GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS PETER HINTZ Abstract. We prove a strong localized gluing result for the general relativistic constraint

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GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS

PETER HINTZ

Abstract. We prove a strong localized gluing result for the general relativistic constraint

equations (with or without cosmological constant) in n≥3 spatial dimensions. We glue

an -rescaling of an asymptotically ﬂat data set (ˆγ, ˆ

k) into the neighborhood of a point

p∈Xinside of another initial data set (X, γ, k), under a local genericity condition (non-

existence of KIDs) near p. As the scaling parameter tends to 0, the rescalings x

of normal

coordinates xon Xaround pbecome asymptotically ﬂat coordinates on the asymptotically

ﬂat data set; outside of any neighborhood of pon the other hand, the glued initial data

converge back to (γ, k). The initial data we construct enjoy polyhomogeneous regularity

jointly in and the (rescaled) spatial coordinates.

Applying our construction to unit mass black hole data sets (X, γ, k) and appropriate

boosted Kerr initial data sets (ˆγ, ˆ

k) produces initial data which conjecturally evolve into

the extreme mass ratio inspiral of a unit mass and a mass black hole.

The proof combines a variant of the gluing method introduced by Corvino and Schoen

with geometric singular analysis techniques originating in Melrose’s work. On a technical

level, we present a fully geometric microlocal treatment of the solvability theory for the

linearized constraints map.

Contents

1. Introduction ........................................................... 2

1.1. Context ............................................................. 4

1.2. Gluing and asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3. Exponential weights and geometric singular analysis . . . . . . . . . . . . . . . . . . 11

1.4. Outlook: evolution of glued initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Outline of the paper ...................................................... 12

Acknowledgments ......................................................... 13

2. Geometric and analytic background ................................. 13

2.1. b- and scattering structures; blow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2. 00-structures ........................................................ 15

2.3. Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4. Estimates for b- and 00-diﬀerential operators . . . . . . . . . . . . . . . . . . . . . . . . . 19

Date: October 26, 2022.

2010 Mathematics Subject Classiﬁcation. Primary 83C05, 35B25, Secondary 35C20, 35N10, 83C57.

arXiv:2210.13960v1 [math.AP] 25 Oct 2022

2 PETER HINTZ

3. Geometry and analysis on the total gluing space .................. 27

3.1. Analysis on the total gluing space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4. The constraints map and its linearization .......................... 31

4.1. Asymptotically ﬂat initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2. Manifolds with punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3. Constraints map on the total gluing space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5. Gluing construction ................................................... 56

5.1. Formal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2. True solution ........................................................ 61

6. Applications ........................................................... 64

6.1. Gluing Kerr initial data into a given data set . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.2. Gluing a given data set into subextremal Kerr and Kerr–(anti) de Sitter

initial data ................................................................ 66

A. Quasilinear elliptic Douglis–Nirenberg systems .................... 68

References ................................................................. 69

1. Introduction

Let n≥3. For a smooth Riemannian n-manifold (X, γ) and a symmetric 2-tensor kon

X, the constraint equations for γ, k are

(Rγ− |k|2

γ+ (trγk)2= 2Λ,

δγk+ d(trγk)=0.(1.1)

Here, Rγis the scalar curvature of γ,δγis the negative divergence, and Λ ∈Ris the

cosmological constant. We say that (X, γ, k) is an initial data set if (1.1) holds. Given a

Lorentzian manifold (M, g) of dimension (n+ 1) and signature (−,+,...,+), and given an

embedded spacelike hypersurface X⊂M, the ﬁrst and second fundamental form γand k

of Xsatisfy (1.1) provided gsatisﬁes the Einstein vacuum equations

Ein(g)+Λg= 0,Ein(g) := Ric(g)−1

2Rgg. (1.2)

The fundamental theorems of Choquet-Bruhat and Geroch [CB52,CBG69] show, con-

versely, that given (X, γ, k) satisfying (1.1), there exists a maximal globally hyperbolic

spacetime (M, g), unique up to isometries, which solves the Einstein vacuum equations and

into which Xembeds with ﬁrst and second fundamental form given by γand k, respectively.

