Event horizon gluing and black hole formation in vacuum: the very slowly rotating case

In this paper, we initiate the study of characteristic event horizon gluing in vacuum. More precisely, we prove that Minkowski space can be glued along a null hypersurface to any round symmetry sphere in a Schwarzschild black hole spacetime as a $C^2$ solution of the Einstein vacuum equations. The method of proof is fundamentally nonperturbative and is closely related to our previous work in spherical symmetry [KU22] and Christodoulou's short pulse method [Chr09]. We also make essential use of the perturbative characteristic gluing results of Aretakis-Czimek-Rodnianski [ACR21a; CR22]. As an immediate corollary of our methods, we obtain characteristic gluing of Minkowski space to the event horizon of very slowly rotating Kerr with prescribed mass $M$ and specific angular momentum $a$. Using our characteristic gluing results, we construct examples of vacuum gravitational collapse to very slowly rotating Kerr black holes in finite advanced time with prescribed $M$ and $0\le |a|\ll M$. Our construction also yields the first example of a spacelike singularity arising from one-ended, asymptotically flat gravitational collapse in vacuum.


Introduction
Characteristic gluing is a powerful new method for constructing solutions of the Einstein field equations for a spacetime metric g and coupled matter fields by gluing together two existing solutions along a null hypersurface.The setup of characteristic gluing is depicted in Fig. 1 below and we will repeatedly refer to this diagram for definiteness.
(M, g, . . . ) Figure 1: Penrose diagram depicting the setup of characteristic gluing.The null hypersurface C is declared to be "outgoing." In Fig. 1, the two dark gray regions R 1 and R 2 carry Lorentzian metrics and matter fields which satisfy (1.1).The goal is to embed these regions into a spacetime (M, g, . . . ) which satisfies (1.1) globally, in the configuration depicted in Fig. 1.The characteristic gluing problem reduces to constructing characteristic data along a null hypersurface C going between spheres S 1 ⊂ R 1 and S 2 ⊂ R 2 , so that after constructing the light gray regions in Fig. 1 by solving a characteristic initial value problem, the resulting spacetime is of the desired global regularity.
Characteristic gluing is a useful tool for constructing spacetimes that share features of two existing solutions, and therefore display interesting behavior.This technique was recently pioneered for the Einstein vacuum equations Ric(g) = 0 (1.2) by Aretakis, Czimek, and Rodnianski [ACR21a], and we will give an overview of their work in Section 1.1.1below, including the obstruction-free gluing of Czimek-Rodnianski [CR22], which we will make crucial use of.Characteristic gluing was also recently used by the present authors to disprove the so-called third law of black hole thermodynamics in the Einstein-Maxwell-charged scalar field model in spherical symmetry [KU22]; see already Section 1.2.We refer the reader to [KU22] for a general formalism for the characteristic gluing problem.The present work is the first in a series of papers aimed at extending [KU22] to the Einstein vacuum equations (1.2).
The most basic question in the study of characteristic gluing is the following: Question 1. Which spheres S 1 and S 2 in which vacuum spacetimes can be characteristically glued as in Fig. 1 as a solution of the Einstein vacuum equations (1.2)? Are there any nontrivial obstructions?If so, can they be characterized geometrically?
For example, a genuine obstruction arises from Raychaudhuri's equation (see already (A.5)), which implies that S 2 cannot be strictly outer untrapped if S 1 is (marginally) outer trapped.Another genuine obstruction arises from the rigidity of the positive mass theorem, which implies that if R 2 is Minkowski space, then R 1 is either Minkowski space or must be singular or incomplete in some sense.
Our first theorem shows that the characteristic gluing of Minkowski space to (positive mass) Schwarzschild solutions is completely unobstructed, provided that we aim to glue to a symmetry sphere in Schwarzschild.In the statements of our theorems, refer to Fig. 1.
Theorem 1.Let M > 0. Let S 2 be any non-antitrapped symmetry sphere in the Schwarzschild solution of mass M .Then S 2 can be characteristically glued to a sphere S 1 as depicted in Fig. 1, to order C 2 as a solution of the Einstein vacuum equations (1.2), where S 1 is a spacelike sphere in Minkowski space which is arbitrarily close to a round symmetry sphere.
For the precise statement of this theorem, see already Theorem A in Section 3.2 below.Our method also immediately generalizes to very slowly rotating Kerr, and gives the following particularly clean statement about event horizon gluing: Theorem 2. There exists a constant 0 < a 0 1 such that if S 2 is a spacelike section of the event horizon of a Kerr black hole with mass M > 0 and specific angular momentum a satisfying 0 ≤ |a|/M ≤ a 0 , then S 2 can be characteristically glued to a sphere S 1 as depicted in Fig. 1, to order C 2 as a solution of the Einstein vacuum equations (1.2), where S 1 is a spacelike sphere in Minkowski space which is close to a round symmetry sphere.
This theorem is a special case of Theorem B in Section 3.3 below.
Remark 1.1.Strictly speaking, Theorem B applies to Kerr coordinate spheres on the event horizon.However, it is easy to see that any spacelike section on the event horizon can be connected to a Kerr coordinate sphere.
Remark 1.2.There is an apparent asymmetry in the statements of Theorem 1 and Theorem 2 about the allowable S 2 's.In fact, in Theorem B below, we show that any Kerr coordinate sphere can be connected to a sphere in Schwarzschild with smaller mass, but the maximum value of allowed angular momentum depends on the sphere in a non-explicit way that we prefer to explain later in the paper, see already Section 2.2.
Remark 1.3.In Theorem 1, the bottom sphere S 1 can be made arbitrarily close to an exact symmetry sphere in Minkowski, whereas in Theorem 2, the closeness to an exact symmetry sphere is limited by the size of a/M .Remark 1.4.The C 2 regularity of the spacetime metric in Theorem 1 and Theorem 2 is due to limited regularity in the direction transverse to C. The regularity of the metric in directions tangent to C can be made arbitrarily high (but finite).By using Theorem 2 and solving the Einstein equations backwards, we can construct examples of gravitational collapse to a black hole of prescribed very small angular momentum.The proof will be given in Section 4 below.
Corollary 1 (Gravitational collapse with prescribed M and 0 ≤ |a| M ).Let a 0 be as in Theorem 2. Then for any mass M > 0 and specific angular momentum a satisfying 0 ≤ |a|/M ≤ a 0 , there exist oneended asymptotically flat Cauchy data for the Einstein vacuum equations (1.2) on Σ ∼ = R 3 , satisfying the constraint equations, such that the maximal future globally hyperbolic development (M 4 , g) contains a black hole BH .= M \ J − (I + ) and has the following properties: • The Cauchy surface Σ lies in the causal past of future null infinity, Σ ⊂ J − (I + ).In particular, Σ does not intersect the event horizon H + .= ∂(BH) or contain trapped surfaces.
• For sufficiently late advanced times v ≥ v 0 , the domain of outer communication, including the event horizon H + , is isometric to that of a Kerr solution with parameters M and a.For v ≥ v 0 , the event horizon of the spacetime can be identified with the event horizon of Kerr.
For the relevant Penrose diagram, consult Fig. 2 below.By performing characteristic gluing as in Theorem 1 of Minkowski space to spheres in a Schwarzschild solution lying just inside the horizon and using Cauchy stability, we also obtain: Corollary 2 (Gravitational collapse with a spacelike singularity).There exist one-ended asymptotically flat Cauchy data for the Einstein vacuum equations (1.2) on Σ ∼ = R 3 , satisfying the constraint equations, such that the maximal future globally hyperbolic development (M 4 , g) contains a black hole BH .= M \ J − (I + ) and has the following properties: • The Cauchy surface Σ lies in the causal past of future null infinity, Σ ⊂ J − (I + ).In particular, Σ does not intersect the event horizon H + .= ∂(BH) or contain trapped surfaces.
• For sufficiently late advanced times v ≥ v 0 , the domain of outer communication, together with a full double null slab lying in the interior of the black hole, is isometric to a portion of a Schwarzschild solution as depicted in Fig. 3.The double null slab terminates in the future at a spacelike singularity, isometric to the "r = 0" singularity in Schwarzschild.
We will sketch the proof of this result in Section 4 below.
Figure 3: Example of gravitational collapse with a piece of a spacelike singularity emanating from timelike infinity i + .Characteristic gluing is performed along the textured line segment, where the top gluing sphere has radius very close to the Schwarzschild radius of the black hole to be formed.

