Mode, Linear, and Nonlinear Stability Interfaces

Stability is not a single yes-or-no property. A black hole may have no exponentially growing separated modes while a proof of bounded evolution is still missing; a field may remain bounded but fail to decay; a metric perturbation may decay only after gauge and parameter modes are removed; and an exactly extremal horizon may support growth of transverse derivatives even when the scalar itself stays small. This chapter separates those statements, records the hypotheses of the principal theorems, and explains what must be added before a linear estimate can support a nonlinear argument.

Helpful background. Black-Hole Geometry and Mechanics fixes Schwarzschild, Kerr, horizons, and extremality; PDEs and Green Functions for Gravitation supplies wave operators, energy estimates and causal evolution; Formulations of General Relativity supplies constraints and gauge reduction; Mathematical Relativity supplies the language of global evolution, norms, and theorem hypotheses.

Check that you can distinguish initial data from boundary conditions, and a coordinate change from a physical perturbation. The PDE and formulation introductions provide the shortest preparation routes. The first and third pages below develop the chapter’s spectral and energy arguments.

A hierarchy of stability statements

For the shared time convention $e^{-i\omega t}$, mode stability excludes a nonzero separated solution with $\operatorname{Im}\omega\gt0$ and the physical future-horizon and infinity boundary conditions. It is a spectral exclusion, not an estimate for arbitrary initial data. Uniform boundedness instead controls every solution in a specified energy class,

$$E[\Psi](\Sigma_\tau)\leq C E_k[\Psi](\Sigma_0),$$

where $E$ is the stated nonnegative energy, $E_k$ is the initial norm with the stated additional derivatives or weights, and $C$ is independent of time and the solution. The field, foliation, parameter range and boundary conditions must also be fixed; boundedness need not lose derivatives. Integrated local energy decay controls a spacetime bulk norm, normally with a degeneracy or derivative loss at trapping. Quantitative decay adds a rate in an identified region and norm. Each step requires information absent from the preceding one: in particular, real-frequency, zero-frequency, trapping, redshift, and high-frequency estimates cannot be replaced by the statement that the upper half-plane contains no mode. Nor does integrated decay alone give a pointwise rate without additional regularity and propagation estimates. Local energy can decay while outgoing radiation still carries nonzero energy on a complete slice ending at spatial infinity. A surface ending on a later null-infinity cut measures a different quantity.

For linearized gravity, asymptotic stability has an additional quotient. A perturbation can contain a pure diffeomorphism and stationary variations of mass and angular momentum. The meaningful claim is therefore convergence to a linearized member of the Kerr family, modulo controlled gauge, rather than decay of every metric component to zero. The Schwarzschild theorem proves this kind of boundedness and inverse-polynomial decay for the smooth asymptotically flat characteristic data, high-order weights, and gauge normalizations in Dafermos, Holzegel, and Rodnianski 2019, §§ 10.3.2–10.4, Theorems 3–4 and Corollaries 10.2–10.3, pp. 111–116.

Finally, nonlinear orbital stability means that the vacuum development of small constrained data remains near the Kerr family in specified norms, allowing controlled identifications of the spacetimes. Nonlinear asymptotic stability adds convergence of intrinsic parameters and decay of a radiative remainder. A finite comparison uses a controlled diffeomorphism, not merely an additive Lie derivative. Mass and spin are intrinsic parameters; a boost or center trajectory is a separate frame issue, and a moving center need not converge in the original frame. The proof must also state its future region, regularity and geometric completeness: all coordinate time is not infinite affine length. Linear decay is an input, not the nonlinear theorem.

Six pages in order

The pages move from the cleanest spectral argument to the nonlinear research boundary.

