End Point of the Ultraspinning Instability and Violation of Cosmic Censorship

TL;DR

This paper investigates the nonlinear evolution of ultraspinning instabilities in six-dimensional Myers-Perry black holes, revealing that they lead to naked singularities and violate cosmic censorship.

Contribution

It identifies the end state of the ultraspinning instability and demonstrates the formation of naked singularities in higher-dimensional black holes.

Findings

Formation of concentric black rings connected by membranes

Thinning of membrane segments over time

Violation of weak cosmic censorship in D=6

Abstract

We determine the end point of the axisymmetric ultraspinning instability of asymptotically flat Myers-Perry black holes in D = 6 spacetime dimensions. In the non-linear regime, this instability gives rise to a sequence of concentric rings connected by segments of black membrane on the rotation plane. The latter become thinner over time, resulting in the formation of a naked singularity in finite asymptotic time and hence a violation of the weak cosmic censorship conjecture in asymptotically flat higher-dimensional spaces.

Tables

Table 1. Table 1: Growth rates of the first unstable mode. Errors are ± 3 % plus-or-minus percent 3 \pm 3\% for a / μ 1 3 ≥ 1.7 𝑎 superscript 𝜇 1 3 1.7 a/\mu^{\frac{1}{3}}\geq 1.7 and ± 25 % plus-or-minus percent 25 \pm 25\% for a / μ 1 3 = 1.6 𝑎 superscript 𝜇 1 3 1.6 a/\mu^{\frac{1}{3}}=1.6 .

$a/{\mu }^{\frac{1}{3}}$	1.6	1.7	1.8	1.9	2.0
$\Im \varpi {\mu }^{\frac{1}{3}}$	$0.020$	$0.130$	$0.213$	$0.262$	$0.299$

Table 2. Table 2: Properties of the generations. The ratio of length to thickness of the i th superscript 𝑖 th i^{\text{th}} generation membrane was measured just before the formation of the ( i + 1 ) th superscript 𝑖 1 th (i+1)^{\text{th}} generation. The time it takes to form the next generation decreases with factors 0.07 0.07 0.07 , 0.41 0.41 0.41 and 0.26 0.26 0.26 .

Gen.	1	2	3	4	5
$t_i/{\mu }^{\frac{1}{3}}$	31.8	36.45	36.78	36.916	36.952
$L_i/Z_{AH,i}$	540	530	370	510	$>370$

Equations

$$d{s}^2=$$

$$+(r^2+a^2){\sin }^2\theta d{\varphi }^2+r^2{\cos }^2\theta d{\Omega }_{(2)}^2,$$

$$r=\rho {\left(1+\frac{1}{4}\frac{r_h^3}{{\rho }^3}\right)}^{\frac{2}{3}},$$

$$\chi ={\chi }_0\left\{1+A{J}_0\left[j_{0,k}\sin \left(\frac{\pi }{2}\sigma \right)\right]\exp \left[-{\left(\frac{{\chi }_0}{{\chi }_h}-\frac{{\chi }_h}{{\chi }_0}\right)}^2\right]\right\},$$

$$t_c<t_0+(t_1-t_0)\sum _i{X}^i<t_0+(t_1-t_0)/(1-X_{\max }).$$

$$Z_{\text{AH}}=\alpha (t_c-t),$$

$${\partial }_t{\beta }^i-F{\left({\partial }_t{\beta }^i\right)}_{t=0}=F({\hat{\Gamma }}^i-{\hat{\Gamma }}_{t=0}^i)-\eta ({\beta }^i-{\beta }_{t=0}^i),$$

$${\partial }_t{\beta }^i=F({\hat{\Gamma }}^i-f(t){\hat{\Gamma }}_{t=0}^i)-\eta ({\beta }^i-{\beta }_{t=0}^i)+{\beta }^k{\partial }_k{\beta }^i,$$

$$f(t)=\exp \left[-\left({\delta }_1{r}_{\mathrm{hor}}^2/r^2+{\delta }_2\right)t^2/{\mu }^{\frac{2}{3}}\right],$$

$$\Theta =({\gamma }^{ab}-s^a{s}^b)(-k_{ab}-K_{ab})\equiv 0,$$

$$x^1-F(x^2,\dots ,x^{d-1})=0,$$

$$x^i=X^i(u^{\alpha }),$$

$$T_{(\alpha )}^i=\frac{\partial X^i}{\partial u^{\alpha }},S^i=\star {\left(T_{(1)}\wedge \cdots \wedge T_{(d-1)}\right)}_j{\gamma }^{ji},$$

$$k_{\alpha \beta }=$$

$$=$$

$$s_i{h}^{\alpha \beta }\left(\frac{{\partial }^2{X}^i}{\partial u^{\alpha }\partial u^{\beta }}+{\Gamma }_{jk}^i\frac{\partial X^j}{\partial u^{\alpha }}\frac{\partial X^k}{\partial u^{\beta }}\right)+({\gamma }^{ij}-s^i{s}^j)K_{ij}=0,$$