The characteristic gluing problem
The study of the characteristic gluing problem was initiated by Aretakis for the linear scalar wave equation on general spacetimes (M 3+1 , g) in [Are17].Aretakis showed that there is always a finite-dimensional (but possibly trivial) space of obstructions to the characteristic gluing problem.More precisely, he showed that there are at most finitely many (possibly none) conserved charges that are computed from the given solutions at S 1 and S 2 in Fig. 1 that determine whether characteristic gluing can be performed.These charges are conserved along C for any solution of (1.3).This gives a definitive answer1 to Question 1 for (1.3):There is a precise characterization of which spheres can be glued-the matching of all conserved charges is both necessary and sufficient.

Characteristic gluing for the Einstein vacuum equations
Characteristic gluing for the Einstein vacuum equations (1.2) was initiated by Aretakis, Czimek, and Rodnianski in a fundamental series of papers [ACR21a; ACR21b; ACR21c].They study the perturbative regime around Minkowski space, that is, when both spheres S 1 and S 2 in Fig. 1 are close to symmetry spheres in Minkowski space.The strategy employed is to linearize the Einstein equations around Minkowski space in the framework of Dafermos-Holzegel-Rodnianski [DHR], solve the characteristic gluing problem for the linearized Einstein equations, and then conclude a small-data nonlinear gluing result by an implicit function theorem argument.
In the course of their argument, they discover that the linearized Einstein equations around Minkowski space in double null gauge possess infinitely many conserved charges.However, it turns out that all but ten of these charges are due to gauge invariance of the Einstein equations (cf. the pure gauge solutions of [DHR]).The remaining charges, which we define precisely in Definition 2.9 below, are genuine obstructions to the linear characteristic gluing problem, and must therefore be assumed to be equal at S 1 and S 2 in order for the inverse function scheme to yield a genuine solution.
The conserved charges of Aretakis-Czimek-Rodnianski can be identified with the ADM energy, linear momentum, angular momentum, and center of mass.This identification is used in [ACR21c] to give a new proof of the spacelike gluing results of [Cor00; CS06; CS16] using characteristic gluing.
Later, Czimek and Rodnianski [CR22] made the fundamental observation that the linear conservation laws can be violated at the nonlinear level by certain explicit "high frequency" seed data for the characteristic initial value problem.2They then use these high frequency perturbations to adjust the linearly conserved charges in the full nonlinear theory, so that the main theorem of [ACR21a] applies.The result, which we state as Theorem 5 in Section 2.3.2 below, is that two spheres close to Minkowski space can be glued if the differences of the conserved charges satisfy a certain coercivity condition.Roughly, the assumption is that the change in the Hawking mass be larger than the changes in the other conserved charges and that the change in angular momentum is itself much smaller than the distance of S 1 and S 2 to spheres in Minkowski space.Their result has the remarkable corollary of obstruction-free spacelike gluing of asymptotically flat Cauchy data to Kerr in the far region.
We note at this point that the analysis of [ACR21a; ACR21b; ACR21c; CR22] is limited to C 2 regularity in the ingoing direction u, but allows for arbitrarily high regularity in v and angular directions.3It is not clear whether their analysis (especially [CR22]) can be generalized to higher order transverse derivatives.
The linearized characteristic gluing problem for (1.2) was redone in Bondi gauge and extended to incorporate a cosmological constant and different topologies of the cross sections of the null hypersurface C by Chruściel and Cong [CC22].This work also addresses linearized characteristic gluing of higher order transverse derivatives.
Question 2. Is there a general geometric characterization of conservation laws associated to the linearized Einstein equations around a fixed background?Is there always a finite number of conservation laws?Is the generic spacetime free of conservation laws at the linear level?
One might also wonder if there is a precise connection between the conservation laws observed in the null setting with the cokernel of the linearized constraint map studied in the spacelike gluing problem [Cor00; CS06; CD03].We refer the reader to [CC23] for more discussion about these issues.

