1.Weak Cosmic Censorship
One of the main open problems in general relativity is the Weak Cosmic Censorship conjecture, proposed by Penrose in 1969 [Pen69]. It states that, for generic initial data, naked singularities do not form. That is to say, singularities are generically hidden inside black holes and invisible to far-away observers who exist for all time: null infinity \(\mathcal I\) is future complete [GH78, Chr99b]. The conjecture is motivated by the fact that a future incomplete \(\mathcal I\) would threaten the predictability of general relativity as a physical theory.
Christodoulou proved a suitable formulation of the conjecture for the model problem of the spherically symmetric Einstein-scalar field system in 3+1 dimensions. The argument of [Chr99a] relies on a novel blueshift effect to show that, while explicit naked singularity spacetimes can be constructed [Chr94, Cho93], all naked singularities are in fact unstable to black hole formation and the set of naked singularity spacetimes is indeed non-generic in a suitable sense. However, the instability mechanism in [Chr99a] required working in a topology defined by a low-regularity class of solutions. Thus, the question of Weak Cosmic Censorship in a smoother topology remains open even in this setting.
This paper initiates the mathematical study of gravitational collapse for the circularly symmetric Einstein-scalar field system in 2+1 dimensions with \(\Lambda<0\), and proves Weak Cosmic Censorship in a topology of arbitrary smoothness: in \(C^k\), for any \(k\ge2\).
2.The model for gravity in 2+1 dimensions
The paper considers the Einstein-scalar field system with a negative cosmological constant:
\[\mathrm{Ric}_{\mu\nu}-\tfrac12 R\,g_{\mu\nu}+\Lambda g_{\mu\nu}=\nabla_\mu\phi\,\nabla_\nu\phi-\tfrac12 g_{\mu\nu}\lvert\nabla\phi\rvert^2,\qquad \square_g\phi=0.\]In 2+1 dimensions this is perhaps the most elementary model for gravitational collapse that allows for the dynamical formation of singularities [PC00, JGC15]. It was first studied in numerical relativity by Pretorius and Choptuik [PC00].
The simplest solution is anti-de Sitter (AdS) spacetime, with \(\phi=0\) and
\[g_{\mathrm{AdS}}=-\big(1+\lvert\Lambda\rvert r^2\big)dt^2+\big(1+\lvert\Lambda\rvert r^2\big)^{-1}dr^2+r^2d\sigma_1^2.\]It is circularly symmetric, contains no singularities, is geodesically complete, and has a timelike asymptotic boundary, null infinity \(\mathcal I\).
The paper studies circularly symmetric solutions, written in global double null coordinates as
\[g=-\Omega^2(u,v)\,du\,dv+r^2(u,v)\,d\sigma_1^2,\qquad \phi=\phi(u,v).\]We consider maximal developments of \(C^k\) asymptotically AdS characteristic data on an outgoing null cone \(C_0^+\), obtained by solving an initial-boundary value problem with reflective boundary conditions at \(\mathcal I\), adapting the 3+1-dimensional theory of Holzegel–Smulevici [HS12].
The BTZ family
The Bañados–Teitelboim–Zanelli (BTZ) spacetimes [BTZ92] are a 1-parameter family of circularly symmetric static solutions with \(M\geq0\):
\[g_M=-\big(1-M+\lvert\Lambda\rvert r^2\big)dt^2+\big(1-M+\lvert\Lambda\rvert r^2\big)^{-1}dr^2+r^2d\sigma_1^2,\quad \phi_M=0.\]For \(M=0\) this is AdS. For \(M\in(0,1)\) the metric has a conical singularity at \(r=0\), and for \(M=1\) a cusp singularity. For \(M>1\) the spacetime has no center, but it extends to a two-ended black hole. See the survey [Car95] for more on these solutions.
A space slice of the BTZ solutions
\(t=\text{const}\) sliceNot a spacetime diagram. Drag to rotate.
- Cone angle
- 180.0°
- 1 − M
- 0.250
- Horizon
- none\(r=\) \(/\sqrt{\lvert\Lambda\rvert}\)
AdS spacetime. A regular center: the only member of the BTZ family with one. Conical singularity at \(r=0\). The slice is a cone of angle \(2\pi\sqrt{1-M}\) = 180°. Cusp singularity at \(r=0\). The cone angle has closed up, and the tip has become an infinitely long spike. No center. The spacetime extends to a two-ended black hole. The slice through the horizon is a throat joining two ends, with the horizon circle \(r=\sqrt{(M-1)/\lvert\Lambda\rvert}\) at its waist.