1. Mode Stability of Schwarzschild proves the positive-potential identity for Regge–Wheeler and Zerilli modes and separates radiative harmonics from gauge and parameter perturbations.
2. Whiting Mode Stability of Kerr shows why the Kerr ergoregion defeats the direct energy proof and how an integral transformation restores positivity for the separated problem.
3. Boundedness and Decay of Linear Fields distinguishes energy boundedness, Morawetz control, redshift estimates, trapping loss, and pointwise decay for all solutions in the theorem’s data class.
4. Canonical Energy and Linearized-Gravitational Stability formulates a quadratic diagnostic on constrained, zero-charge data with controlled boundary conditions, and distinguishes positivity, coercivity and growth arguments.
5. Extremal Horizons and Aretakis Instability derives horizon-charge conservation. For the spherical mean on extremal Reissner–Nordström, nonzero charge together with horizon-field decay gives first-derivative nondecay and second-derivative growth.
6. From Linear Decay to Nonlinear Stability identifies the gauge, constraint, modulation, derivative-loss, and backreaction estimates needed in a nonlinear bootstrap and states the current evidence boundary.

The order prevents two common category errors. One cannot promote a mode calculation directly to evolution decay, and one cannot promote a test-field estimate directly to a theorem for the dynamical metric. Canonical energy and the extremal example then test where a simple ascending ladder is too crude: the relevant quotient, norm, derivative order, and spacetime region always matter.

Two reader routes

Route	Pages	Question to carry
Spectral to linear evolution	1 → 2 → 3 → 4	Which additional frequency, energy, gauge, and reconstruction estimates turn mode exclusion into controlled gravitational evolution?
Horizon and nonlinear boundary	3 → 5 → 6	Which estimates degenerate at extremality, and which linear estimates remain robust enough to close a quasilinear bootstrap?

The first route is suited to readers coming from master equations and quasinormal modes. Its recurring diagnostic is logical implication: after each theorem, ask whether it controls one separated solution or every solution in a normed data class. Whiting’s result excludes unstable Teukolsky modes on subextremal Kerr Whiting 1989, pp. 1301–1305, whereas the scalar Kerr theorem establishes boundedness for finite-energy data and quantitative decay with additional weighted regularity Dafermos, Rodnianski, and Shlapentokh-Rothman 2014, v2, Theorems 3.1–3.2 and Corollary 3.1, pp. 15–17, PDF. The latter locators belong to that preprint edition, not the 2016 journal pagination. Those claims are related but not interchangeable.

The second route is suited to readers interested in the extremal limit or the Einstein equation. Its diagnostic is uniformity: ask whether constants persist as surface gravity tends to zero, whether the norm controls horizon-transverse derivatives, and whether inhomogeneous estimates survive small changes of the geometry. Aretakis’s conserved section charges exhibit an exact-extremal nondecay obstruction under the general theorem’s geometric hypotheses Aretakis 2015, arXiv v2, Proposition 3.1 and Theorem 1, pp. 8–9 and 11–12, PDF. They do not contradict a theorem for each fixed subextremal parameter whose constants deteriorate toward extremality.

Recurring examples and decisive distinctions

Schwarzschild gravitational harmonics with $\ell\geq2$ provide the clean positive-potential example. The integrated radial identity excludes an upper- half-plane mode, while the exceptional $\ell=0,1$ sectors remind us that mass, angular momentum, center, and gauge perturbations require separate treatment. Canonical energy needs a more careful domain than “remove all stationary directions.” On a stationary asymptotically flat vacuum background with a bifurcation surface, the axisymmetric construction in the fourth page uses constraint-satisfying data with zero first-order ADM mass, momentum and angular momentum, the stated horizon gauge and falloff. Quotient only the actual stationary degeneracies in that space, including admissible gauge directions with vanishing boundary terms. A stationary-family variation outside the zero-charge space need not have zero canonical energy. Positivity controls the canonical-energy norm itself; a uniform coercivity estimate is needed to control a stronger chosen norm. Neither gives decay by itself Hollands and Wald 2013, arXiv v4, Proposition 4, pp. 20–21, and §§4.1–4.2, pp. 33–35, PDF.

Negative canonical energy also requires interpretation. The original future-flux argument rules out relaxation only when the proposed admissible stationary limit makes the raw energy converge and both cut corrections vanish. It does not automatically produce a growing mode. Prabhu and Wald’s additional spectral argument explains the reflection symmetry, positive kinetic form, gauge-fixed data and restricted Hilbert-space domain used for the stronger exponential-growth result; that result is not automatically a discrete normal-mode eigenfunction Prabhu and Wald 2015, arXiv v2, Proposition 6.2 and Theorem 2, pp. 35–36, PDF.