Characteristic gluing in spherical symmetry
The present authors have studied the characteristic gluing problem for the Einstein-Maxwell-charged scalar field system in spherical symmetry [KU22].Our main theorem can be stated as follows: Theorem 3 (Theorem 2 in [KU22]).Let k ∈ N be a regularity index, q ∈ [−1, 1] a charge to mass ratio, and e ∈ R \ {0} a fixed coupling constant.For any M f sufficiently large depending on k, q, and e, and any 0 ≤ M i ≤ 1 8 M f , and 2M i < R i ≤ 1 2 M f , there exist spherically symmetric characteristic data for the Einstein-Maxwell-charged scalar field system with coupling constant e gluing the Schwarzschild symmetry sphere of mass M i and radius R i to the Reissner-Nordström event horizon with mass M f and charge e f = qM f up to order k.
Our proof is fundamentally nonperturbative in that we work directly with the nonlinear equations and do not require a perturbative analysis.This is made possible by three ingredients: (i) The Hawking mass is glued by judiciously initiating the transport equations at S 1 or S 2 and directly exploiting gauge freedom in the form of boosting the double null gauge by hand.
(ii) The charge of the Maxwell field is glued by exploiting a monotonicity property of Maxwell's equation specific to spherical symmetry.
(iii) Transverse derivatives of the scalar field are glued by exploiting a parity symmetry of the Einstein-Maxwell-charged scalar field system specific to spherical symmetry and invoking the Borsuk-Ulam theorem.
The argument in the present paper has two crucial ingredients: the implementation of idea (i) above in the context of the Einstein vacuum equations, and the obstruction-free characteristic gluing of Czimek-Rodnianski [CR22], which replaces the Borsuk-Ulam argument in vacuum.See already Section 1.3 for the outline of our proof.

The third law of black hole thermodynamics in vacuum
The third "law" of black hole thermodynamics is the conjecture that a subextremal black hole cannot become extremal in finite time by any continuous process, no matter how idealized, in which the spacetime and matter fields remain regular and obey the weak energy condition [BCH73;Isr86].The main application of our previous characteristic gluing result in [KU22] is the following definitive disproof of the third law: Theorem 4 (Theorem 1 in [KU22]).The "third law of black hole thermodynamics" is false.More precisely, subextremal black holes can become extremal in finite time, evolving from regular initial data.In fact, there exist regular one-ended Cauchy data for the Einstein-Maxwell-charged scalar field system which undergo gravitational collapse and form an exactly Schwarzschild apparent horizon, only for the spacetime to form an exactly extremal Reissner-Nordström event horizon at a later advanced time.
We refer the reader to [KU22] for an extensive discussion of the history and physics of the third law.The black holes in Theorem 4 are constructed in two stages: First the scalar field is used to form an exact Schwarzschild apparent horizon in finite time, which is then charged up to extremality by exploiting the coupling of the scalar field with the electromagnetic field.Theorem 1 above can be viewed as a generalization of this first step.We conjecture that the second step can also be generalized to vacuum: Conjecture 1.The Schwarzschild symmetry sphere of mass M i and radius R i can be characteristically glued to any non-antitrapped Kerr coordinate sphere with radius R f R i in a Kerr solution with mass M f M i and specific angular momentum If this conjecture holds, Schwarzschild can be spun up to extremality.Arguing as in [KU22] and Corollary 1 in the present paper, this would imply Conjecture 2. The "third law of black hole thermodynamics" is already false in vacuum.More precisely, there exist regular one-ended Cauchy data for the Einstein vacuum equations (1.2) which undergo gravitational collapse and form an exactly Schwarzschild apparent horizon, only for the spacetime to form an exactly extremal Kerr event horizon at a later advanced time.
Remark 1.6.By using negatively charged pulses in [KU22], we can design characteristic data that also "discharges" the black hole.It would be very interesting to find a mechanism that can both "spin up" and "spin down" a Kerr black hole, or move the rotation axis without changing the angular momentum much.
Remark 1.7.It is not possible to have a solution of the pure Einstein-Maxwell equations which behaves like one of the solutions in Theorem 4. This is because the vacuum Maxwell equation d F = 0 always gives rise to a conserved electric charge (4π) −1 S F , even outside of spherical symmetry.On Schwarzschild, this charge is zero, and on Reissner-Nordström, it equals the charge parameter e.
Remark 1.8.Similarly, if a vacuum spacetime has an axial Killing field Z, then the Komar angular momentum (16π) −1 S dZ is conserved.Therefore, Conjecture 2 cannot be proved in axisymmetry.

Outline of the proof
In this section we give a very brief outline of the proof of Theorem 1.The gluing is performed in two stages and should be thought of as being performed backwards in time.First, a fully nonperturbative mechanism is used to connect the exact Schwarzschild sphere S 2 of mass M and radius R to a sphere data set S * which is very close to a Schwarzschild sphere of mass 0 < M * R and radius ≈ R. See already Proposition 3.1 below.In the second stage of the gluing, we use the main theorem of [CR22], which we state below as Theorem 5, as a black box.In order to satisfy the necessary coercivity conditions required for obstruction-free characteristic gluing, we choose 0 < ε M * R, where ε measures the closeness of S * to the (M * , R)-Schwarzschild sphere.
The nonperturbative mechanism which glues S * to S 2 involves the injection of two pulses 4 of gravitational waves (described mathematically by the shear χ) along C = [0, 1] v × S 2 , of amplitude δ 1/2 = O(ε ), together with a choice of outgoing null expansion tr χ at S 2 such that |tr χ| δ.In order to fix the Hawking mass of S 2 to be M , the ingoing null expansion tr χ is then chosen to be ≈ −δ −1 at S 2 . 5The Hawking mass of S * is fixed to be arbitrarily close to M * by tuning χ and using the monotonicity of Raychaudhuri's equation; see already Lemma 3.6. 6We then step through the null structure equations and Bianchi identities as in [Chr09, Chapter 2] and establish a δ-weighted hierarchy for the sphere data at S * ; see already Lemma 3.8.Finally, we boost the cone by δ, which brings S 2 to a reference Schwarzschild sphere and S * within ε of a reference Schwarzschild sphere with mass M * .This construction may be thought of as a direct adaptation, in vacuum, of the idea used to prove Schwarzschild event horizon gluing in spherical symmetry for the Einstein-scalar field system in [KU22, Section 4.1].