Since the paper only considers spacetimes with a regular center, the only member of the BTZ family in this class is AdS. Nonetheless, the entire BTZ family plays an important role in the asymptotic behavior of the solutions at \(\mathcal I\). It also already exhibits the threshold \(M=1\), which reappears as the mass gap.
3.Global structure of spacetimes
The moduli space \(\mathfrak M\) consists of \(C^k\) asymptotically AdS characteristic data on \(C_0^+\). To each data set we associate its unique maximal development. Adapting the work of Kommemi [Kom13] and of Holzegel–Smulevici [HS12] to 2+1 dimensions, the paper shows that its Penrose diagram is one of the following four.
The possible Penrose diagrams of the maximal development
Click inside a diagram to send light both ways: outgoing and ingoing.
(a)Black hole spacetime, spacelike-only singularity
\(\mathfrak M^{\mathrm{spacelike}}_{\mathrm{black}}\)
An event horizon \(\mathcal H\) forms, and the singularity inside the black hole is spacelike.
(b)Black hole spacetime, locally naked singularity
\(\mathfrak M^{\mathrm{loc.naked}}_{\mathrm{black}}\)
A null component \(\mathcal B_0\) arises out of \(b_\Gamma\) before the spacelike singularity \(\mathcal B\).
(c)Non-collapsing spacetime
\(\mathfrak M_{\mathrm{non}}\)
\(\Gamma\) and \(\mathcal I\) are future complete. AdS spacetime is an example.
(d)Naked singularity spacetime
\(\mathfrak M_{\mathrm{naked}}\)
The first singularity \(b_\Gamma\) is not hidden behind an event horizon: \(\mathcal B_0\) reaches \(\mathcal I\).
The spacetimes (b) and (d) are locally naked singularity spacetimes. Light from points arbitrarily close to \(b_\Gamma\) reaches the observers near \(\mathcal B_0\). Equivalently, locally naked singularity spacetimes contain no trapped surfaces in a neighborhood of \(b_\Gamma\). We denote the corresponding set of initial data by \(\mathfrak M_{\mathrm{loc.naked}}\).
We denote by \(\mathfrak M_{\mathrm{black}}\subset\mathfrak M\) the set of initial data whose maximal development is a black hole spacetime. Equivalently, the maximal development contains a trapped surface. Since having a trapped surface is an open condition, \(\mathfrak M_{\mathrm{black}}\) is open.
4.The main result
Theorem 1.1 · Weak Cosmic Censorship
For any \(k\ge2\), the maximal development of generic \(C^k\) asymptotically AdS characteristic data does not contain naked singularities.
In particular, naked singularity spacetimes are non-generic and are unstable to black hole formation. Additionally, locally naked singularities are unstable to the formation of spacelike-only singularities.
Precise statement
More precisely, \(\mathfrak M_{\mathrm{generic}}:=\mathfrak M_{\mathrm{black}}\sqcup\mathrm{int}\big(\mathfrak M_{\mathrm{non}}\big)\) is open and dense in \(\mathfrak M\), and:
\[\mathfrak M_{\mathrm{naked}}\subset\mathrm{cl}\big(\mathfrak M_{\mathrm{black}}\big)\setminus\mathfrak M_{\mathrm{black}}\subset\mathfrak M\setminus\mathfrak M_{\mathrm{generic}}\] \[\mathfrak M_{\mathrm{loc.naked}}\subset\mathrm{cl}\big(\mathfrak M^{\mathrm{spacelike}}_{\mathrm{black}}\big)\setminus\mathfrak M^{\mathrm{spacelike}}_{\mathrm{black}}\]Theorem 1.1 in pictures
Naked singularity spacetime
Black hole spacetime with a locally naked singularity
Both kinds of locally naked singularity spacetimes, (d) and (b) of Figure 3, are perturbed by an arbitrarily small amount in \(C^k\). Trapped surfaces then form arbitrarily close to \(b_\Gamma\), and the maximal development is a black hole with a spacelike-only singularity, as in Figure 3(a). Click inside either diagram to send light outward.
The paper chooses a more elementary notion of genericity than Christodoulou [Chr99a]. It proves that naked singularity data lie in the complement of an open and dense subset of \(\mathfrak M\), rather than that they have positive codimension. The reason is the need to extend initial data perturbations globally and satisfy the boundary conditions at \(\mathcal I\).