Subextremal Kerr is the recurring test of nonpositive stationary energy. Superradiant scattering can amplify a real-frequency wave without producing an exponentially growing homogeneous mode. Trapping can force loss in a Morawetz estimate without violating boundedness. Damped quasinormal modes can govern an intermediate response without furnishing a complete basis. Each phenomenon blocks an invalid implication while remaining compatible with the more carefully stated theorem.

Extremal Reissner–Nordström supplies the sharp derivative-order example. For a neutral massless test scalar, take its spherical mean and the regular ingoing coordinates of the fifth page. This average does not require spherical symmetry of the full field or assert growth at every angular point. If the mean decays on the horizon and its spherical charge is nonzero, its first radial derivative tends to a nonzero limit and its second grows linearly in advanced time Aretakis 2011, Part II, arXiv v1, Theorems 4–6 and §3, pp. 6–9, PDF. Conservation alone is insufficient: a constant nonzero scalar is an exact local solution with nonzero spherical charge and no derivative growth. It is not a member of the compact decaying global data class. For more general extremal geometries, the theorem concerns section norms and needs additional geometric and decay hypotheses for higher-derivative blow-up; the RN pointwise coefficient is not universal. Calling the solution simply “stable” or “unstable” loses the information needed to decide whether a claimed estimate is true.

What nonlinear stability theorems establish

Evidence cutoff: 2026-09-12. The published Klainerman–Szeftel program treats sufficiently small rotation and sufficiently small high-regularity admissible initial layers, with its weighted curvature, connection and frame norms. Its conclusion includes complete future null infinity and convergence toward nearby Kerr. The generally covariant modulated construction fixes geometric normalization; it is not a symmetry restriction on the perturbations Klainerman and Szeftel 2023, author-final PDF, §§3.1, 3.3.6 and 3.4, pp. 117–121 and 137–144. These are the author-final edition’s printed pages, not the journal pages.

Hintz’s full-subextremal preprint provides, for each fixed subextremal Kerr background, a concrete sufficient class of smooth constrained data on a regular horizon-crossing slice: exactly Kerr outside a compact set and small in a sufficiently high Sobolev norm. The constructed future region of the maximal development settles toward nearby Kerr in its specified gauge, with distinct compact-region and near-infinity estimates Hintz 2026, v2, Theorem 1.1, pp. 1–3, PDF. The broader inverse-power/logarithmic data class and the slow-rotation data classes are not nested; wider parameter coverage does not imply inclusion of every earlier data class. The sixth page explains the separate constraint-damping and tame-wave companion inputs.

At this cutoff the Hintz arXiv record lists the 3 August 2026 version and no journal reference. That observable publication status does not disclose private refereeing, establish consensus or refute the theorem. This overview does not independently verify either full nonlinear proof, assert arbitrary small-data stability, or provide uniform estimates at extremality. A new version, substantive technical critique, erratum, or changed journal record calls for a fresh comparison of the precise hypotheses and conclusions.

Chapter review

Classify the conclusion

Classify the strongest statement justified in each case: (a) no physical separated solution exists with $\operatorname{Im}\omega\gt0$; (b) every solution in a finite-energy class satisfies $E(\tau)\leq CE(0)$; (c) a metric perturbation approaches a linearized Kerr perturbation modulo pure gauge; (d) small constrained vacuum data generate a future exterior approaching a nearby Kerr metric; (e) a scalar decays on an extremal horizon but its second transverse derivative grows linearly.

Solution

(a) is mode stability only. It says nothing by itself about completeness, transient growth, or arbitrary-data evolution. (b) is uniform linear boundedness, not decay. (c) is linearized gravitational asymptotic stability, with gauge and stationary parameter modes handled explicitly. (d) is nonlinear asymptotic stability toward the Kerr family, provided the theorem states the norms, gauge, regularity, parameter range and regions needed to interpret “approaching.” Geometric completeness is a separate conclusion to check, not a consequence of that word. Orbital stability alone would require remaining nearby without convergence. (e) is an Aretakis derivative instability: it is compatible with boundedness or decay in weaker norms and away from the horizon. The five answers differ because field, data class, norm, region, derivative order, and linear versus nonlinear dynamics are all part of a stability statement.

