Extremal Horizon Instability Interface

Degeneracy removes the horizon redshift estimate and permits conserved transverse data. For linear waves this produces Aretakis behavior: a field can decay while one transverse derivative fails to decay and higher derivatives grow along the event horizon. That statement is neither a universal destruction theorem nor a complete stability result. The answer depends on the field equations, symmetry class, derivative norm, exact or near extremality, and whether backreaction is included.

Helpful background. Exact Solutions of Einstein’s Equations supplies extremal Reissner–Nordström and Kerr backgrounds; Causality and Global Structure supplies event horizons and late-time limits; The Einstein Equation supplies backreaction and curvature; Black-Hole Perturbation Theory develops the detailed evolution estimates; Black-Hole Geometry and Mechanics supplies the degenerate Killing geometry isolated here.

The local conservation mechanism

Consider a spherically symmetric massless scalar $\psi(v,r)$ on extremal Reissner–Nordström, with $L=GM$ and

$$ds^2=-D(r)dv^2+2\,dv\,dr+r^2d\Omega_2^2, \qquad D(r)=\left(1-\frac{L}{r}\right)^2.$$

The wave equation can be written

$$2r\,\partial_v\partial_r(r\psi) +\partial_r\!\left[(r-L)^2\partial_r\psi\right]=0.$$

At the horizon, both $(r-L)^2$ and its first derivative vanish. Therefore

$$H_0=\left.\partial_r(r\psi)\right|_{r=L}$$

is independent of advanced time $v$. For generic data $H_0\ne0$. If $\psi$ itself decays, then $\partial_r\psi\to H_0/L$ on the horizon. Differentiating the wave equation once in $r$ consequently forces $\partial_r^2(r\psi)$ to acquire a term proportional to $-H_0v/L^2$. The conserved first derivative seeds linear growth one derivative higher.

This calculation isolates the role of the double zero. For a subextremal horizon, $D'(r_+)\ne0$ and the horizon equation instead contains the redshift term controlled by $\kappa\gt0$. Aretakis proves the corresponding nondecay and blow-up results under explicit axisymmetric horizon hypotheses in Aretakis 2015, §2.1, assumptions A1–A4, p. 4; §§3–4, Proposition 3.1 and Theorems 1–2, pp. 8–12.

What “instability” must specify

Several inequivalent questions sit behind that word.

Quantity and regularity. Pointwise decay of $\psi$, bounded energy, and growth of $\partial_r^2\psi$ can all hold simultaneously. Stability in a $C^0$ or degenerate energy norm does not imply stability in a norm controlling two transverse derivatives.

Test field or geometry. A scalar on a fixed background does not include backreaction. Electromagnetic and gravitational perturbations require gauge-invariant variables and constraint-compatible data. Curvature components may grow even when scalar polynomial invariants do not.

Symmetry and mode content. Spherical, axisymmetric, and general nonaxisymmetric data are different problems. A theorem for one azimuthal mode does not automatically control an infinite superposition.

Exact or near extremality. The exact horizon has $\kappa=0$ and an exact conservation law. A near-extremal horizon has small positive $\kappa$; it can display a long Aretakis-like transient before the redshift becomes effective, but the exact conserved charge is absent.

Full exterior or throat. NHEK and the full Kerr exterior have different boundaries. A mode statement on NHEK requires throat boundary conditions and does not by itself prove a statement for asymptotically flat Kerr.

Extremal-horizon results through 12 August 2026

The following boundary is deliberately dated. It compares theorem, perturbative, fixed-background approximation, numerical, and open-conjecture evidence available through 2026-08-12 and gives no universal verdict for all extremal black holes. The evidence classes cannot be pooled: each row has a different theory, data class, observable, and standard of support.

Result	Evidence class and domain	Established conclusion	Limitation
Aretakis 2015, assumptions A1–A4 and Theorems 1–2	Analytic theorem within the linear fixed-background approximation, for a massless scalar on smooth axisymmetric extremal-horizon backgrounds satisfying the stated hypotheses	A conserved horizon quantity obstructs decay of one transverse derivative and seeds higher-derivative growth.	Exact extremality, regular data, and the stated horizon geometry; no metric backreaction.
Lucietti and Reall 2012, §2.2, pp. 4–8	Linear electromagnetic and gravitational perturbation theory, in gauge-invariant Teukolsky variables on fixed exactly extremal Kerr	Perturbation variables exhibit analogous horizon nondecay or growth.	Fixed asymptotically flat background and linearized fields; not a nonlinear vacuum evolution theorem.
Gajic 2023, §§4 and 12, pp. 20–23 and 104–108	Analytic linear scalar theorem on the fixed asymptotically flat extremal Kerr exterior, for solutions supported on a bounded set of azimuthal modes	Precise late-time asymptotics include stronger azimuthal instabilities and an imprint at null infinity.	Preprint; does not control arbitrary infinite-mode data or backreaction.
Angelopoulos, Kehle, and Unger 2026, Theorems 1–2, pp. 23–30	Nonlinear theorem for spherical Einstein–Maxwell–neutral-scalar evolution on a codimension-one stable set	Low-regularity metric convergence and scalar decay coexist with a nonlinear Aretakis hierarchy and Ricci nondecay or growth.	Spherical, neutral-scalar, asymptotically extremal data; not Kerr or generic untuned data.
Gelles and Pretorius 2026, §§III–V, especially Figs. 7–10, 12–14, and 18, arXiv PDF pp. 3–16	Fully nonlinear numerical spherical Einstein–Maxwell–charged-scalar evolution tuned to the extremal threshold	The constructed event horizon has growing energy density, charge hair, and scalar curvature that appears to grow without bound just inside.	Numerical and family-dependent, at finite resolution and time; neither a theorem nor evidence for generic vacuum Kerr.
Dafermos 2025, §6, Conjectures 6.1–6.2 and Corollaries 6.1–6.2	Perspective and conjectural synthesis for extremal Reissner–Nordström and Kerr dynamics	Exact extremality is framed as a dynamically tuned threshold, and the general stability problem remains open.	Organizes open problems and possible outcomes; supplies neither a proof nor a numerical endpoint.
Kunduri and Lucietti 2013, §2.1, pp. 11–12	Exact local geometric scaling for a smooth degenerate horizon; using a throat solution as a proxy for the full exterior is an additional approximation	The limiting throat is an exact spacetime on bounded scaled-coordinate regions and may have enhanced symmetry.	Parent asymptotics and global boundaries are not inherited; throat boundary conditions do not prove full-exterior decay or instability.