A conjectural picture of the moduli space \(\mathfrak M\)
Conjecture, based on numericsThis conjectural picture comes from numerical work on critical collapse. Theorem 1.1 implies that naked singularities are on the threshold, or they are surrounded by \(\mathfrak M_{\mathrm{black}}\). Click a point or a curve to see its Penrose diagram.
Numerical relativity works on critical collapse ([Cho93] in 3+1 dimensions, see also the review [GMG07]; [PC00] and [JGC15] in 2+1 dimensions) suggest the above conjectural picture of the moduli space. Take an initial data profile \(\Phi\) and any scaling parameter \(p\). For small \(p\) the solution does not collapse, for large \(p\) it forms a black hole, and at the threshold \(p=p^*\) it is expected to form a naked singularity. In this picture the threshold is a codimension-1 set of naked singularity spacetimes, consistent with Theorem 1.1. The theorem also covers naked singularities off the threshold, if they exist, showing they are surrounded by \(\mathfrak M_{\mathrm{black}}\).
5.Idea of the proof
The overall strategy goes back to Christodoulou [Chr99a]. The main step is to show that locally naked singularities are unstable to the formation of spacelike-only singularities, and for this it suffices to perturb them so that trapped surfaces form arbitrarily close to \(b_\Gamma\). The proof relies on two fundamental results about circularly symmetric solutions: a mass gap, and a quantitative criterion for the formation of trapped surfaces. Together they show that every first singularity at the center has unbounded blueshift, and the blueshift instability then produces the trapped surfaces.
The structure of the proof
Mass gap
No singularities form in the domain of dependence of a disk bounded by a circle with Hawking mass \(m<1\). Corollary: on the past light cone of a first singularity at the center, \(m\ge1\), and \(m\to1^+\) at the singularity.
Formation of trapped surfaces criterion
If the energy content of a thin annular region on an outgoing null cone is large compared to its size, \(\eta_0>\delta_0\), a marginally trapped surface forms.
Unbounded blueshift
On the past light cone \(C_0^-\) of \(b_\Gamma\), the mass gap gives \(m\ge1\) and the absence of trapped surfaces gives \(m\le1+\lvert\Lambda\rvert r^2\). Hence the blueshift satisfies \(\mathbf b(u)\ge2\log(u_0/u)\). This holds for any first singularity at the center.
Bounds for locally naked singularity spacetimes
Locally naked singularities have no trapped surfaces near \(b_\Gamma\), so the contrapositive of Theorem 1.3 gives \(\eta\le\delta\) in a region \(\mathcal P\) near \(b_\Gamma\), which bounds the energy of the scalar field.
The blueshift instability
For any \(n>k+1\) and \(\lambda\ne0\), the evolution of the perturbed data contains a sequence of trapped surfaces converging to \(b_\Gamma\). The blueshift amplifies the perturbation, and along a curve \(\mathcal C_\lambda\), the formation of trapped surfaces criterion \(\eta_\lambda>\delta_\lambda\) holds.
Asymptotically AdS characteristic data
The perturbation is constructed on a compact subset of \(C_0^+\). For \(\lambda\) small it extends to asymptotically AdS data. This is a nontrivial step, necessary because of the timelike boundary \(\mathcal I\).
Proof of Theorem 1.1
Cauchy stability reduces the problem to the case where \(C_0^-\) intersects \(C_0^+\). The instability then gives \(\mathfrak M_{\mathrm{loc.naked}}\subset\mathrm{cl}(\mathfrak M^{\mathrm{spacelike}}_{\mathrm{black}})\), and the rest follows from the decomposition of \(\mathfrak M\) and the openness of \(\mathfrak M_{\mathrm{black}}\).
5.1.The mass gap
The paper defines the Hawking mass by
\[m=-g(\nabla r,\nabla r)-\Lambda r^2+1,\]translated by 1 as for the BTZ family. It is non-negative for solutions arising in gravitational collapse, and it vanishes on a circle exactly when the solution is AdS in the domain of dependence of the disk bounded by that circle.
A circle is trapped exactly when \(m>1+\lvert\Lambda\rvert r^2\).
The Bondi mass is \(M=\lim_{r\to\infty}m|_{C_0^+}\), and the reflective boundary conditions are \(m|_{\mathcal I}=M\).