Relation to Christodoulou's short pulse method
After the boost, one can interpret the above approximate nonperturbative gluing mechanism as a "short pulse" data set defined on [0, δ] × S 2 as in [Chr09], but fired backwards.In this context, our equation (3.9) below should be compared with the condition (4.1) in [LY15] (see also [LM20;AL20]).This condition guarantees that certain components of the sphere data at S * are a posteriori closer to spherical symmetry.
Likewise, our Corollary 1 can be compared with the main theorem of Li and Mei in [LM20].In particular, we also prove trapped surface formation starting from Cauchy data outside of the black hole region.Their proof utilizes the trapped surface formation mechanism of Christodoulou [Chr09] and Corvino-Schoen spacelike gluing [CS06].
Our proof of trapped surface formation starting from Cauchy data is fundamentally different from [LM20] because it does not appeal to Christodoulou's trapped surface formation mechanism [Chr09].In fact, the only aspect of the evolution problem we require is Cauchy stability.Furthermore, we can directly prescribe the (very slowly rotating) Kerr parameters of the black hole to be formed.In particular, we may take a = 0, which guarantees the existence of a spacelike singularity, see Corollary 2. However, our data is of limited regularity (but still in a well-posed class).Nevertheless, by appealing to Cauchy stability once again, Corollary 1 has the further corollary of trapped surface formation starting from an open set of Cauchy data.

Characteristic initial data and characteristic gluing
In this section, we give a brief review of the characteristic gluing problem for the Einstein vacuum equations (1.2) in double null gauge [ACR21a; ACR21b; ACR21c; CR22].We follow the conventions of [CR22] unless otherwise stated.
If the reader is unfamiliar with double null gauge, we defer to [Chr09] for exposition.We have collected formulas and notions used explicitly in the present paper in Appendix A.

Sphere data, null data, and seed data
The terminology used in this paper is in agreement with [CR22], which we will be using as a black box, and therefore differs slightly from our previous paper [KU22].We hope this facilitates the reader in understanding exactly how the main notions and results from [CR22] are being used here.

Sphere data
Given a solution (M 4 , g) of the Einstein vacuum equations (1.2) and a sphere S in a double null foliation, the 2-jet of g can be determined from knowledge of the metric coefficients, Ricci coefficients, and curvature components.However, the equations themselves, such as the Codazzi equation (A.20) allow some of these degrees of freedom to be computed from the others, just in terms of derivatives tangent to S. This leads to the following definition: Definition 2.1 (C 2 sphere data, [ACR21b, Definition 2.4]).Let S be a 2-sphere.Sphere data x on S consists of a choice of round metric γ on S and the following tuple of S-tensors x = (Ω, / g, Ω tr χ, χ, Ω tr χ, χ, η, ω, Dω, ω, Dω, α, α), (2.1) where Ω is a positive function, / g a Riemannian metric, Ω tr χ, χ, Ω tr χ, ω, Dω, ω, Dω are scalar functions, η is a vector field, χ, χ, α and α are symmetric / g-traceless 2-tensors.
sphere data, [ACR21b, Definition 2.28]).Let S be a 2-sphere and m ≥ 0 an integer.Higher order sphere data x on S consists of a choice of round metric γ on S, the tuple (2.1), together with ( Dα, . . ., Dm α, D 2 ω, . . ., D m+1 ω), (2.2) where Dj α are symmetric / g-traceless 2-tensors and D j ω scalar functions.We will write x = (x low , x high ), where x low is a C 2 sphere data set and x high denotes the tuple (2.2).
When sphere data is obtained from a geometric sphere in a vacuum spacetime, one has to make a choice of normal null vector fields L and L. See Lemma A.1 below, for instance.As is well known, the null pair {L, L} can be "boosted" by the transformation where λ ∈ R + .This boost freedom was also quite useful in the preceding paper [KU22].
Definition 2.3 (Boosted sphere data).Let x be a sphere data set as in Definition 2.2 and λ ∈ R + .Then the λ-boosted sphere data set is the This is the effect that the boost (2.3) has on the metric coefficients, Ricci coefficients, and curvature components in double null gauge.
There is a norm x X m defined on C 2 u C 2+m v sphere data sets employed in [ACR21b; CR22], which is just a sum of high order (in the angular variable θ) Sobolev norms of the sphere data components [ACR21b, Definition 2.5].We will show very strong pointwise smallness for arbitrary numbers of angular derivatives later and thus will not need the exact form of these norms in order to apply the result of [CR22].
Definition 2.4 (Sphere diffeomorphisms).Given a diffeomorphism ψ : S 2 → S 2 , we let ψ act on C 2 u C 2+m v data sets by pullback on each component.

Null data
Definition 2.5 (Ingoing and outgoing null data [ACR21b, Definition 2.6]).Let v 1 < v 2 .An outgoing null data set is given by an assignment v → x(v), where x(v) is a C 2 sphere data set.We may say that the null data lives on the null hypersurface . An ingoing null data set is defined in the same way, but with the formal variable v replaced with u and η replaced by η in (2.1).
Higher order null data is defined in the obvious way, with x(v) being C 2 u C 2+m v sphere data.Null data on its own is not assumed to satisfy the null structure equations and Bianchi identities.
There are several norms on null data that are employed in [ACR21a; ACR21b; ACR21c; CR22].These include the standard norm X defined on ingoing and outgoing null data the high regularity norm X + defined on ingoing null data, and the high frequency norm X h.f.defined on outgoing null data using in obstructionfree characteristic gluing.As we will not need the precise forms of these norms in the present work, we refer the reader to [ACR21b, Definition 2.7] for details.

Christodoulou seed data
We will employ the following method, originating in the work of Christodoulou [Chr09], for producing solutions of the null structure and Bianchi equations along a null hypersurface C.
For definiteness, we seek a solution of the null constraints on the null cone segment C where D / g .= / L L / g as in Appendix A. 7 Let / g 1 be a Riemannian metric on S 1 which is conformal to / g(1), tr χ 1 and tr χ 1 be functions on S 1 , η 1 be a 1-form on S 1 , and χ1 and α 1 be / g 1 -traceless symmetric 2-tensors on S 1 .
The condition (2.4) on the representative / g of K can be imposed without loss of generality, i.e., K always contains a representative satisfying (2.4).Indeed, let / g ∈ [ / g] and let ψ .= exp( 7 Concretely, this means / g = / g(v) is a smooth 1-parameter family of Riemannian metrics on S 2 .We identify S 2 with Sv ⊂ C. Since b = 0, / L L / g = L L / g and relative to any angular coordinates ϑ A defined on S 1 extended to C according to Lϑ A = 0, (D / g) AB = ∂v( / g AB ). 8 That is, the quantities that may be freely prescribed.
Remark 2.1.In Lemma 3.1 below, we will directly construct a specific / g satsifying the volume form condition D(dµ / g ) = 0, which easily implies (2.4) by the first variation formula for area.
Outline of the proof of Lemma 2.1.Let φ 1 be the positive function on S 1 satisfying / g 1 = φ 2 1 / g 1 .We make the ansatz on C, where φ is now a positive function on C agreeing with φ 1 on S 1 .
We define relative to any Lie-transported angular coordinate system on the spheres.We set and let φ be the unique solution of the second order ODE with initial conditions (φ 1 , ∂ v φ 1 ).If φ remains strictly positive on all of C, then we let v 0 be any strictly negative number.If however φ has a zero on C, then we take v 0 to be the supremum of v ∈ [0, 1] for which inf Sv φ ≤ 0. This definition gives (2.7).
We now set χ .
From here, the full null data along C can be determined by stepping through the null structure and Bianchi equations in the right order, as in [Chr09].We will outline this procedure in the proof of Lemma 3.8 below.

