Linear Estimates as Nonlinear Inputs

A linear estimate is useful to nonlinear stability only if its norm, derivative count, degeneracies, and time weights are compatible with the quadratic errors of the Einstein equation. Mode stability rules out exponentially growing separated solutions, but it does not by itself give the spacetime decay or low-frequency control needed to integrate nonlinear forcing. The required package combines bounded energy, horizon redshift, trapping-aware local decay, asymptotic radiation estimates, and a finite-dimensional description of zero modes.

Required background. PDEs and Green Functions for Gravitation supplies energy identities, Green operators, and decay. Helpful background. Black-Hole Geometry and Mechanics supplies Kerr causal geometry; Black-Hole Perturbation Theory supplies separated and gauge-invariant variables; Formulations of General Relativity supplies the reduced Einstein system; Computational Relativity supplies nonlinear comparisons.

Energy and integrated decay estimates

Write a gauge-fixed perturbation equation schematically as

$$L_{g_b}h=F,$$

where $L_{g_b}$ is the linearized operator about a subextremal Kerr metric $g_b$. A nonlinear proof needs an estimate of the form

$$\sup_{0\le s\le t}E_N[h](s)^{1/2} +\left(\int_0^t D_N[h](s)\,ds\right)^{1/2} \le C\left( E_{N+k}[h](0)^{1/2} +\int_0^t\|F(s)\|_{\mathcal Y_N}\,ds \right).$$

Here $E_N$ is a nondegenerate quadratic high-order energy, $D_N$ is a quadratic local decay density, and $k\ge0$ records derivative loss from trapping or the chosen formulation. The source norm $\mathcal Y_N$ includes the derivatives and spatial weights required by the chosen estimate; the displayed $L^1$ time norm is one illustrative forcing norm. All three terms scale linearly under $(h,F)\mapsto\lambda(h,F)$. This matters: adding a linear source norm directly to a quadratic energy would give an inconsistent estimate. Hiding $k$ or a degenerating weight can likewise make a formal bound useless in the quasilinear iteration.

Near the event horizon, a redshift multiplier controls derivatives transverse to the horizon. Near the trapped set, a Morawetz estimate controls energy integrated over a spatial region and over time, but degenerates or loses derivatives at trapping. Far away, an $r^p$ hierarchy uses different radial weights to control radiation through null infinity and obtain time decay. These geometric components and their relation to Kerr linear decay are summarized in Klainerman and Szeftel 2022, §§2.4–2.6 and 3.2, pp. 10–18, PDF.

Decay strong enough for quadratic errors

Measure time and length in units of the fixed background scale $GM_0$ in the following schematic estimates. Suppose the nonlinear equation is

$$L_{g_b}h=Q(\nabla h,\nabla h)+\text{higher order}.$$

An energy inequality typically contains

$$\frac{dE_N}{dt} \lesssim \|\nabla h(t)\|_{L^\infty}E_N(t) +\cdots.$$

If $\|\nabla h(t)\|_{L^\infty}\le C\varepsilon(1+t)^{-1-\delta}$ with $\delta\gt 0$, then

$$\int_0^\infty\|\nabla h(t)\|_{L^\infty}dt\lt \infty,$$

and Grönwall’s inequality gives uniform high-order control for sufficiently small $\varepsilon$. Borderline $(1+t)^{-1}$ decay produces a logarithm and demands null structure, weighted hierarchies, renormalization, or controlled mild growth. Thus the phrase “the perturbation decays” is too weak; the theorem needs the exact integrable hierarchy.

Weak decay estimates can still start a Nash–Moser scheme if they are tame and if later asymptotic analysis improves the radiative remainder. Hintz constructs forward estimates for wave-type operators on dynamical backgrounds, first obtaining weak decay with high regularity and then eliminating slowly decaying pure-gauge pieces Hintz 2026, §§1.5.1–1.5.4, pp. 33–42, and §11, pp. 218–295, PDF.

Zero modes are part of the estimate

At zero frequency, $L_{g_b}$ has nondecaying solutions. Physical states vary Kerr mass, angular momentum, boost, or center of mass; other states are pure gauge. A correct linear statement has the form

$$h=h_{\rm rad} +\delta b^A\,\partial_{b^A}g_b +\mathcal L_Xg_b,$$

with decay required only of $h_{\rm rad}$. Projecting away the entire zero-frequency space would accidentally remove allowed physical endpoints. Keeping it all inside the error would prevent decay.

The constraint propagation operator must also be controlled. Gauge fixing solves a reduced wave system; to recover Einstein evolution, the gauge constraint and its compatible initial normal derivative must vanish, and uniqueness for their homogeneous propagation equation must keep them zero. Damping is a useful design choice, not a substitute for those conditions. Hintz analyzes the gauge potential, constraint, and linearized Einstein spectral problems separately before combining them Hintz 2026, §§6–10, pp. 113–217, PDF.

Structural checklist and limitations

For nonlinear closure, ask:

• Is the energy nondegenerate at the horizon?
• Where does local decay lose derivatives or weights?
• Are superradiant frequencies controlled by a positive substitute energy?
• Are radiative, physical zero, and pure-gauge modes separated?
• Is source-to-solution decay strong enough for quadratic time integrability?
• Do estimates persist with tame constants when the background metric slowly changes?
• Does the constraint subsystem remain controlled?

Linear estimates derived only for a fixed exact background do not automatically apply to the evolving metric. Conversely, a nonlinear calculation that settles numerically does not supply the uniform norms required by a theorem. The interface is precise: the linear theory must be stable under the small coefficient changes generated by the nonlinear bootstrap.

Exercise: an integrable coefficient

Suppose the dimensionless energy obeys $E'(t)\le C\varepsilon(1+t)^{-1-\delta}E(t)$, with $\delta\gt0$. Find the resulting uniform bound and compare it with $\delta=0$.

Solution

Integrating the coefficient gives

$$\int_0^t(1+s)^{-1-\delta}\,ds =\frac{1-(1+t)^{-\delta}}{\delta}\le\frac1\delta,$$

so $E(t)\le E(0)e^{C\varepsilon/\delta}$. To keep the growth factor close to one, smallness must be compared with $\delta$, not merely with unity. At $\delta=0$ the integral is $\ln(1+t)$ and $E(t)\le E(0)(1+t)^{C\varepsilon}$; this permits mild growth but supplies no uniform bound. An estimate can therefore deteriorate as its decay exponent approaches a borderline value even while every fixed positive $\delta$ works.

References

• Hintz, Peter, “Nonlinear Stability of Subextremal Kerr Black Holes”, arXiv:2606.28253v2, 2026. Open PDF.
• Klainerman, Sergiu, and Jérémie Szeftel, “Brief Introduction to the Nonlinear Stability of Kerr”, arXiv:2210.14400v2, 2022. Open PDF.

Further reading

• Dafermos, Mihalis, Igor Rodnianski, and Yakov Shlapentokh-Rothman, “Decay for Solutions of the Wave Equation on Kerr Exterior Spacetimes III”, Annals of Mathematics 183, 787–913, 2016. Open PDF.