We consider the problem of gluing the initial data of a small asymptotically ﬂat black

hole (such as a Schwarzschild or Kerr black hole with small mass), or more generally of

a rescaled asymptotically ﬂat initial data set (Rn\ˆ

K◦,ˆγ, ˆ

k) with cosmological constant

GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS 3

0, into a neighborhood of a point p∈Xin a given generic (near p) smooth initial data

set (X, γ, k) with arbitrary cosmological constant Λ ∈R. Here ˆ

K⊂Rnis compact (and

possibly empty); and the asymptotic ﬂatness condition, in standard coordinates ˆx∈Rn,

means that ˆγ(∂ˆxi, ∂ˆxj)→δij and k(∂ˆxi, ∂ˆxj)→0 as |ˆx|→∞, with appropriate rates of

convergence. (See Deﬁnition 4.4 for the precise deﬁnition used in this paper.)

Theorem 1.1 (Main result, rough version).Assume that Xis generic in a connected

neighborhood U◦of p(in the sense that it does not admit any KIDs in U◦, see Deﬁni-

tion 4.12). Then there exist ]>0and a family (X, γ, k),∈(0, ]), of initial data sets

with cosmological constant Λwith the following properties.

(1) On X\ U◦⊂X, we have (γ, k)=(γ, k).

(2) In geodesic normal coordinates x= (x1, . . . , xn)∈Rnaround p∈X, the manifold

Xis equal to B(0,1) \ˆ

K◦, and we have smooth convergence (γ, k)→(γ, k)as

&0in |x|> δ for any δ > 0. (That is, the matrix coeﬃcients (γ)ij =γ(∂xi, ∂xj)

and k(∂xi, ∂xj)converge to those of γand k.)

(3) The tensors γ|ˆx= (γ|ˆx(∂xi, ∂xj))i,j=1,...,n and k|ˆxconverge, smoothly and lo-

cally uniformly in ˆx∈Rn\ˆ

K◦, to ˆγ|ˆx= (ˆγ|ˆx(∂ˆxi, ∂ˆxj)) and ˆ

k|ˆx, respectively, as

&0.

To explain part (3), we ﬁrst note that 2(ˆγ, ˆ

k) is a rescaled asymptotically ﬂat data

set: asymptotically ﬂat coordinates for it are ˆx, which as &0 become local coordinates

xon Xnear p. Note that the ADM mass of 2(ˆγ, ˆ

k) is times that of (ˆγ, ˆ

k). Since in

the coordinates ˆx=x

we have ∂ˆxi=∂xi, part (3) states that (γ, k)≈2(ˆγ, ˆ

k) in x-

coordinates when |x|..1See Figure 1.1. In the region .|x|.1, the family (γ, k)

transitions from the -rescaling 2(ˆγ, ˆ

k) to the original data set (γ, k) in an appropriate

manner; we describe this more precisely in §1.2 below.

A natural choice for (ˆγ, ˆ

k) for n= 3 is the initial data set of a (boosted) Schwarzschild or

Kerr black hole [Sch16,Ker63]; see §6.2. To illustrate Theorem 1.1 in this case, consider the

initial data (ˆγ, ˆ

k) of an unboosted mass ˆ

m>0 Schwarzschild black hole in polar coordinates

ˆx= ˆrω,ω∈S2,

ˆγ=1−ˆ

ˆrdˆr2+ ˆr2gS2,ˆ

k= 0,

on R3\ˆ

K◦where ˆ

Kis a bounded closed ball of radius >max(0,2ˆ

m). Then

2ˆγ=1−ˆ

rdr2+r2gS2, r =ˆr,

is the metric of a mass ˆ

mSchwarzschild black hole. The initial data set (γ, k) thus

describes a mass ˆ

mblack hole glued into the given data set (X, γ, k).