Reference sphere data for the Kerr family
, where S 2 carries standard spherical polar coordinates ϑ and ϕ.The Kerr family of metrics is the smooth two-parameter family of Lorentzian metrics on M * , where M ≥ 0 is the mass, a ∈ R is the specific angular momentum, When a = 0, g M,a reduces to the Schwarzschild metric where γ .= dϑ 2 + sin 2 ϑ dϕ 2 .When M = 0, g M,a reduces to the Minkowski metric m .= −dv 2 + 2 dv dr + r 2 γ.
The metrics g M,a solve the Einstein vacuum equations (1.2).The spacetime (M * , g M,a ) is time-oriented by ∂ v for r 1.The vector field ∂ v is Killing-the Kerr family is stationary.If |a| ≤ M and M > 0, these metrics describe black hole spacetimes.For 0 ≤ |a| < M , the black hole is said to be subextremal, and for 0 < |a| = M , extremal.
Remark 2.2.In the context of the Schwarzschild solution, the coordinates (v, r, ϑ, ϕ) are called ingoing Eddington-Finkelstein coordinates.Indeed, defining and (t, r, ϑ, ϕ) are called Schwarzschild coordinates.In the context of the Kerr solution, the coordinates (v, r, ϑ, ϕ) are called Kerr-star coordinates.For the relation to the perhaps more familiar Boyer-Lindquist coordinates, see [ONe95].The advantage of defining the Kerr family g M,a directly in these coordinates is that we may view it as a smooth two-parameter family of Lorentzian metrics on the fixed smooth manifold M * , even across the horizons located at r ± = M ± √ M 2 − a 2 when M > 0.
Remark 2.3.The spacetimes (M * , g M,a ) defined here do not cover the entire maximal analytic extensions of the Minkowski, Schwarzschild, and Kerr solutions.Most importantly, (M * , g M,a ) includes the portion of the future event horizon H + .= {r = r + } strictly to the future of the bifurcation sphere.
We will now define the reference sphere data for the Kerr family.We will use the notion of sphere data x[g, i, L] generated by a Lorentzian metric g on a smooth manifold M, an embedding i : S 2 → M, and a choice of null vector field L defined along and orthogonal to i(S 2 ), which is defined in Lemma A.1 below.Note that Y .= −∂ r is a future-directed null vector field for (M * , g M,a ).We also define the family of embeddings Definition 2.7 (Reference sphere data).Let M ≥ 0, a ∈ R, R > 0, and m ≥ 0 be an integer.The reference Kerr sphere data set of mass M , specific angular momentum a, and radius9 R is the (2.17) The reference Schwarzschild data sets are defined by and the reference Minkowski data sets are defined by m R .= s 0,R . (2.19) We will colloquially refer to k M,a,R as a "Kerr coordinate sphere" and s M,R (resp., m R ) as a "(round) Schwarzschild symmetry sphere" (resp., "(round) Minkowski symmetry sphere").
In the notation of Section 2.1.1,one can show that (2.21) A similarly simple expression is neither available nor needed for Kerr.Indeed, we have the Lemma 2.2.For any integer m ≥ 0, k M,a,R is a smooth three-parameter family of C 2 u C 2+m v sphere data sets.In particular, lim Proof.The metrics g M,a are defined on the fixed smooth manifold M * .By inspection of (2.15), g M,a varies smoothly in M and a.Therefore, the smooth dependence of k M,a,R on the parameters and (2.22) follow from the smooth dependence of the sphere data generated by (g, i, L) on g, i, and L; see Lemma A.1.
We conclude this section with several remarks.
Remark 2.4.As was already mentioned, the Kerr family is stationary.Defining i R (ϑ, ϕ) = (v, R, ϑ, ϕ) for any v ∈ R leads to the same sphere data.
Remark 2.5.We always take the Kerr axis to point along the poles of the fixed identification of the Kerr coordinate spheres with the usual unit sphere.
Remark 2.6.The induced metric / g M,a,R in k M,a,R is not conformal to the round metric γ (defined relative to the Kerr angular coordinates).For this reason we have slightly modified the setup in Section 2.1.3by imposing (2.4) instead of simply dµ / g = dµ γ as in [Chr09, Chapter 2].See already Lemma 3.1 below.Remark 2.7.The induced metric / g M,a,R is given in Kerr angular coordinates by / g M,a,R = Σ dϑ 2 + ρ 2 sin 2 ϑ dϕ 2 . (2.23) To show that this extends smoothly over the poles relative to the smooth structure defined by the Kerr angular coordinates, we note the identity (2.24) Now sin 2 ϑ dϕ is a globally defined smooth 1-form on S 2 since it is the γ-dual of the globally defined vector field ∂ ϕ , so the right-hand side of (2.24) can be extended smoothly over the poles.

Perturbative characteristic gluing
Since the characteristic gluing results of [ACR21a; ACR21b; ACR21c; CR22] pass through linear theory, the conserved charges in Minkowski space play an important role.In Section 2.3.1, we give the definition of conserved charges.In Section 2.3.2,we state the main result of [CR22] in the form which we will directly apply it.