Theorem 1.2 · Mass gap
No singularities form in the domain of dependence of a disk bounded by a circle with Hawking mass strictly less than 1.
Proof idea
For the proof of Theorem 1.2, the bound on \(m\) first gives quantitative control of the geometry. A non-concentration estimate then shows that the scalar field cannot blow up at a faster than self-similar rate, so \(r\partial_r\phi\) is bounded. This improves by a small power of \(r\): \(r^{1-\rho}\partial_r\phi\) is bounded for some small \(\rho>0\). These bounds propagate the local well-posedness norm from the initial data, so no singularities form.
The mass gap
Bondi mass \(M\) = < 1. Every circle on \(C_0^+\) has \(m<1\), so no singularities form: the solution is non-collapsing (Corollary 1.2). This proves the conjecture of Bizoń and Jałmużna. \(m(S)\) = < 1. By Theorem 1.2, no singularities form in the domain of dependence of the disk bounded by \(S\) (shaded). \(m(S)\) = ≥ 1. Theorem 1.2 says nothing about this disk. It applies to every smaller disk with \(m<1\), up to the dashed outline, so a singularity can only form outside that region.
Example data on \(C_0^+\), with Hawking mass \(m=Mr^2/(1+r^2)\) increasing from 0 at the center to the Bondi mass \(M\) at \(\mathcal I\) (units \(\lvert\Lambda\rvert=1\)). Move the circle \(S\) along \(C_0^+\), or drag it in the diagram. While \(m(S)<1\), the domain of dependence of the disk bounded by \(S\) is regular. When \(M<1\) this holds for every circle.
The main use of the mass gap is the following description of first singularities at the center:
Corollary 1.1
Let \(b_\Gamma=(u_*,v_*)\) be a first singularity at the center \(\Gamma\). Then \(m(u,v_*)\to1^+\) as \(u\to u_*^-\).
Globally, the Hawking mass is non-decreasing along \(C_0^+\) and tends to the Bondi mass at \(\mathcal I\). As a corollary, we obtain the mass gap in terms of the Bondi mass:
Corollary 1.2
Solutions with Bondi mass less than 1 do not form singularities. More precisely, if \(\mathfrak D\in\mathfrak M\) has \(M<1\), then \(\mathfrak D\in\mathrm{int}\big(\mathfrak M_{\mathrm{non}}\big)\). In particular, small perturbations of AdS spacetime are non-collapsing.
This proves a conjecture of Bizoń and Jałmużna [BJ13], who gave numerical evidence for very small Bondi mass. It marks a sharp contrast with AdS spacetime in 3+1 dimensions, which is unstable to black hole formation, see [DH06, BR11, Mos20, Mos23].
5.2.Formation of trapped surfaces criterion
The second ingredient is a quantitative criterion for trapped surface formation, inspired by Christodoulou's work in 3+1 dimensions [Chr91]. Intuitively, if the energy content of an annular region bounded by two circles on the outgoing null cone \(C_0^+\) is large compared to the size of the region, then a trapped surface forms in evolution.
Theorem 1.3 · Formation of trapped surfaces
Let \(C_1=\{v=v_1\}\) be an ingoing cone along which \(r\to0\) and \(m\ge1\), and let \(C_2=\{v=v_2\}\) with \(v_2>v_1\). On \(C_0^+\) set
\[\delta_0=\frac{r(u_0,v_2)}{r(u_0,v_1)}-1,\qquad \eta_0=\frac{m(u_0,v_2)-m(u_0,v_1)}{\lvert\Lambda\rvert\,r^2(u_0,v_2)}.\]If \(\delta_0>0\) is small enough and \(\eta_0>\delta_0\), then a marginally trapped surface forms on \(C_2\), at radius at least \(\frac{5\delta_0/4}{1+\delta_0}\,r(u_0,v_2)\).
Proof idea
The proof adapts Christodoulou's argument [Chr91] to 2+1 dimensions: the monotonicity properties of the system give a differential inequality for \(\eta=\lvert\Lambda\rvert^{-1}r^{-2}(u,v_2)\big(m(u,v_2)-m(u,v_1)\big)\) along \(C_2\).
Formation of trapped surfaces
\(\eta_0\) = > \(\delta_0\) = . A marginally trapped surface forms on \(C_2\), at radius at least \(r_2\). \(\eta_0\) = ≤ \(\delta_0\) = . The energy content is not large compared to the width, and the criterion gives no trapped surface.
