Linearized Gravity on Kerr

Decay of the extreme Weyl scalars is indispensable for Kerr stability, but it is not yet decay of a metric perturbation. Curvature variables omit stationary mass and angular-momentum changes, do not fix infinitesimal diffeomorphisms, and must be integrated through differential identities before the remaining connection and metric components are known. A rigorous method therefore has four parts: estimate the Teukolsky fields, reconstruct radiative metric data, add the nonradiative completion, and impose a globally controlled gauge.

Required background. Analysis for Mathematical Relativity supplies weighted and microlocal estimates, while Partial Differential Equations and Green Functions supplies hyperbolic propagation and reconstruction identities.

Helpful background. Einstein Equations fixes the constrained system, Causality and Global Structure fixes the exterior, and Black-Hole Geometry and Mechanics together with Black-Hole Perturbation Theory supplies Kerr tetrads and Teukolsky variables.

Gauge-invariant curvature data

Let $g_{M,a}$ be a four-dimensional subextremal Kerr metric, with physical mass $M\gt0$, $a=J/M$, $\lvert a\rvert\lt GM$, and vacuum perturbation $h$ satisfying $D\operatorname{Ric}_{g_{M,a}}(h)=0$. Use a principal null tetrad $(\ell,n,m,\bar m)$ with $g(\ell,n)=-1$ and $g(m,\bar m)=1$, all other inner products zero, and define

$$\Psi_0=-C(\ell,m,\ell,m), \qquad \Psi_4=-C(n,\bar m,n,\bar m).$$

The background extreme scalars vanish. Their first variations $\dot\Psi_0$ and $\dot\Psi_4$ are invariant under infinitesimal coordinate gauge and tetrad rotations about this principal background. They still have spin/boost weights under a finite rescaling of the background tetrad, so estimates must fix a horizon-regular normalization. The appropriately rescaled extreme fields obey the spin $+2$ and $-2$ Teukolsky equations. They encode the radiative degrees of freedom and separate into radial and angular ordinary differential equations. The decoupling is derived in Teukolsky 1973, §§II–IV and Eqs. (4.7)–(4.8).

This extraction has a kernel. A linearized Kerr perturbation changes $M$ or $a$ but has vanishing linearized extreme scalars in an adapted principal tetrad. A pure-gauge perturbation $\mathcal L_Xg_{M,a}$ also represents no physical radiation. Consequently,

$$(\dot\Psi_0,\dot\Psi_4)=0 \quad\not\Longrightarrow\quad h=0.$$

Any injective reconstruction statement must quotient gauge and separately identify the finite-dimensional stationary kernel.

Teukolsky estimates and transformed variables

The spin equations inherit horizon redshift, Kerr trapping, superradiance, and a complex first-order structure. Teukolsky–Starobinsky identities relate the two extreme spins. Differential transformations produce Regge–Wheeler-type variables with more coercive energy identities, after which one seeks boundedness, integrated local-energy decay, and weighted estimates at null infinity. The hierarchy must control enough derivatives to invert the transformations without losing the regularity needed for the metric.

The 2025 published theorem of Andersson, Bäckdahl, Blue, and Ma is conditional in the full subextremal range: given the required integrated energy estimates for the Teukolsky master equation, metric quantities obey integrated and pointwise decay estimates. In the slowly rotating range those inputs are available, so their result is unconditional there Andersson et al. 2025, Theorems 1.1–1.4 and §1.2.

A distinct June 2025 preprint by Häfner, Hintz, and Vasy claims an unconditional full-subextremal theorem. For constraint-satisfying initial perturbations with eight controlled angular and scaled-radial derivatives and decay $\dot\gamma=O(r^{-1-\alpha})$, $\dot k=O(r^{-2-\alpha})$, $0<\alpha<1$, it decomposes the solution into a linearized Kerr perturbation, a controlled pure-gauge term, and a decaying remainder Häfner, Hintz, and Vasy 2025, Theorem 1.1 and §8. As of 2026-08-12, that is a recent v1 preprint claim rather than a peer-reviewed replacement for the conditional theorem. The two statements must therefore be reported with both their hypotheses and publication status.