Conserved charges
Definition 2.8 (Spherical harmonics).For ∈ N 0 and m = − , . . ., , let Y m denote the standard real-valued spherical harmonics on the unit sphere (S 2 , γ).We also define the electric and magnetic 1-form spherical harmonics by for ≥ 1 and |m| ≤ .By a standard abuse of notation, we will use the same symbol for the vector-valued spherical harmonics, with the understanding that γ is used to raise the index.
Definition 2.9 (Linearly conserved charges).Let x be C 2 sphere data and define the 1-form B and scalar function m by The conformal factor φ is defined as the unique positive function on S 2 such that dµ / g = φ 2 dµ γ , where γ is the distinguished choice of round metric on S. Then the charges E, P, L, and G (where the latter three are vectors in R 3 indexed by m ∈ {−1, 0, 1}) are defined by Here the modes are defined by
Theorem 5 (Czimek-Rodnianski obstruction-free characteristic gluing).For any C E > 0 and integer m ≥ 0, there exist constants C * > 0 and ε 0 > 0 such that the following holds.Let x be ingoing null data on an ingoing cone C = [− 1 100 , 1 100 ] u × S 2 solving the null structure equations and Bianchi identities, and x 2 be C 2 u C 2+m v sphere data.Let x 1 be the sphere data in x corresponding to u = 0. Let (∆E, ∆P, ∆L, ∆G) .
= (E, P, L, G)(x 2 ) − (E, P, L, G)(x 1 ) be the difference of the conserved charges of x 2 and x 1 .If the data sets satisfy the smallness condition for some 0 < ε < ε 0 , where m is reference Minkowski null data 10 and m 2 is reference Minkowski sphere data, and the following "coercivity" conditions on the charge differences and (2.27) then there is a solution x ∈ X (C) of the null structure equations along a null hypersurface The sphere data x 1 is obtained by applying a sphere diffeomorphism and a transversal sphere perturbation to x 1 inside of C. See [ACR21b; CR22] for the precise definitions of these terms.
10 That is, reference Minkowski sphere data defined along the ingoing cone C. See [ACR21b] for details.
Remark 2.8.The matching condition (2.29) is to order C 2+m in directions tangent to the cone.Since all hypotheses in Theorem 5 are open conditions, we immediately have: Corollary 3. If the sphere data set x 2 satisfies the hypotheses of Theorem 5, there exists an ε * > 0 such that if x2 is another sphere data set such that for some 0 ≤ ε < ε * , then the conclusion of the theorem holds for x2 in place of x 2 .
3 Proofs of the main gluing theorems 3.1 Gluing an almost-Schwarzschild sphere to a round Schwarzschild sphere with a larger mass In this subsection, we prove the main technical lemma of our paper.In essence, we show how to decrease the mass of a Schwarzschild sphere (going backwards in time) by an arbitrary amount, with an arbitrary small error.
Proposition 3.1.Given any 0 ≤ M * < M , R > 0, integer m ≥ 0, and any ε > 0, there exists a δ > 0 and null data x on C [0,1] 1 = [0, 1] × S 2 solving the null structure equations and Bianchi identities such that where b is the boost operation defined in Definition 2.3 and X m is the sphere data norm appearing in Theorem 5.
The proof of the proposition is given at the end of this subsection.We first give a general construction of seed data / g compatible with the hypotheses of Lemma 2.1.
Lemma 3.1.Let C be as in Section 2.1.3.Let γ be a Riemannian metric on S 2 .There exists an explicitly defined smooth assignment γ → h, where h is a traceless (1, 1)-S-tensor field along C, such that for any defines a Riemannian metric for each v, satisfies condition (2.4), and / g(1) = γ.Here γ is defined along C according to Dγ = 0. We have and the inverse metric is given by Proof.We first fix some cutoff functions.Let χ ∈ C ∞ c (0, 1 2 ) be nonnegative and not identically zero.Let Let p 1 be the north pole of S 2 , p 2 the south pole, and set 2 ) be a coordinate chart covering U 2 , and set As matrices, these tensor fields are given by diag(1, −1) in the respective coordinate systems.We now claim that the symmetric (0, 2)-tensor fields are nowhere vanishing on their respective domains of definition.This follows from the fact that where no summation is implied.Since γ is positive definite, we must at each point have both h i 11 and h i 22 nonvanishing, so h i is always nonvanishing.Let h i be the (1, 1)-tensor field obtained by dualizing h i with γ.We then finally define It is clear that tr h = 0 and that h is symmetric, where is taken relative to γ.
We now show that (3.3) defines a Riemannian metric.Viewing h as an endomorphism T S 2 → T S 2 , the power series converges and defines a smooth family of endomorphisms.
To verify that / g is symmetric, we examine (3.7) term by term: where we used the symmetry of h repeatedly.That / g is positive definite follows easily from the fact that at the origin of a normal coordinate system for γ, / g AB is the matrix exponential of a symmetric matrix, and hence positive definite.
To show that (2.4) is satisfied, we use Jacobi's formula to calculate det / g = det γ exp(λ tr h) = det γ relative to any coordinate system, where we used tr h = 0. We conclude that the volume form of / g satisfies dµ / g = dµ γ . (3.8) Observe that since D(dµ γ ) = 0 by definition of γ along C, (3.8) implies To prove (3.4), we use the fact that h(v) and h(v ) commute for any v and v sufficiently close to simply differentiate (3.3): The formula (3.5) is immediately seen to hold.
Remark 3.1.By the Poincaré-Hopf theorem, the shear χ must vanish at some point on each S v ⊂ C. Equivalently, any h for which (3.3) satisfies condition (2.4), must vanish at some point on each S v .In order to satisfy (3.9) below, this zero cannot stay along the same generator of C. The simplest solution to this problem is the two-pulse configuration above.
With this general construction out of the way, we begin the proof of Proposition 3.1 in earnest.We specialize now to the case of γ = γ, the round metric on the unit sphere.
Convention.We now introduce a parameter δ > 0 satisfying 0 < δ < δ 0 , where δ 0 > 0 is a sufficiently small fixed parameter only depending on M * , M, R and the fixed choices of χ, U 1 , U 2 , f 1 , and f 2 .We will further use in this section the notation that implicit constants in , , and ≈ may depend M * , M, R and χ, U 1 , U 2 , f 1 , and f 2 .We also use the notation j , j , and ≈ j if the implicit constants in , , and ≈ depend on an additional parameter j. for j ≥ 0.
The remaining sphere data at v = 1 is now specified as follows: Combining everything and using the null structure and Bianchi equations to solve the rest of the system, we have Lemma 3.8.For 0 < δ ≤ δ 0 we have at v = 0 for every j ≥ 0 and χ(0) = 0, α(0) = 0. (3.26) The terms in (3.25) are displayed in the order in which they are estimated.
Proof.The proof follows the procedure of [Chr09, Chapter 2], which we now outline.The first term is estimated using (3.22).The second term was estimated in (3.24).The third term is estimated using the formula Note that the first and third terms are estimated by δ 1/2 on the whole cone, but are improved at v = 0.
To estimate the fourth term, the transport equation (A.9) combined with the Codazzi equation (A.20) and (A.1) yields Now |η| can be estimated using Grönwall's inequality and (3.22).To estimate the fifth term, the transport equation (A.15) is combined with the Gauss equation (A.19) to give Grönwall gives |tr χ| δ −1 , which then easily implies the desired estimate by Grönwall applied to To estimate the sixth term, apply Grönwall directly to (A.17).The first variation formula (2.12), the second variation formula (A.Combined with the Bianchi identity (A.24), this yields from which the desired estimate follows by Grönwall.The twelfth term is estimated by integrating (A.11) and the thirteeth term is estimated by integrating (A.23).
We are now ready to prove the main result of this subsection.
Proof of Proposition 3.1.Let x low (v), v ∈ [0, 1], be the null data constructed above.We have defined and we set x high (1) .= 0. Immediately from the definition of the boost b δ in Definition 2.3, we have (3.1).Since Ω 2 = 1 along C and χ is compactly supported away from v = 0, we have x high (0) = 0.The boost b δ changes every positive power of δ on the left-hand side of (3.25) into a negative power, so that b δ (x(0)) − s M * ,R C j j δ 1/2 for any j ≥ 0. Therefore, taking j sufficiently large, we have where X m is the sphere data norm appearing in Theorem 5. Now (3.2) follows follows by taking δ sufficiently small.

