A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar ﬁeld system Jaydeep Singh1

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A construction of approximately self-similar naked singularities

for the spherically symmetric Einstein-scalar ﬁeld system

Jaydeep Singh ∗1

1Department of Mathematics, Princeton University,

Washington Road, Princeton, NJ 08544, United States of America

December 8, 2022

Abstract

In this work we investigate the stability and instability properties of a class of naked singularity

spacetimes. The ﬁrst rigorous study of naked singularities in the spherically symmetric Einstein-scalar

ﬁeld system is due to Christodoulou [7], who identiﬁed singularity formation in the class (gk, φk)of

k-self-similar solutions, for any k2∈(0,1

3). Here we extend the construction to produce examples of

interior and exterior regions of naked singularity spacetimes locally modeled on k-self-similar solutions,

without requiring exact self-similarity.

The main result is a global stability statement under ﬁne-tuned data perturbations, for a class of

naked singularity spacetimes satisfying self-similar bounds. Given the well-known blueshift instability for

suitably regular naked singularities in the Einstein-scalar ﬁeld model, we require non-generic conditions

on the data perturbations. In particular, the scalar ﬁeld perturbation along the past lightcone of the

singular point Ovanishes to high order near O. Technical diﬃculties arise from the singular behavior

of the background solution, as well as regularity considerations at the axis and past lightcone of the

singularity. The interior region is constructed via a backwards stability argument, thereby avoiding

activating the blueshift instability. The extension to the exterior region is treated as a global existence

problem to the future of O, adapting techniques developed for vacuum spacetimes in [23].

Contents

1 Introduction 2

1.1 Christodoulou’s k-self-similarsolutions............................... 4

1.2 Further constructions of naked singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Related works and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Guidetothepaper.......................................... 9

1.5 Acknowledgements .......................................... 9

2 Preliminaries 9

2.1 Outline of spherical symmetry and the solution manifold Q................... 9

2.2 Formulations of the Einstein-scalar ﬁeld system and consequences . . . . . . . . . . . . . . . . 11

2.3 Solution classes and local existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Asymptotic ﬂatness and naked singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Einstein-scalar ﬁeld system for diﬀerences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Integrationlemmas.......................................... 18

2.7 Notationandconstants ....................................... 21

∗jaydeeps@math.princeton.edu

arXiv:2210.11325v2 [gr-qc] 8 Dec 2022

3 Main Results 21

3.1 Admissible background spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Admissibledata............................................ 25

3.3 Statementoftheorems........................................ 26

3.4 Proofoutline ............................................. 28

4 Interior solution: Proof of Theorem 4 33

4.1 Approximateinteriors ........................................ 33

4.2 Limitingsolution........................................... 46

5 Exterior solution: Proof of Theorem 5 48

5.1 Localexistence ............................................ 49

5.2 RegionI................................................ 49

5.3 Changeofgauge ........................................... 54

5.4 RegionII ............................................... 54

5.5 RegionIII............................................... 59

5.6 RegionIV............................................... 63

5.7 Concluding the proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A Construction of (gk, φk)67

A.1 Consequences of k-self-similarity .................................. 67

A.2 Theinteriorsolution ......................................... 69

A.3 Theexteriorsolution......................................... 78

References 87

1 Introduction

As examples of singular solutions arising from regular, asymptotically ﬂat initial data, naked singularities

are model cases for studying the formation of gravitational singularities. In this paper we restrict attention

to the spherically symmetric setting, and consider solutions to the Einstein-scalar ﬁeld system:

(Ricµν −1

2Rgµν = 2Tµν [φ],

gµν φ= 0.(1.1)

The main results of this paper establish the existence and stability under ﬁnely-tuned initial data perturba-

tions of a class of naked singularity solutions to (1.1) satisfying self-similar bounds.

The problem of singularity formation arises already in the simplest case of spherically-symmetric vacuum

spacetimes, as the famous Schwarzschild solution illustrates. Here the singularity assumes the form of a

spacelike {r= 0}boundary, across which curvature invariants blowup and the solution fails to be extendible

as a regular spacetime. Despite this breakdown in regularity, the presence of the event horizon decouples the

singularity from the causal futures of observers who remain in the black hole exterior region. In particular,

“inﬁnitely far away” observers along I+exist for inﬁnite proper time.

In contrast, the spacetimes considered in this paper contain an incomplete I+(see Figure 1). Imagined

observers along I+thus reach the future lightcone of the singularity Oin ﬁnite proper time, rendering the

singularity “naked”. The region of spacetime uniquely determined by initial data abruptly comes to an end

for these distant observers, even though they see no local breakdown in regularity.

The conceptual issues raised by these pathological spacetimes were taken up by Penrose in [21], leading

to the conjecture that naked singularity formation should be non-generic. In practice, such a statement

must be accompanied by a choice of matter model, a class of initial data and solutions, and a notion of

genericity. Setting aside these issues, one form of this so-called weak cosmic censorship (WCC) conjecture,

adapted from [8], is the following:

Conjecture (Weak Cosmic Censorship).The maximal development of generic regular, asymptotically ﬂat,

initial data admits a complete I+.