Reconstruction and nonradiative completion

There are two complementary reconstruction routes. A Hertz-potential or radiation-gauge construction applies an adjoint Teukolsky operator to a scalar potential and obtains a radiative metric perturbation. A physical-space route uses the linearized null structure and Bianchi equations to integrate inward from controlled curvature quantities. Both require compatibility identities, horizon and infinity regularity, and boundary data for transport equations.

With the spin axis fixed, the reconstructed field is schematically completed by

$$h=h_{\mathrm{rad}} +\delta M\,\partial_M g_{M,a} +\delta a\,\partial_a g_{M,a} +\mathcal L_Xg_{M,a}.$$

Here $\delta M$ and $\delta a$ are fixed by conserved or asymptotic charges; $X$ is chosen to enforce the desired gauge and regularity. A formulation that keeps the asymptotic angular axes fixed must also retain spin-direction variations: replace the scalar spin term by $\sum_{i=1}^3\delta a^i\,\partial_{a^i}g_{M,\mathbf a}$ in a smooth vector-parameter description of Kerr. This is especially important at Schwarzschild, where the three odd dipoles cannot be represented by one fixed-axis variation. Which asymptotic rotations or boosts count as gauge depends on the theorem’s boundary normalization. Radiation gauges can have string or patch singularities for generic sources, so a formula that works mode by mode is not automatically a global finite-energy reconstruction. The completion must also retain the linearized constraints.

Proof architecture and Schwarzschild check

A complete proof begins by normalizing admissible initial data and propagating the linearized constraints. It estimates gauge-invariant curvature, removes the stationary kernel using charge projections, reconstructs the connection and metric, and finally proves that the chosen gauge vector is controlled by the same initial norm. The last step ensures that bounded curvature has not been paired with an uncontrolled coordinate distortion.

At $a=0$, the method must reduce to the established Schwarzschild theorem: Regge–Wheeler estimates control transformed curvature, the null hierarchy reconstructs the metric, and the remainder is linearized Kerr plus pure gauge. That independent limiting-case check is carried out in Dafermos, Holzegel, and Rodnianski 2019, Main Theorem and §§7–13. Failure to recover the mass and odd dipole remainders in this limit signals an incomplete reconstruction.

Limitations and failure modes

This page describes the theorem-level pipeline, not a claim that every method proves every estimate under identical hypotheses for every $\lvert a\rvert\lt GM$. The range of spin, regularity, asymptotic weights, choice of gauge, and conditional Teukolsky input must accompany each result. Curvature decay alone is a counterexample to neither stationary parameter drift nor gauge growth because it does not see those sectors. Linear control also does not estimate the quadratic terms of the vacuum equations or prove nonlinear stability.

Metric reconstruction can fail at algebraically special frequencies or global boundaries unless kernel and compatibility conditions are handled. Likewise, a radiation gauge that is regular near future null infinity need not be regular across the future horizon. These are mathematical limitations, not merely choices of notation.

Check the missing information

Suppose $\dot\Psi_0=\dot\Psi_4=0$ for a regular linearized vacuum perturbation. Why can one not conclude that $h$ vanishes, and what data complete the reconstruction?

Solution

The extreme scalars annihilate pure-gauge perturbations and do not distinguish all tangent directions to the Kerr family. Thus $h$ may contain a mass or angular-momentum variation and a Lie derivative $\mathcal L_Xg$. One must determine the stationary coefficients from charges or boundary data, choose and control $X$, and then solve the remaining reconstruction identities. Only after quotienting gauge and fixing the stationary parameters can vanishing radiative curvature imply a trivial radiative perturbation.

References

• Andersson, Lars, Thomas Bäckdahl, Pieter Blue, and Siyuan Ma. “Stability for Linearized Gravity on the Kerr Spacetime.” Annals of PDE 11 (2025): 11. DOI.
• Dafermos, Mihalis, Gustav Holzegel, and Igor Rodnianski. “The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations.” Acta Mathematica 222 (2019): 1–214. DOI.
• Häfner, Dietrich, Peter Hintz, and András Vasy. “Linear Stability of Kerr Black Holes in the Full Subextremal Range.” arXiv:2506.21183v1 (2025). Versioned arXiv.
• Teukolsky, Saul A. “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations.” Astrophysical Journal 185 (1973): 635–647. DOI.