Gluing Minkowski space to any round Schwarzschild sphere
Theorem A. Let M > 0, R > 0, and k ∈ N.For any ε > 0 there exists a solution x of the null constraints on a null cone C [0,1] such that x(1) equals s M,R after a boost and x(0) can be realized as a sphere in Minkowski space in the following sense: There exists a C k spacelike 2-sphere S in Minkowski space and a choice of L and L on S such that the induced C 2 u C k v sphere data on this sphere equals x(0) after a boost and a sphere diffeomorphism.
Remark 3.2.The sphere S can be made arbitrarily close to round and the sphere diffeomorphism can be made arbitrarily close to the identity.1.We now aim to use Theorem 5 to connect x 2 .= b δ (x(0)) to a sphere in Minkowski space.Let x be the usual ingoing Minkowski null data passing through the unit sphere at u = 0. 11 By a direct computation, s M * ,2 − m 2 X m ≈ M * .It follows that if ε is sufficiently small, where C 1 does not depend on ε .We must estimate the conserved charge deviation vector (∆E, ∆P, ∆L, ∆G) = (E, P, L, G)(b δ (x(0))).
We then compute where O(ε Let and C * and ε 0 as in Theorem 5 for the choice Then (2.25) and (2.26) are satisfied, sufficiently small, so (2.27) is satisfied, and finally if ε is sufficiently small, so (2.28) is also satisfied.By applying Theorem 5, we obtain a null data set for which the bottom sphere x 1 is a sphere diffeomorphism of a genuine Minkowski sphere data set and satisfies which can be made arbitrarily small and hence completes the proof of the theorem.

Gluing Minkowski space to any Kerr coordinate sphere in very slowly rotating Kerr
In this section, we perform Kerr gluing for small angular momentum essentially as a corollary of the Schwarzschild work.
Theorem B. For any k ∈ N, there exists a function a 0 : (0, ∞) 2 → (0, ∞) with the following property.Let M > 0 and R > 0. If 0 ≤ |a| ≤ a 0 (M, R)M , there exists a solution x of the null constraints on a null cone C [0,1] 0 such that x(1) equals k M,a,R after a boost and x(0) can be realized as a sphere in Minkowski space in the following sense: There exists a C k spacelike 2-sphere S in Minkowski space and a choice of L and L on S such that the induced C 2 u C k v sphere data on this sphere equals x(0) after a boost and a sphere diffeomorphism.Proof.Again, it suffices to prove the theorem for R = 2 and M > 0 fixed but otherwise arbitrary.Let x(v) and δ be the associated null data set and boost parameter constructed in Proposition 3.1, where M and R are as in the statement of the present theorem and ε is sufficiently small that the argument of Theorem A applies.