$${\square }_{\mathcal{H}}{u}^{\alpha }=H^{\alpha }(u^{\beta }),$$

$${\gamma }_{ij}{h}^{\beta \gamma }\left(\frac{{\partial }^2{X}^i}{\partial u^{\beta }\partial u^{\gamma }}\frac{\partial X^j}{\partial u^{\alpha }}+{\Gamma }_{jk}^i\frac{\partial X^j}{\partial u^{\alpha }}\frac{\partial X^k}{\partial u^{\beta }}\frac{\partial X^l}{\partial u^{\gamma }}\right)=H_{\alpha }(u^{\beta }),$$

$${\gamma }_{ij}{h}^{\beta \gamma }\left(\frac{{\partial }^2{X}^i}{\partial u^{\beta }\partial u^{\gamma }}{S}^j+{\Gamma }_{jk}^i{S}^j{T}_{(\beta )}^k{T}_{(\gamma )}^l\right)=\parallel S^k\parallel ({\gamma }^{ij}-s^i{s}^j)K_{ij},$$

$${\gamma }_{ij}{h}^{\beta \gamma }\left(\frac{{\partial }^2{X}^i}{\partial u^{\beta }\partial u^{\gamma }}{T}_{(\alpha )}^j+{\Gamma }_{jk}^i{T}_{(\alpha )}^j{T}_{(\beta )}^k{T}_{(\gamma )}^l\right)=H_{\alpha }.$$

$$d{s}_{{\mathbf{E}}_4}^2=d{U}^2+d{Z}^2+Z^{2}d{\Omega }_{(2)}^2.$$

Full text

*Introduction.—*The recent detection of gravitational waves from black hole binary mergers Abbott et al. (2016a, b) has provided the first direct observation of these objects. The current observational data are compatible with the predictions of general relativity, and they suggest that the end point of such mergers is a Kerr black hole (BH) Kerr (1963). These observations provide evidence that the Kerr BH in vacuum is non-linearly stable, at least within a certain range of the angular momentum. However, a mathematically rigorous understanding of the stability of the generic Kerr BH, as well as a thorough understanding of its dynamics under arbitrary perturbations, is still lacking. In fact, recent work suggests that novel and nontrivial dynamics may be present very close to extremality (e.g., Aretakis (2013); Yang et al. (2015); Gralla et al. (2016)).

Higher dimensional BHs, however, can be unstable under gravitational perturbations. This was first shown by Gregory and Laflamme (GL) for black strings and black $p$-branes Gregory and Laflamme (1993). Determining the end point of this instability has been a subject of intense study due to the potential implications on the weak cosmic censorship conjecture (WCC) in such spacetimes. With the aid of numerical relativity (NR), Lehner and Pretorius (2010) found that the GL instability gives rise to a self-similar structure of bulges connected by ever thinner string segments, which all undergo the GL instability. Eventually, the black string pinches off in finite asymptotic time, resulting in a naked singularity. Since no fine-tuning of the initial data was required, this result constituted a violation of the WCC, albeit in spacetimes with compact extra dimensions.

Contrary to the $D=4$ case, asymptotically flat BHs in higher dimensions can carry arbitrarily large angular momenta. At very large angular momenta, BHs become highly deformed and resemble black branes, which are known to be unstable under the GL instability Emparan and Myers (2003). This observation highlighted the possibility that higher dimensional asymptotically flat BHs can be unstable under gravitational perturbations. This indeed turned out to be the case. For instance, the black rings of Emparan and Reall (2002) suffer from various types of instabilities Elvang et al. (2006); Figueras et al. (2011); Santos and Way (2015); Tanabe (2016); Figueras et al. (2016); Tanabe , including the GL instability. The non-linear evolution of the latter was studied in a very recent work by three of us Figueras et al. (2016), where it was found that, for sufficiently thin rings, the evolution of the instability is similar to that of the GL instability of black strings. Hence, a naked singularity should form in finite asymptotic time, thus violating the WCC in higher-dimensional asymptotically flat spaces. However, the calculations in Figueras et al. (2016) were computationally highly demanding, which limited the extent to which the instability could be explored. It was therefore not possible to estimate the timescale of a possible pinch-off or to determine whether the process is self-similar as for black strings.

Ref. Emparan and Myers (2003) conjectured (and Dias et al. (2009) later confirmed) that rapidly spinning Myers-Perry (MP) BHs Myers and Perry (1986)111MP BHs are the higher-dimensional analogues of Kerr BHs. in $D\geq 6$ are unstable under a GL-type of instability, which is referred to as the “ultraspinning instability”. As with the GL case, there exist zero modes that connect MP BHs with different families of “bumpy” BHs Dias et al. (2014a); Emparan et al. (2014). In this Letter, we report on the final stages of the evolution of the ultraspinning instability of singly spinning MP BHs in 6 dimensions. We restrict ourselves to the instability that deforms the horizon without breaking any of the rotational symmetries of the background, i.e., the axisymmetric one. The imposed symmetries reduce the problem to a system of $(2+1)$-dimensional PDEs, which is significantly more computationally tractable than the one described in Figueras et al. (2016). This allows us to elucidate the dynamics of the ultraspinning instability in full detail.