I+

Figure 1: Formation of a globally naked singularity from regular Cauchy data on Σ. Observers (represented

by arrows) leave the maximal development associated to data, despite not encountering a breakdown in

regularity.

For the spherically symmetric Einstein-scalar ﬁeld system, a positive resolution of (WCC) was given by

Christodoulou in the series of works [6], [9]. The foundational work [6] identiﬁed limited-regularity solution

classes in which well-posedness and continuation criteria hold. These classes provide enough ﬂexibility to

construct unstable perturbations, and [9] established the existence of an unstable two-dimensional plane of

directions in the space of initial data for suitably regular naked singularities1. Note that the “genericity”

assumption in the statement of (WCC) was shown to be necessary in the work [7], which provided examples of

absolutely continuous naked singularity spacetimes. See also the work [17] providing a more robust instability

proof.

A key role in the argument is played by the dynamics along the backwards light cone of O, given by

{v= 0}in appropriate double null coordinates. Along this null surface the evolution equations for transversal

double null unknowns (e.g. ∂vφ) satisfy ordinary diﬀerential equations (ODEs), with coeﬃcients and source

terms determined by the quantities intrinsic to the surface. An important discovery of [9] was that such

ODEs exhibit a blueshift instability. For generic perturbations of the value of ∂vφat a single point, say

(u, v)=(−1,0), the instability leads to rapid growth of the scalar ﬁeld and trapped surface formation.

An additional feature of the proof in [9] is that the unstable perturbations are supported on the exterior

region of the naked singularity (i.e. {v≥0}). As a result, the argument says little about the stability

properties of the interior. Some questions left unanswered by the analysis of [9] include:

(i) Are there examples of naked singularity interiors that are “stable” in a suitable sense, either generically

or up to perturbations of ﬁnite codimension?

(ii) Can the support of the unstable perturbations identiﬁed by [9] be extended into the region {v= 0}?

(iii) Under generic perturbations, is the existence of a central singularity stable? Or can appropriate per-

turbations de-singularize the solution?

This work addresses a simpler question of constructing additional examples of naked singularity interiors.

Our main result extends the construction of k-self-similar naked singularities in [7] outside the class of exact

self-similarity. The proof moreover guarantees the existence of stable directions in the space of initial data

for the forwards problem, but does not give any estimate for the dimension of a stable manifold.

We take a perturbative approach to constructing the interior regions. Namely, given a self-similar naked

singularity as in [7], or more generally any admissble spacetime (see Section 3.1 for the full list of require-

ments), our ﬁrst result establishes the existence of ﬁne-tuned data perturbations along the backwards light

cone of O, for which the naked singularity interior is backwards stable. The class of allowed data is explicit,

and is non-generic in the space of all initial data.

1More precisely, [9] shows a co-dimension 1instability result for absolutely continuous (AC) naked singularities. In the more

general bounded variation (BV) setting, he identiﬁes an unstable two-dimensional plane of perturbuations, subject to an inﬁnite

blueshift assumption along the past lightcone of the singularity.

A preliminary statement of the backwards stability result for naked singularity interiors is given below.

For precise statements of all results, see Section 3.3.

Theorem 1. Let (Q(in), g, φ)be a given admissible2naked singularity interior. Choose data for the scalar

ﬁeld along the past light cone of Osatisfying

∂uφ(u, 0) = ∂uφ(u, 0) + f0(u),(1.2)

where f0(u) = O(|u|α−1)as u→0, and where αis a large constant depending on the background spacetime.

See Figure 2a).

There exists an 0depending on the background solution and f0such that for ≤0, there exists a

bounded variation solution to the spherically symmetric Einstein-scalar ﬁeld system in Q(in)achieving the

data for ∂uφalong {v= 0}. The solution satisﬁes appropriate gauge and regularity conditions, as well as

self-similar bounds.

Finally, the solution is asymptotic to the background solution as u→0, with rates controlled by α.

We emphasize that the property of being a naked singularity is global, and relies on the behavior of the

solution in the asymptotically ﬂat region of the exterior. It is thus necessary to verify that there exists a

choice of outgoing data for the scalar ﬁeld along {u=−1, v ≥0}for which the above interior regions extend

to genuine naked singularities. In particular, the exterior construction is a forwards problem. Our second

main result constructs the exterior region globally up to I+.

Theorem 2. Fix an admissible background spacetime (Q, g, φ). Choose data for the scalar ﬁeld along {v= 0}

as in (1.2), and let (Q(in), g, φ)denote the solution constructed by Theorem 1.