We do not concern ourselves here with the construction of appropriate initial data sets

(X, γ, k). We recall that Beig–Chru´sciel–Schoen [BCS05] demonstrated the genericity of

the assumption on the absence of KIDs in a large number of settings; moreover, Moncrief

1The −1-scaling of the second fundamental form in (γ, k)≈(2ˆγ, −12ˆ

k) arises from the scaling

properties of (1.1), see Lemma 4.16; heuristically, it follows from the fact that the future unit normal for

the embedding of (X, γ, k) into the spacetime M=Rt×Xis, near p, scaled by −1relative to the future

unit normal of the embedding of ( ˆ

X, ˆγ, ˆ

k), ˆ

X=Rn

ˆx, into Rˆ

t×ˆ

X; that is, the time scale ˆ

tof the small black

hole is related to the time scale tof the ambient spacetime Mby ˆ

t=t

.

4 PETER HINTZ

U◦

(γ, k)

U◦X/2

U◦

ˆ

(γ, k)=(γ, k)(γ, k)≈(γ, k)(γ, k)≈2(ˆγ, ˆ

X

Figure 1.1. Illustration of Theorem 1.1. The data set (γ, k) is unchanged

outside of U◦, and in U◦\ {p}it converges smoothly and locally uniformly

to (γ, k) as &0. Near pon the other hand, the glued initial data are close

to the -rescaling 2(ˆγ, ˆ

k) of the asymptotically ﬂat data set (ˆγ, ˆ

k).

[Mon75] showed that the absence of KIDs is equivalent to the absence of Killing vector

ﬁelds in the evolving spacetime. KIDs on (X, γ, k) near pare nonzero elements, deﬁned

in a neighborhood of p, of the cokernel of the linearization of the constraint equations

around (γ, k). While this cokernel is necessarily ﬁnite-dimensional, it may be non-trivial.

However, a non-trivial cokernel is typically an obstruction for localized gluing constructions

or deformations of initial data sets; for instance, the rigidity part of the Positive Mass

Theorem [SY79] in the time-symmetric setting (k= 0, and γis scalar-ﬂat) implies that

one cannot compactly perturb the Euclidean metric to a non-isometric scalar-ﬂat metric.

(See however the work by Czimek–Rodnianski [CR22] on how to overcome the presence of a

cokernel in characteristic gluing problems by taking advantage of the nonlinear nature of the

constraint equations.) In this paper, we impose the local genericity condition on (X, γ, k)

in order to obtain a local gluing result in Theorem 1.1 via an appropriate solvability theory

for the linearized constraints map.

The initial data of a subextremal Kerr (or Kerr–de Sitter or Kerr–anti de Sitter) black

hole do not satisfy the genericity condition required in Theorem 1.1. Nonetheless, we

show in §6.2 how to prove Theorem 1.1 also in this case, provided U◦intersects the black

hole interior; see Theorem 6.2. (The main idea in the proof is to eliminate the cokernel by

allowing for a violation of the constraint equations deep inside the black hole.) In particular,

we are thus able to glue a small black hole into a unit mass black hole initial data set.

1.1. Context. Starting with the construction by Majumdar–Papapetrou [Maj47,Pap45]

of electrovacuum spacetimes via the superposition of extremally charged black holes (with

Kastor–Traschen [KT93] performing a similar construction in Λ >0), several constructions

of initial data sets containing several black hole regions have been proposed. By solving

GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS 5

the constraint equations using explicit ansatzes mainly involving superpositions of scalings

and translations of the harmonic function |x|−1, Brill–Lindquist [BL63] constructed initial

data for the Einstein–Maxwell equations with any ﬁnite number Nof charged wormholes

(Einstein–Rosen bridges) at arbitrary points in R3with arbitrary masses; the resulting

data have N+ 1 asymptotically ﬂat regions. Misner’s time-symmetric ‘matched throat’

vacuum initial data identify all but 2 asymptotically ﬂat ends; see Lindquist [Lin63] for

the Einstein–Maxwell case. There also exist hybrid approaches, such as the one developed

by Brandt–Bruegmann [BB97] which involves an explicit prescription for the conformal

class of k(with γconformally Euclidean) together with a numerical scheme for ﬁnding the

conformal factor.

