Characteristic gluing to extremal Kerr horizons and the third law of black hole thermodynamics in vacuum

Stefan Czimek

AI summary · arXiv: 2610.35836v1 · math.AP

Characteristic gluing to extremal Kerr horizons and the third law of black hole thermodynamics in vacuum

Stefan Czimek

Original paper Submitted to arXiv on 26 Oct. 2026 84 pages, all comments welcome License: CC BY 4.0 Also in: fr

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The essentials

The paper proves exact characteristic gluing from Kerr exterior spheres to extremal Kerr horizons and uses it to construct a vacuum counterexample to the black-hole third law.

Question

Can large, non-axisymmetric vacuum data on a Kerr exterior sphere be glued exactly to an extremal Kerr horizon section? If so, can this establish that a non-extremal black hole can become extremal in finite advanced time in vacuum gravity?

Method

The author solves the vacuum null-constraint hierarchy along one outgoing null hypersurface, starting from a Boyer–Lindquist sphere and targeting an extremal Kerr horizon of sufficiently large mass. The construction raises angular momentum to the extremal value using non-axisymmetric shear pulses, damps unmatched incoming data along a long neck, and solves the endpoint matching problem using an all-frequency linear isomorphism and a nonlinear contraction argument. The gluing theorem assumes angular regularity s ≥ 14; its application constructs one-ended asymptotically flat vacuum initial data on a hypersurface diffeomorphic to R³.

Findings

Result Every Boyer–Lindquist sphere outside a Kerr black hole can be glued along one outgoing null hypersurface to an extremal Kerr horizon sphere of sufficiently large mass, with full sphere data matched exactly.
See the passage in the paper
every Boyer–Lindquist sphere outside a Kerr black hole, in particular every symmetry sphere of a Schwarzschild exterior, can be glued along a single outgoing null hypersurface to a horizon sphere of an extremal Kerr black hole of sufficiently large mass, with the full sphere data matched exactly at both ends.
Result The construction raises angular momentum to extremality by non-axisymmetric gravitational radiation; in axisymmetry the angular momentum is conserved.
See the passage in the paper
In vacuum the only way to reach extremal Kerr is to spin the black hole up to $J=M_{f}^{2}$, and angular momentum can only be carried by non-axisymmetric gravitational radiation: in axisymmetry the Komar angular momentum is conserved (Corollary 3.10 is the null-hypersurface version of this).
Result The long non-expanding neck exponentially damps the unmatched incoming data before the final matching stage.
See the passage in the paper
Along such a segment the incoming data decay like $e^{-2\omega v}$ and $e^{-4\omega v}$, so the neck damps the uncontrolled transversal data of Steps 1 and 2 by $O(e^{-2\omega T})$.
Result The endpoint linearised matching problem is invertible at all angular frequencies through a factorisation involving angular Teukolsky–Starobinsky operators.
See the passage in the paper
Together with the shape of the terminal section and the boost this gives an isomorphism at all angular frequencies; the key is a factorisation of $\mathcal{P}$ through an angular Teukolsky–Starobinsky product.
Result The resulting vacuum development contains an exactly Schwarzschild apparent horizon first and an exactly extremal Kerr event horizon later, providing a rigorous counterexample to the vacuum third law.
See the passage in the paper
There exist one-ended asymptotically flat vacuum initial data $(\Sigma,g_{0},k_{0})$, $\Sigma\cong\mathbb{R}^{3}$, $(g_{0},k_{0})\in H^{7/2-}_{\mathrm{loc}}\times H^{5/2-}_{\mathrm{loc}}$, whose maximal globally hyperbolic development first contains an exactly Schwarzschild apparent horizon and later an exactly extremal Kerr event horizon.

Limitations

The theorem treats extremal Kerr targets, not sub-extremal Kerr targets; its matching argument depends on degeneracy of the extremal horizon.
See the passage in the paper
Target spheres in sub-extremal Kerr, $|a_{f}|<M_{f}$, are not treated here: the matching at the target uses the degeneracy of the extremal horizon, which leaves the incoming second fundamental form of a horizon section undetermined by its intrinsic geometry (Remark 4.4).
The construction requires the final extremal Kerr mass to be sufficiently large relative to the initial sphere; the paper does not claim that an area inequality alone is enough.
See the passage in the paper
“Sufficiently large” is relative to $S_{1}$: the construction needs the target area to be much larger than the area of $S_{1}$; we do not claim that the area inequality $8\pi M^{2}>|S_{1}|$ alone suffices.
The constructed metric need not be C² at the central crease of the global construction.
See the passage in the paper
The metric is $C^{2}$ across the characteristic gluing interfaces, but need not be $C^{2}$ at the central crease of the construction, as in [23, Remark 4.1].