Gravitational collapse to a Kerr black hole of prescribed mass and angular momentum
In this section we give the proof of Corollary 1 and the sketch of the proof of Corollary 2. Recall the fractional Sobolev spaces H s , s ∈ R, and their local versions H s loc .Recall also the notation f ∈ H s− loc which means f ∈ H s loc for every s < s.We refer the reader back to Fig. 2 for the Penrose diagram associated to the following result.
Corollary 1.There exists a constant a 0 > 0 such that the following holds.For any mass M > 0 and specific angular momentum a satisfying a/M ∈ [−a 0 , a 0 ], there exist one-ended asymptotically flat Cauchy data (g 0 , k 0 ) ∈ H 7/2− × H 5/2− for the Einstein vacuum equations (1.2) on Σ ∼ = R 3 , satisfying the constraint equations such that the maximal future globally hyperbolic development (M 4 , g) has the following properties: • Null infinity I + is complete.
• The Cauchy surface Σ lies in the causal past of future null infinity, Σ ⊂ J − (I + ).In particular, Σ does not intersect the event horizon H + .= ∂(BH) or contain trapped surfaces.
• For sufficiently late advanced times v ≥ v 0 , the domain of outer communication, including the event horizon, is isometric to that of a Kerr solution with parameters M and a.For v ≥ v 0 , the event horizon of the spacetime can be identified with the event horizon of Kerr.
Remark 4.1.The spacetime metric g is in fact C 2 everywhere away from the region labeled "Cauchy stablity" in Fig. 5 below.Near the set Ḣ+ − (see [HE73, p. 187] for notation), the spacetime metric might fail to be C 2 , but is consistent with the regularity of solutions constructed in [HKM76] with s = 7 2 −.See also [Chr13] for the notion of the maximal globally hyperbolic development in low regularity.
Proof.We refer the reader to Fig. 5 for the anatomy of the proof, which is essentially the same as Corollary 1 in [KU22].The region to the left of H + is constructed using our gluing theorem, Theorem 3.3, and local existence (in this case we appeal to [Luk12]).See [KU22, Proposition 3.1].The region to the right of the horizon, save for the part labeled "Cauchy stability" in Fig. 5, is constructed in the same manner.These two regions can now be pasted along u = 0 and the resulting spacetime will be C 2 .
We can now use a Cauchy stability argument to construct the remainder of the spacetime.A very similar argument in carried out in [KU22, Lemma 5.1], but the lower regularity of our gluing result in the present paper forces us to use slightly more technology here.As in [KU22, Lemma 5.1], we take the induced data (g * , k * ) on a suitably chosen spacelike hypersurface Σ * passing through the bottom gluing sphere.See , the locus where the null geodesic generators "end").The event horizon does not necessarily end in a point since the distinguished Minkowski sphere is not necessarily round.
Having constructed the spacetime, we can finally extract a Cauchy hypersurface Σ, which completes the proof.
Lemma 4.1.Let f and g be functions defined on B 2 ⊂ R 3 , the ball of radius two, such that f (ii) Suppose that f = g = 0 identically on B 1 .For 0 < ε < 1 2 , let θ ε be a cutoff function which is equal to one on B 1+ε and zero outside of B 2+ε .Then f ε .= θ ε f and g for any s < 3.
Proof.The proof of (i) follows in a straightforward manner from the physical space characterization of fractional Sobolev spaces (such as in [DPV12]) and is effectively an elaboration of the fact that the characteristic function of B 1 lies in H 1/2− .Proof of (ii): Using Taylor's theorem as in [KU22, Lemma 5.1], we see that (f ε , g ε ) H 2 ×H 1 (B2) → 0 as ε → 0. By iterating Hardy's inequality, we see that We now obtain (4.3) by interpolation.
Remark 4.2.In fact (4.3) holds for s < 7 2 , but this requires a fractional Hardy inequality.We now sketch the proof of Corollary 2 and refer the reader back to Fig. 3 for the associated Penrose diagram.
Sketch of the proof of Corollary 2. Using Theorem A, we glue Minkowski space to a round Schwarzschild sphere of mass 1 and radius R = 2 − ε for 0 ≤ ε 1.As ε → 0 (perhaps only along a subsequence ε j → 0), the gluing data converge to the horizon gluing data used in the proof of Corollary 1, in an appropriate norm.It then follows by Cauchy stability that the spacetimes constructed by solving backwards as in the proof of Corollary 1 contain the full event horizon, for ε sufficiently small.

A.1.2 Algebra and calculus of S-tensors
Let (M, g) be a spacetime equipped with a double null gauge as above.For vector fields on M, we define the orthogonal projection to S vector fields by Π : T M → T S, ΠX .= X + 1 2 g(X, e 3 )e 4 + 1 2 g(X, e 4 )e 3 which we extend componentwise to contravariant tensors of higher rank.We note that Π•i = id on T S, where i : T S ⊂ T M is the natural inclusion.By duality, this defines a "promotion" operator Π * : T * S → T * M which extends componentwise to covariant S-tensors and satisfies i * • Π * = id on T * S.
We now define projected Lie derivatives / L L and / L L on S-tensors.If X is an S-vector field, then / L L X .= L L X, / L L X .= L L X are already S-vector fields.If ξ is an S-1-form, then where we have explicitly written the "promotion" operation which will be consistently omitted in the sequel.The operation is extended to general S-tensor fields via the Leibniz rule.As a shorthand, we write The symbol / ∇ acts on functions and S-vector fields as the induced covariant derivative on the spheres and is extended to general S-tensors by the Leibniz rule.
We will frequently make use of the following notation: Let ξ, η be S-1-forms and θ, φ symmetric covariant S-2-tensor fields.We then define where / ε is the induced volume form on S u,v .The notation / g AB denotes the inverse of the induced metric / g AB .Indices of S-tensors are raised and lowered with / g and / g −1 .

A.1.3 Ricci and curvature components
The Ricci components are given by the null second fundamental forms

Figure 2 :
Figure 2: Penrose diagram for Corollary 1.The textured line segment is where the gluing data constructed in Theorem 2 live.
Figure 4: Two-step process for the proof of Theorem 1.
Definition 2.6 (The Kerr family of metrics).Let M *

Proof.
By scaling, it suffices to prove the theorem when R = 2.We first use Proposition 3.1 to connect b δ (s M,R ) to the sphere data set b δ (x(0)) with M * = ε 1/4

Fig. 5 .
Figure 5: Penrose diagram for the proof of Corollary 1.The diagram does not faithfully represent the geometry of the spacetime near the "bottom" of the event horizon Ḣ+− (i.e., the locus where the null geodesic generators "end").The event horizon does not necessarily end in a point since the distinguished Minkowski sphere is not necessarily round.

A. 1 . 1
Double null gaugeLet W ⊂ R 2 u,v be a domain and define M 3+1 .= W ×S 2 .Denote S u,v .= {(u, v)}×S 2 ⊂ M. The distinguished foliation of M by these spheres carries a tangent bundle T S and cotangent bundle T * S .= (T S) * .An Stensor (field) is a section of a vector bundle consisting of tensor products of T S and T * S. Let / g be a positive-definite (0, 2) S-tensor field, let Ω 2 be a positive function on M, and b be an S-vector field.Under these assumptions, the formulag = −4Ω 2 du dv + / g AB (dϑ A − b A dv)(dϑ B − b B dv)defines a Lorentzian metric on M, where (ϑ 1 , ϑ 2 ) are arbitrary local coordinates on S 2 and / g AB (resp., b A ) are the components of / g (resp., b) relative to this coordinate basis.The coordinate functions u and v satisfy the eikonal equation, i.e., g µν ∂ µ u∂ ν u = 0 and g µν ∂ µ v∂ ν v = 0. Consequently, the hypersurfaces C u .= {u = const.}and C v .= {v = const.}are null hypersurfaces.We time orient (M, g) by declaring ∂ u + ∂ v + b A ∂ ϑ A to be future-directed.The vector fields L .= −2(du) and L .= −2(dv) are future-directed null geodesic vector fields.We set L .= Ω 2 L and L .= Ω 2 L , which then satisfy Lu = 0, Lv = 1, Lu = 1, Lv = 0. Finally, we set e 4 .= ΩL , e 3 .= ΩL .
denotes a function all of whose angular derivatives are ε .It follows that for ε sufficiently small,