Change the comparison metric

Take an exact Kerr solution with slightly different mass or spin from the chosen reference Kerr metric. Must the difference decay to zero? Does this give a nonlinear instability or a zero direction of every canonical-energy construction?

Solution

The stationary parameter difference need not decay. The spacetime is already Kerr and is asymptotic to itself; it is not unstable merely because the reference member was chosen differently. A finite diffeomorphism and the correct nearby parameters specify the comparison, while intrinsic parameters must be distinguished from a moving asymptotic frame.

Nor can an arbitrary mass or spin variation be declared a canonical-energy zero direction. It usually violates the fixed first-order charge conditions of the fourth page’s domain. Only degeneracies lying in the specified constrained, boundary-controlled space are quotiented there.

Test a missing hypothesis

A proposed argument says that any nonzero extremal-horizon charge forces second transverse derivatives of a scalar to grow. Test it on a constant nonzero scalar. What must be checked before applying the RN growth result to a general extremal horizon or to a nearly extremal family?

Solution

A constant solves the massless wave equation. Its RN spherical charge is nonzero, but all its derivatives vanish. The missing assumption is decay of the horizon field: the exact differentiated transport equation includes a field source as well as the conserved charge, and the two cancel for this solution. The constant is a local countercheck, not compact decaying initial data for the global theorem.

For a general extremal horizon, check the geometric hypotheses, derivative frame, section norm and required lower-order decay; do not transplant a pointwise RN coefficient. For a near-extremal family, check parameter-uniform constants and the forced subextremal transport equation. A result for each fixed positive surface gravity need not survive exchanging late-time and extremal limits.

From linear stability to nonlinear evolution

This chapter distinguishes mode stability, bounded linear evolution, decay, linear asymptotic stability, canonical-energy diagnostics, extremal horizon instability, and nonlinear orbital versus asymptotic stability. Continue to Mathematical Relativity for detailed nonlinear proof architecture, or Black-Hole Dynamics for finite-amplitude evolution and endpoint formation. For limit ordering and throat dynamics, continue to Near-Horizon and Near-Extremal Dynamics.

References

• Aretakis, S. (2011). “Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations II.” Annales Henri Poincaré 12, 1491–1538. DOI. Open PDF, arXiv v1.
• Aretakis, S. (2015). “Horizon instability of extremal black holes.” Advances in Theoretical and Mathematical Physics 19, 507–530. DOI. Open PDF, arXiv v2.
• Dafermos, M., Holzegel, G., and Rodnianski, I. (2019). “The linear stability of the Schwarzschild solution to gravitational perturbations.” Acta Mathematica 222, 1–214. DOI.
• Dafermos, M., Rodnianski, I., and Shlapentokh-Rothman, Y. (2016). “Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $\lvert a\rvert\lt M$.” Annals of Mathematics 183, 787–913. DOI. Open PDF, 2014 arXiv v2.
• Hintz, P. (2026). “Nonlinear stability of subextremal Kerr black holes.” arXiv:2606.28253v2, 339 pp., 3 August 2026. arXiv.
• Hollands, S., and Wald, R. M. (2013). “Stability of black holes and black branes.” Communications in Mathematical Physics 321, 629–680. DOI. Open PDF, arXiv v4.
• Klainerman, S., and Szeftel, J. (2023). “Kerr stability for small angular momentum.” Pure and Applied Mathematics Quarterly 19(3), 791–1678. DOI. Open PDF, author-final text dated 14 July 2023.
• Prabhu, K., and Wald, R. M. (2015). “Black hole instabilities and exponential growth.” Communications in Mathematical Physics 340, 253–290. DOI. Open PDF, arXiv v2.
• Whiting, B. F. (1989). “Mode stability of the Kerr black hole.” Journal of Mathematical Physics 30, 1301–1305. DOI.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Near-Extremal and Extremal Stability