The annulus between the circles \(r_1<r_2\) on \(C_0^+\) has relative width \(\delta_0\) and energy content \(\eta_0\). When \(\eta_0>\delta_0\), the marginally trapped surface on \(C_2\) is drawn at a radius above the theorem's lower bound.
Following Christodoulou [Chr99a], the criterion is used in two ways: directly, after the perturbation, and through its contrapositive, to get a priori control of the scalar field for any locally naked singularity spacetime.
5.3.Locally naked singularity spacetimes
Let \(b_\Gamma\) be a first singularity at the center whose past light cone \(C_0^-\) intersects \(C_0^+\), and choose double null coordinates such that
\[b_\Gamma=(0,0),\qquad C_0^-=\{v=0,\ 2r=-u\},\qquad C_0^+=\{u=u_0,\ 2r=v-u_0\}.\]The renormalized outgoing derivative of the scalar field \(\theta=\sqrt r\,\partial_v\phi/\partial_vr\) satisfies
\[\partial_u\theta=\frac{2\lvert\Lambda\rvert r(-\partial_ur)}{1+\lvert\Lambda\rvert r^2-m}\,\theta-\frac{1}{2\sqrt r}\,\partial_u\phi,\]and the blueshift is the integrating factor of this equation:
\[\mathbf b(u,v)=\int_{u_0}^{u}\frac{2\lvert\Lambda\rvert r(-\partial_ur)}{1+\lvert\Lambda\rvert r^2-m}(u',v)\,du',\qquad \mathbf b(u)=\mathbf b(u,0).\]Near \(C_0^-\) the mass gap gives \(m\ge1\), and the absence of trapped surfaces gives \(m\le1+\lvert\Lambda\rvert r^2\). Thus, we get Proposition 5.1:
\[\mathbf b(u)\ \ge\ 2\log\frac{u_0}{u}.\]So all first singularities at the center have infinite blueshift, whether or not they are locally naked.
Infinite blueshift from the mass gap
On \(C_0^-\), the Hawking mass stays between the floor \(m=1\) (mass gap) and the ceiling \(m=1+\lvert\Lambda\rvert r^2\) (no trapped surfaces), and it increases with \(r\). Drag the four points to choose such a profile (units \(\lvert\Lambda\rvert=1\), \(r(u_0,0)=1\)). The right panel shows \(\mathbf b(u)=\int_{u_0}^u\frac{\lvert\Lambda\rvert r}{1+\lvert\Lambda\rvert r^2-m}\,du'\) for that profile. It always stays above \(2\log(u_0/u)\), with equality exactly when \(m\equiv1\).
Moreover, the absence of trapped surfaces near \(b_\Gamma\) gives, by the contrapositive of Theorem 1.3, bounds for the energy of the scalar field in a region \(\mathcal P\) near \(b_\Gamma\), whose size depends on the strength of the blueshift, proving Proposition 5.2:
\[\int_0^v\theta^2(u,v')\,dv'\lesssim\frac{v\lvert u\rvert}{(1+\lvert\Lambda\rvert r^2-m)(u,0)}.\]In \(\mathcal P\) one also has \(e^{\mathbf b(u,v)}\sim e^{\mathbf b(u)}\) and \(\partial_vr(u,v)\sim e^{-\mathbf b(u)}\).
5.4.The blueshift instability
Consider perturbations of the initial data on \(C_0^+\) supported on \(\{v\ge0\}\), so that they leave the causal past of \(b_\Gamma\) unchanged. For any integer \(n>k+1\) and any \(\lambda\ne0\), set
\[r_\lambda=r,\ \ \phi_\lambda=\phi\ \text{ for } v\le0,\qquad r_\lambda=r,\ \ \theta_\lambda=\theta+\lambda v^n\ \text{ for } 0\le v\le v_0'.\]The main role of the blueshift is to amplify this perturbation: the main contribution in \(\theta_\lambda-\theta\) comes from the integrating factor acting on \(\lambda v^n\),
\[\int_0^v\Big\lvert(\theta_\lambda-\theta)(u,v')-e^{\mathbf b_\lambda(u,v')}(\theta_\lambda-\theta)(u_0,v')\Big\rvert^2dv'\ \le\ \lambda^2v^{2n+\frac43}e^{2\mathbf b(u)}\ \ll\ \lambda^2v^{2n+1}e^{2\mathbf b(u)}.\]The blueshift instability, step by step
Growth. The perturbation \(\theta_\lambda-\theta=\lambda v^n\) on \(C_0^+\) is carried along the ingoing null cones toward \(b_\Gamma\), and the blueshift amplifies it:
\[\theta_\lambda-\theta\approx e^{\mathbf b_\lambda(u,v)}\,\lambda v^n,\]up to an error of lower order in \(L^2\), in the region \(\mathcal P_\lambda\) where the perturbed solution is controlled. Near \(b_\Gamma\) the amplification factor is at least \((u_0/u)^2\).