Why it matters

The authors present the gluing theorem as proving Kehle and Unger’s conjecture for extremal Kerr targets and enabling the first rigorous counterexample to the third law in vacuum gravity. The result extends non-perturbative characteristic gluing to non-axisymmetric vacuum data and shows that extremality can be reached in finite advanced time in the constructed development.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Black hole mechanics; Extremal Kerr; Vacuum Einstein equations; Characteristic gluing; Third law of black hole mechanics

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Comprehensive Summary
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Original abstract

The paper asks whether a vacuum black hole that is initially non-extremal can become extremal in finite advanced time. Israel’s formulation of the third law of black hole mechanics says that it cannot. Earlier constructions showed that this conclusion can fail in models coupled to matter, while the only known vacuum construction reached a slowly rotating final Kerr black hole. The paper proves a characteristic gluing theorem that reaches an extremal Kerr horizon and uses it to construct a vacuum counterexample to the third law. Its central result is a non-perturbative, non-axisymmetric solution of the vacuum null constraint equations on a single outgoing null hypersurface. The solution begins at any Boyer–Lindquist sphere outside a Kerr black hole—including a symmetry sphere in Schwarzschild—and ends at a section of an extremal Kerr horizon, provided the final mass is sufficiently large. The sphere data match exactly at both ends, including the incoming second fundamental form and transversal curvature. The construction is non-perturbative in that it does not rely on small data near Minkowski space. Instead, its perturbative arguments are carried out around explicit geometric configurations. The method proceeds in several stages. First, the authors build a long bridge to the intrinsic geometry of an extremal horizon. Since angular momentum cannot be added in axisymmetry in vacuum, they use non-axisymmetric gravitational-wave shear pulses to raise the angular momentum to its extremal value, $J=M^2$. A focusing tail then brings the horizon area to its target value while making the outgoing expansion vanish at every angle. The remaining uncontrolled incoming data are damped along a long, non-expanding neck. Their influence decays exponentially, at rates involving $\exp(-2\omega T)$ and $\exp(-4\omega T)$, where $T$ is the neck length and $\omega$ is the chosen non-affinity. In a final collar, a small shear seed adjusts the incoming data. The authors show that the linearised matching problem is invertible at every angular frequency, using a factorisation of a fourth-order angular operator through a Teukolsky–Starobinsky product. A nonlinear contraction argument then yields exact matching. An exact weighted trace identity and a calculation of the second variation establish that the endpoint boost is positive. Thus, the endpoint is a genuine future-horizon section rather than merely a formal matching target. To determine the sign, the authors construct a moment-free seed whose leading quadratic effect makes the relevant functional negative. The reported normalised static coefficient is approximately $0.49842$, below $0.5$. This allows them to choose a seed with strictly negative second variation and, consequently, a positive boost, up to errors of order the cube of the seed amplitude and $\exp(-2\omega T)$. The data can be realised in a vacuum spacetime with one-sided $C^4$ regularity and $C^2$ matching across the gluing interfaces. As an application, the paper constructs one-ended, asymptotically flat vacuum initial data on a space topologically equivalent to $\mathbb{R}^3$. The maximal development contains an exactly Schwarzschild apparent-horizon segment, with trapped spheres just inside it, followed later by an exactly extremal Kerr event horizon. The initial slice lies outside the event horizon and contains no trapped surfaces. After a sufficiently late Cauchy level, the escaping region and event horizon coincide with those of extremal Kerr. The result is a rigorous counterexample to the vacuum third law as formulated in the paper.

Can a vacuum black hole become extremal in finite time?

One of the familiar rules of black hole physics says that an ordinary black hole cannot be pushed all the way to extremality in a finite time. Extremal black holes sit at a limit: for a Kerr black hole, the spin reaches its maximum when its angular momentum satisfies J = M squared. But is that limit truly out of reach in a vacuum spacetime, where no matter is present?

In “Characteristic gluing to extremal Kerr horizons and the third law of black hole thermodynamics in vacuum,” Stefan Czimek gives a striking answer. The paper constructs a vacuum spacetime in which an initially non-extremal black hole evolves to an extremal Kerr horizon after a finite interval of advanced time. The result provides a counterexample to Israel’s formulation of the third law of black hole mechanics.