Choose data for the scalar ﬁeld along {u=−1, v ≥0}satisfying

∂vφ(−1, v) = ∂vφ(−1, v) + g0(v) + O(),(1.3)

where g0(v)is a free function vanishing suﬃciently quickly as v→0and v→ ∞. The additional O()terms

are explicitly determined by the interior solution (Q(in), g, φ). See Figure 2b).

There exists 0small depending on the background solution and g0such that for ≤0, there exists a

solution in the exterior region Q(ex)attaining the data along {v= 0} ∪ {u=−1, v ≥0}. The solution

satisﬁes appropriate gauge and regularity conditions, self-similar bounds, and is asymptotically ﬂat. The

exterior and interior spacetimes glue across {v= 0}as a solution of bounded variation, and the resulting

solution admits an incomplete I+.

1.1 Christodoulou’s k-self-similar solutions

A detailed discussion of the solutions constructed in [7], including their global expression in double null gauge

and sharp regularity properties, is given in Appendix A. In this introduction we review some motivating

aspects.

Underlying k-self-similarity and expression in Bondi coordinates

We begin by deﬁning k-self-similarity. The spherically symmetric Einstein-scalar ﬁeld system admits a two

parameter symmetry group R+×R, where a∈R+, b ∈Ract on solutions (r, m, φ)via

r→ar, m →am, φ →φ+b.

One can also write the action on the quotient metric as

gµν →a2gµν .

2Roughly, admissible spacetimes admit an appropriate double null gauge, lie in a ﬁxed regularity class, and satisfy quantitative

self-similar bounds. For details, see Deﬁnition 3.1.

{r= 0}

{u=−1}

∂uφ∼∂uφ(u) + f0(u)

(Q(in), g, φ)

∂vφ∼∂vφ(u) + f0(u)

∂vφ∼∂vφ(v) + g0(v) + O()

{u=−1}

(Q(ex), g, φ)

I+

{u= 0}

Figure 2: (a) Domain of existence for the interior solutions of Theorem 1. Ingoing data perturbation high-

lighted in blue. (b) Domain of existence for the interior solutions of Theorem 2. Outgoing data perturbation

highlighted in red.

Fixing a parameter k∈R,k-self-similarity formalizes the notion of solutions invariant under the above

scaling transformation. More precisely, k-self-similar solutions admit a homothetic vector ﬁeld S, generating

a one-parameter family of diﬀeomorphisms faof Qunder which the solution transforms as

(f∗

ag)µν =a2gµν , f∗

ar=r, f∗

aφ=φ−klog a.

These equations in turn imply

(LSg)µν = 2gµν , Sr =r, Sφ =−k.

The parameter kis critical for the underlying mechanism of singularity formation. Roughly, nonzero k

corresponds to the logarithmic growth of the scalar ﬁeld along the past lightcone of O.

The construction of [7] takes place in self-similar Bondi coordinates (u, r), where uis an outgoing null

coordinate and rthe area radius. With respect to this gauge the homothetic vector ﬁeld can be written as

S=r∂r+u∂u.

The ansatz of k-self-similarity implies the following form for the metric gkand scalar ﬁeld φk:

gk=e2β(u,r)du2−2eβ(u,r)+γ(u,r)dudr, φk=χ(u, r)−klog |u|,(1.4)

where β(u, r) = ˚

β(−r

u), γ(u, r) = ˚γ(−r

u), χ(u, r) = ˚χ(−r

u)reduce to functions of a self-similar parameter

−r

The subclass of solutions with bounded scalar ﬁeld (k= 0) were studied in the earlier work [6] as

prototypical examples of bounded variation solutions. These solutions are called scale-invariant, and can in

fact be written explicitly in double null gauge; however, they do not model a breakdown in regularity relative

to initial data, and are thus not examples of naked singularity formation.

The case with k6= 0 does not lend itself to analytical expressions for the solution. Still, [7] shows that the

ansatz (1.4) reduces the full Einstein-scalar ﬁeld system to an autonomous 2x2 system of ordinary diﬀerential

equations. A detailed phase plane analysis establishes the global properties of solutions, and for the subrange

k2∈(0,1

3), there exist solutions that extend to the future light cone of O. The resulting spacetimes are

not asymptotically ﬂat, as a consequence of self-similarity; however, an appropriate truncation in {r1}

remedies this issue, and leads to a naked singularity spacetime Q. In coordinates Qtakes the form

Q={(u, r)|u∈[−1,0), r ∈[0,∞)}.

We will only consider the region to causal future of a ﬁxed outgoing null ray, e.g. {u=−1}, although

extensions to past null inﬁnity are possible as well.

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Aconstructionofapproximatelyself-similarnakedsingularitiesforthesphericallysymmetricEinstein-scalareldsystemJaydeepSingh*11DepartmentofMathematics,PrincetonUniversity,WashingtonRoad,Princeton,NJ08544,UnitedStatesofAmericaDecember8,2022AbstractInthisworkweinvestigatethestabilityandinstabilitypropert...

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A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar ﬁeld system Jaydeep Singh1

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