The criterion applies. The curve \(\mathcal C_\lambda=\{(u,v_\sharp(u))\}\) is suitably chosen and lies in \(\mathcal P_\lambda\) near \(b_\Gamma\). At \(S_\lambda=(u,v_\sharp(u))\), use Theorem 1.3 with \(C_1=C_0^-\), along which \(m\ge1\) and \(r\to0\), and \(C_2=\{v=v_\sharp(u)\}\). The amplified energy of the scalar field gives
\[\delta_\lambda\lesssim v_\sharp(u)\,\lvert u\rvert^{-1}e^{-\mathbf b(u)},\] \[\eta_\lambda\gtrsim v_\sharp(u)\,\lvert u\rvert^{-1-\frac1{5n}}e^{-\mathbf b(u)},\]so \(\eta_\lambda/\delta_\lambda\gtrsim\lvert u\rvert^{-1/(5n)}\to\infty\). Hence \(\eta_\lambda>\delta_\lambda\), with \(\delta_\lambda\) small, at every point of \(\mathcal C_\lambda\) close enough to \(b_\Gamma\) (thick).
Trapped surfaces form. By Theorem 1.3, a marginally trapped surface \((u_{\mathcal A},v_\sharp(u))\) forms on \(C_2\). This holds for every \(S_\lambda\in\mathcal C_\lambda\) close enough to \(b_\Gamma\), so the spacetime contains a sequence of trapped surfaces converging to \(b_\Gamma\) (Proposition 6.1).
A spacelike singularity. Since trapped surfaces form arbitrarily close to \(b_\Gamma\), the null piece \(\mathcal B_0\) of the background disappears: the singularity \(\mathcal B\) is spacelike-only, starting at \(b_\Gamma\). Once the perturbation is extended to asymptotically AdS data, these data lie in \(\mathfrak M_{\mathrm{black}}^{\mathrm{spacelike}}\).
The smaller \(\lvert\lambda\rvert\), the closer to \(b_\Gamma\) the trapped surfaces form, but the singularity is still spacelike-only.
Concretely, along the curve \(\mathcal C_\lambda=\{(u,v_\sharp(u))\}\) the two estimates above give
\[\int_0^{v_\sharp(u)}\big\lvert\theta_\lambda(u,v')\big\rvert^2dv'\gtrsim\lambda^2\big[v_\sharp(u)\big]^{2n+1}e^{2\mathbf b(u)},\]which implies \(\eta_\lambda\big(u,v_\sharp(u)\big)>\delta_\lambda\big(u,v_\sharp(u)\big)\) for all \(\lvert u\rvert\) small. So the perturbed spacetime contains a sequence of trapped surfaces converging to \(b_\Gamma\) (Proposition 6.1), and its singularity is spacelike-only.
Finally, the perturbation is constructed only on a compact subset of \(C_0^+\). For \(\lambda\) small, Appendix A extends it to asymptotically AdS data, a nontrivial step needed because of the timelike boundary \(\mathcal I\). Cauchy stability and the decomposition of \(\mathfrak M\) complete the proof of Theorem 1.1.
6.Comparison with 3+1 dimensions
| Christodoulou [Chr99a] | This paper | |
|---|---|---|
| Setting | 3+1 dimensions spherical symmetry \(\Lambda=0\) | 2+1 dimensions circular symmetry \(\Lambda<0\) |
| Initial data | \(\theta\) absolutely continuous | \(\phi\in C^k\), any \(k\ge2\) |
| Genericity | positive codimension | complement of an open dense set |
| Blueshift | unbounded, without a rate | unbounded, with \(\mathbf b(u)\ge2\log(u_0/u)\) |
Both proofs use the formation of trapped surfaces criterion and its contrapositive, and both rely on the fact that locally naked singularity spacetimes have unbounded blueshift. In Christodoulou's theorem [Chr99a] the low-regularity topology is necessary to activate the blueshift instability. Here, the different scaling of the problem and the quantitative lower bound \(\mathbf b(u)\ge2\log(u_0/u)\) allow the blueshift instability to work in \(C^k\).