A vacuum route to extremality

Earlier examples had already shown that the third-law conclusion can fail in settings involving matter. A vacuum construction was also known, but it led to a slowly rotating Kerr black hole rather than an extremal one. The new result reaches an extremal Kerr horizon without matter and without assuming the solution is a small perturbation of empty, nearly flat space.

The main technical achievement is a characteristic gluing theorem. It constructs data satisfying the vacuum null constraint equations on a single outgoing null hypersurface. The hypersurface can begin at any Boyer–Lindquist sphere outside a Kerr black hole, including a symmetry sphere in Schwarzschild, and end at a section of an extremal Kerr horizon, if the final mass is sufficiently large.

Crucially, the match is exact at both ends. It includes not only the geometry of the spherical sections, but also the incoming second fundamental form and transversal curvature. This precision makes it possible to join the constructed region to the surrounding spacetime rather than merely approach the desired horizon geometry.

Building the bridge

Moving from an initially non-extremal black hole to extremal Kerr requires controlling several geometric quantities. In axisymmetry, vacuum evolution cannot simply add angular momentum in the way the construction needs. The authors therefore use non-axisymmetric gravitational-wave shear pulses to increase the angular momentum until it reaches the extremal value.

The construction then uses a focusing tail to bring the horizon area to its target value while arranging for the outgoing expansion to vanish at every angle. Some incoming data remain uncontrolled after these steps. To reduce their effect, the authors insert a long, non-expanding neck into the null hypersurface.

Along this neck, the unwanted influence decays exponentially with its length. The relevant suppression includes factors such as exp(-2ωT) and exp(-4ωT), where T is the neck length and ω is a chosen non-affinity parameter. A longer neck therefore makes it easier to manage the data at the final end.

Exact matching and a genuine horizon

Near the endpoint, a small shear seed is used to tune the remaining incoming data. The linearised matching problem must be solvable across all angular frequencies, not just for a restricted set of patterns. The authors establish this invertibility by factoring a fourth-order angular operator through a Teukolsky–Starobinsky product. They then use a nonlinear contraction argument to obtain an exact match.

Matching the geometry alone would not prove that the endpoint is actually a future horizon. The paper also establishes the required sign of the endpoint boost. An exact weighted trace identity and a second-variation calculation show how to obtain a positive boost, confirming that the endpoint is a genuine future-horizon section.

That sign depends on choosing the seed carefully. The authors construct a moment-free seed whose leading quadratic contribution makes the relevant functional negative. A normalised static coefficient in the calculation is approximately 0.49842, just below 0.5. This gap lets the authors choose a seed with strictly negative second variation. The resulting positive boost holds subject to errors of cubic order in the seed amplitude and errors that decay exponentially with the neck length.

From null data to a black-hole spacetime

The null data can be realised in a vacuum spacetime with one-sided C4 regularity and C2 matching across the gluing interfaces. The paper uses this to build one-ended, asymptotically flat vacuum initial data on a space topologically equivalent to ordinary three-dimensional space.

The maximal development has a particularly revealing sequence of features:

It contains an exactly Schwarzschild apparent-horizon segment.
Trapped spheres lie just inside that segment.
Later, the spacetime contains an exactly extremal Kerr event horizon.
The initial slice lies outside the event horizon and contains no trapped surfaces.
After a sufficiently late Cauchy level, the escaping region and event horizon coincide with those of extremal Kerr.

Why the result matters

The construction is non-perturbative in an important sense: it is not built by treating the entire spacetime as a small disturbance of Minkowski space. Instead, its perturbative steps are organised around carefully chosen geometric configurations, including the long neck and the extremal-horizon target.

The result does not say that extremality is easy to reach or that typical black holes do so. It shows something more precise: under the vacuum equations, the third-law prohibition is not absolute in the formulation studied here. With carefully engineered gravitational radiation and exact characteristic gluing, a non-extremal starting point can lead to an extremal Kerr horizon in finite advanced time.

Created on 02 Nov. 2026

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Characteristic gluing to extremal Kerr horizons and the third law of black hole thermodynamics in vacuum

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

Can a vacuum black hole become extremal in finite time?

A vacuum route to extremality

Building the bridge

Exact matching and a genuine horizon

From null data to a black-hole spacetime

Why the result matters

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