For linear scalar fields, Aretakis established horizon conservation laws and the generic nondecay/growth hierarchy for broad axisymmetric extremal-horizon settings. Lucietti and Reall then applied the Teukolsky equations to show analogous behavior for linearized electromagnetic and gravitational perturbations of extremal Kerr Lucietti and Reall 2012, §2.2, especially pp. 4–8. These are genuine linear horizon instabilities, but they do not constitute a nonlinear vacuum evolution theorem.

For nonaxisymmetric scalar waves on extremal Kerr, Gajic derived precise late-time behavior for solutions supported on a bounded set of azimuthal modes and found stronger azimuthal instabilities, including an imprint at null infinity Gajic 2023, §§4 and 12, pp. 20–23 and 104–108. This advances the nonsymmetric linear problem without closing the nonlinear Einstein problem or the most general infinite-mode evolution problem.

The nonlinear picture is subtler rather than uniformly worse. Angelopoulos, Kehle, and Unger proved codimension-one nonlinear asymptotic stability of the extremal Reissner–Nordström family in the spherically symmetric Einstein–Maxwell–neutral-scalar system. For data on their stable set, the metric converges in low regularity to an extremal Reissner–Nordström solution and the scalar decays, while the first transverse derivative remains approximately constant, the second grows, and suitable Ricci components inherit nondecay or growth. The full hypotheses and paired stability/instability statements are Angelopoulos, Kehle, and Unger 2026, §§3.1–3.4, Theorems 1–2, pp. 23–30.

That result is a counterexample to the inference “Aretakis growth prevents all nonlinear late-time convergence.” It is also not unconditional stability: the model is spherically symmetric, includes a neutral scalar, and the asymptotically extremal data form a codimension-one set. The same work explains nearby subextremal evolutions as exhibiting transient Aretakis behavior Angelopoulos, Kehle, and Unger 2026, §1.3, Remark 1.8, p. 15. As of the stated cutoff, a corresponding general nonlinear stability or instability theorem for extremal Kerr in vacuum is unresolved.

Solved diagnostic: apparently conflicting observations

Suppose an evolution has $\psi\to0$, a bounded low-order energy, $\partial_r(r\psi)\to H_0\ne0$, and $\partial_r^2(r\psi)\sim-H_0v/L^2$ on the horizon. Is the evolution stable or unstable?

Solution

Both labels can be correct only after a norm is named. The field decays in the stated pointwise sense, and the stated low-order energy remains bounded; the latter is stability or boundedness in that energy, not asymptotic stability unless the energy norm itself tends to zero. The evolution is not asymptotically stable in a norm requiring the first transverse derivative to decay, and it is unbounded in a norm controlling the growing second transverse derivative. Nothing in the data given proves that the event horizon is destroyed, that a curvature scalar diverges, or that the same behavior holds without symmetry. A complete claim must add the equations, admissible data, gauge-invariant quantity, spacetime region, derivative order, and late-time norm.

This page identifies the geometric seed and the present logical boundary. The derivation of decay rates, Teukolsky estimates, mode sums, transient near-extremal behavior, and nonlinear bootstrap arguments proceeds in Black-Hole Perturbation Theory. The chapter overview returns to the stationary geometry and its classification.

References

• Angelopoulos, Yannis, Christoph Kehle, and Ryan Unger. “Nonlinear Stability of Extremal Reissner–Nordström Black Holes in Spherical Symmetry.” Forum of Mathematics, Pi 14, e3 (2026). DOI. Open PDF.
• Aretakis, Stefanos. “Horizon Instability of Extremal Black Holes.” Advances in Theoretical and Mathematical Physics 19 (2015): 507–530. DOI. Open PDF.
• Dafermos, Mihalis. “The Stability Problem for Extremal Black Holes.” General Relativity and Gravitation 57, 60 (2025). DOI.
• Gajic, Dejan. “Azimuthal Instabilities on Extremal Kerr.” arXiv:2302.06636 [gr-qc] (2023). arXiv.
• Gelles, Zachary, and Frans Pretorius. “Nonlinear Endpoint of Charged Horizon Instabilities.” Physical Review D 113, 124067 (2026). DOI. Open PDF.
• Kunduri, Hari K., and James Lucietti. “Classification of Near-Horizon Geometries of Extremal Black Holes.” Living Reviews in Relativity 16 (2013): 8. DOI.
• Lucietti, James, and Harvey S. Reall. “Gravitational Instability of an Extreme Kerr Black Hole.” Physical Review D 86, 104030 (2012). DOI. Open PDF.