The two approaches also differ. Christodoulou derives refined necessary conditions for a spacetime to be a locally naked singularity, as compatibility conditions between the data on \(C_0^+\) and the solution on \(C_0^-\), and then violates them by perturbing the initial data. This paper starts with the perturbation and carries out most of the argument in the perturbed spacetime, describing in detail how the solution enters the regime where Theorem 1.3 applies.
7.Upcoming work
Constructing examples of naked singularities plays a fundamental role in understanding their general properties and their instability. Moreover, the existence of such examples justifies the need to include genericity conditions in any formulation of the Weak Cosmic Censorship conjecture.
In 3+1 dimensions, Choptuik's numerical work on critical collapse [Cho93] suggests the existence of naked singularities on the threshold between black hole formation and non-collapse. This threshold behavior has never been constructed rigorously. The examples of naked singularities that do exist, by Christodoulou [Chr94] for the spherically symmetric Einstein-scalar field system and by Rodnianski and Shlapentokh-Rothman [RSR23, SR22] for the Einstein vacuum equations, are not proved to be on the threshold, and they arise from low-regularity initial data.
In 2+1 dimensions with \(\Lambda<0\), the numerical works of Pretorius–Choptuik [PC00] and Jałmużna–Gundlach–Chmaj [JGC15] see the same threshold behavior.
On the finite outgoing cone \(C_0^+=\{u=u_0,\ 0\le r<r_0\}\), we construct a 2-parameter family of \(C^3\) characteristic data \(\mathfrak D^q_\lambda\), for any integer \(q\ge4\) and \(\lambda\in(-\epsilon,\epsilon)\), with the following properties:
- For \(\lambda<0\), the development of \(\mathfrak D^q_\lambda\) does not form singularities.
- For \(\lambda=0\), the development is a locally naked singularity spacetime and contains no trapped surfaces. The case \((\lambda,q)=(0,4)\) corresponds to the numerical examples of [PC00, JGC15].
- For \(\lambda>0\), the development contains trapped surfaces.
Threshold examples
Upcoming work with Igor Rodnianski\(\lambda<0\)\(\lambda=0\)\(\lambda>0\)
Move \(\lambda\) to 0 to see the locally naked singularity of the family \(q\) = . Blueshift at \(\lambda=0\), \(q=4\): \(\mathbf b(u)=\big(3-\tfrac{1}{8}\big)\log(u_0/u)+O(1)\), above the general lower bound \(2\log(u_0/u)\). Blueshift at \(\lambda=0\), \(q=5\): \(\mathbf b(u)=\big(3-\tfrac{1}{10}\big)\log(u_0/u)+O(1)\), above the general lower bound \(2\log(u_0/u)\). Blueshift at \(\lambda=0\), \(q=6\): \(\mathbf b(u)=\big(3-\tfrac{1}{12}\big)\log(u_0/u)+O(1)\), above the general lower bound \(2\log(u_0/u)\). Blueshift at \(\lambda=0\), \(q=7\): \(\mathbf b(u)=\big(3-\tfrac{1}{14}\big)\log(u_0/u)+O(1)\), above the general lower bound \(2\log(u_0/u)\).
These are data on a finite cone, so this is a picture of the local problem. Each family \(\lambda\mapsto\mathfrak D^q_\lambda\) is a short curve and it crosses the threshold at \(\lambda=0\), where the development is a locally naked singularity with no trapped surfaces.
For \(\lambda=0\) the blueshift has the exact rate
\[\mathbf b(u)=\Big(3-\frac1{2q}\Big)\log\frac{u_0}{u}+O(1),\]which should be compared with the general lower bound \(2\log(u_0/u)\) of Proposition 5.1.
An immediate consequence is that \(\mathfrak D^q_0\) extends to asymptotically AdS data in \(\mathfrak M_{\mathrm{loc.naked}}\). In particular \(\mathfrak M_{\mathrm{loc.naked}}\neq\emptyset\): locally naked singularities do occur in the class of solutions of this paper, and by Theorem 1.1 each of them is unstable to the formation of spacelike-only singularities.
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