The key mathematical technique allowing for ﬂexible and localized gluing constructions

for solutions of the constraint equations was introduced by Corvino [Cor00] with Schoen

[CS06]; see also [CD03] and §1.2 below. It is based on the observation that the adjoint

of the linearized constraint equations is overdetermined and permits coercive estimates on

function spaces with very strong weights at the boundary of the gluing region. Concretely,

Corvino–Schoen prove that an asymptotically ﬂat data set can be perturbed near inﬁnity

to an exact Kerr data set for a suitable choice of Kerr parameters (mass and angular

momentum). The relevant linear operator in this setting (namely, the linearization of the

constraints map around the trivial Minkowski data) has a nontrivial cokernel, which is

accounted for by appropriately choosing the parameters of the Kerr data set.

This technique was generalized and used by Chru´sciel–Delay [CD03,§8.9] to construct

initial data containing many Kerr black holes. (See [CD02] for the time-symmetric case of

data containing several Schwarzschild black holes.) The initial data of [CD03] are symmetric

under the parity map x7→ −x; given pairwise disjoint balls B(xi,4ri), the data are equal

to Kerr data (with arbitrary parameters, subject to the parity condition) in each of the

B(xi,2ri), and also in R3\SB(xi,4ri). The gluing procedure succeeds when all black hole

masses are suﬃciently small relative to the radii riand the pairwise distances of the xi. We

also recall that Isenberg–Mazzeo–Pollack [IMP02,§9] constructed many-black-hole initial

data by connecting Euclidean spaces to the neighborhood of any ﬁnite number of points

on a given asymptotically ﬂat maximal (i.e. trγk= 0) data set via wormholes. Chru´sciel–

Mazzeo [CM03] proved the presence of multiple black holes in the spacetime development

of the data produced in [CD02,CD03,IMP02] under suitable smallness conditions. See the

hypotheses in [CM03,§3] for details; in particular, the ﬁrst fundamental form must globally

be suﬃciently close to the Euclidean metric.

Another construction of many-black-hole initial data was given by Chru´sciel–Corvino–

Isenberg [CCI11]. Phrasing their main result in a manner which relates more directly to the

present work, [CCI11] glues suﬃciently small rescalings of large compact subsets of Ngiven

asymptotically ﬂat initial data sets (modiﬁed to be exact Kerr data near their respective

asymptotically ﬂat ends) into Ndisjoint balls in R3; the glued data are exact Kerr data,

with carefully chosen small parameters, also near inﬁnity in R3.

The many-black-hole initial data sets obtained by (repeated2) application of Theorem 1.1

combine features of those of [IMP02] (in that the background geometry is arbitrary) with

2One can extend Theorem 1.1 to simultaneously glue any ﬁnite number of asymptotically ﬂat data sets

into neighborhoods of a matching number of distinct points in X, under a local genericity condition near

each of these points. One may also apply Theorem 1.1 to a data set (X, γ, k), for some ﬁxed small  > 0,

in place of (X, γ, k), or more generally iterate a combination of these two procedures ﬁnitely many times.

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GLUINGSMALLBLACKHOLESINTOINITIALDATASETSPETERHINTZAbstract.Weproveastronglocalizedgluingresultforthegeneralrelativisticconstraintequations(withorwithoutcosmologicalconstant)inn3spatialdimensions.Wegluean-rescalingofanasymptoticallyatdataset(^;^k)intotheneighborhoodofapointp2Xinsideofanotherinitial...

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GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS PETER HINTZ Abstract. We prove a strong localized gluing result for the general relativistic constraint.pdf

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GLUING SMALL BLACK HOLES INTO INITIAL DATA SETS PETER HINTZ Abstract. We prove a strong localized gluing result for the general relativistic constraint

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