate local estimate. Different $T_0$ may require different frequencies and data; there is no single forever-localized finite-frequency solution in this argument.

A trapping-degenerate integrand does not have the same positive lower bound along the ray. Alternatively, a packet of wavelength $hM$ typically has an additional squared derivative-energy cost $h^{-2}$ for each extra initial derivative, at fixed packet construction and observation time. A higher-norm right-hand side can therefore grow as concentration increases. The obstruction explains why a loss or degeneracy is necessary; the DRSR estimate, not this scaling argument, proves that the stated loss suffices.

References

• Dafermos, M.; Holzegel, G.; Rodnianski, I. (2019). “The linear stability of the Schwarzschild solution to gravitational perturbations.” Acta Mathematica 222, 1–214. DOI. The journal pagination is used here.
• Dafermos, M.; Rodnianski, I. (2010). “A new physical-space approach to decay for the wave equation with applications to black hole spacetimes.” In P. Exner (ed.), XVIth International Congress on Mathematical Physics, World Scientific, pp. 421–433. DOI. Open PDF, 2009 preprint v1; the equation locators above use that edition.
• Dafermos, M.; Rodnianski, I.; Shlapentokh-Rothman, Y. (2016). “Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a|\lt M$.” Annals of Mathematics 183, 787–913. DOI. Open PDF, 2014 preprint v2; the theorem and equation locators above use that edition.
• Sbierski, J. (2015). “Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes.” Analysis & PDE 8(6), 1379–1420. DOI. Open PDF.