*Numerical methods.—*We solve the $D=6$ vacuum Einstein equation numerically with a $U(1)\times SO(3)$ isometry imposed. The ultraspinning instability lies within this symmetry sector, but other non-axisymmetric instabilities which are not captured by this ansatz do exist. We impose the symmetry using the modified cartoon method Pretorius (2005); Shibata and Yoshino (2010a); Cook et al. (2016). We employ the CCZ4 formulation Alic et al. (2012); Weyhausen et al. (2012) on a Cartesian grid with the redefinition of the constraint damping parameter $\kappa_{1}\to\kappa_{1}/\alpha$, where $\alpha$ is the lapse function Alic et al. (2013). Typically, we choose $\kappa_{1}=0.5$ and $\kappa_{2}=0$. As initial data, we take the 6-dimensional singly spinning MP BH,

[TABLE]

with a new quasi-radial coordinate $\rho$ defined by

[TABLE]

where $\mu$ and $a$ are the mass and rotation parameters respectively, $\Sigma=r^{2}+a^{2}\,\cos^{2}\theta$, $\Delta=r^{2}+a^{2}-\mu/r$, and $r_{h}$ is the largest real root of $\Delta(r_{h})=0$. In our simulations, we set $\mu=1$ and consider MP BHs with $1.5\leq a/\mu^{\frac{1}{3}}\leq 2.0$. The first (ring-shaped) unstable mode sets in at $a/\mu^{\frac{1}{3}}=1.572$ and the second (saturn-shaped) mode sets in at $a/\mu^{\frac{1}{3}}=1.849$ Dias et al. (2009).

We evolve the lapse and the shift using the CCZ4 $(1+\log)$ slicing with an advection term and the variant of the Gamma-Driver shift condition used in Figueras et al. (2016) (see also the Supplemental Material Sup ). Initially, we choose $\alpha=\chi$ and $\beta^{i}=\chi\,\beta^{i}_{\textrm{MP}}$, where $\beta^{i}_{\textrm{MP}}$ is the analytic shift obtained from (1) and $\chi$ denotes the conformal factor. To help stabilize the evolution, we add diffusion terms well inside the apparent horizon (AH) as described in Figueras et al. (2016). The coordinate singularity present in our initial data is regularized by the “turduckening” method Brown et al. (2007, 2009). Since the gauge in (1) is not optimal, we first evolve this initial data until the gauge has settled to spatial harmonic coordinates with respect to the conformal metric. In this new gauge, the shape of the AH flattens and resembles a pancake for rapidly spinning BHs (see Fig. LABEL:fig:evolutionAH), as one would expect on physical grounds Emparan and Myers (2003). We stress that this gauge adjustment process occurs over a short time period, during which we have verified that there is no significant physical evolution.

Once the gauge dynamics has settled, we trigger the ultraspinning instability by perturbing the conformal factor via

[TABLE]

where $A$ is the amplitude of the perturbation, $\chi_{0}$ is the unperturbed conformal factor, $\chi_{h}$ is the value of the unperturbed conformal factor at the horizon, $J_{0}$ is the Bessel function of the first kind, $j_{0,k}$ is the $k^{\textrm{th}}$ zero of $J_{0}$, and $\sigma=\sqrt{x^{2}+y^{2}}/\tilde{R}$. Here $\tilde{R}$ is a parameter that determines the extent of the deformation in the rotation plane, and $x$ and $y$ are our Cartesian coordinates. The expression (3) ensures that the perturbation is localized on the horizon and behaves like $J_{0}$ near the rotation axis, where $J_{0}$ captures the unstable mode reasonably accurately Emparan and Myers (2003). This perturbation introduces constraint violations, but they are small and depend linearly on the amplitude of the perturbation. In our simulations, we check that these constraint violations decay exponentially with time (thanks to the CCZ4 constraint damping terms), and that the physical parameters of the perturbed BH change by less than $1\%$ compared to those of the unperturbed BH.

To understand the end point of the ultraspinning instability, we monitor the geometry of the AH. Most traditional approaches in NR assume that the AH can be given by the level set of a function of the angular coordinates (see Thornburg (2007) for a review). In our current symmetry setting this would mean $r=R(\theta)$, where $r$ is the radial coordinate and $\theta$ is the polar angle on the sphere. However, in the final stages of the ultraspinning instability, this is not a valid assumption as $R(\theta)$ fails to be a single-valued function (see Fig. LABEL:fig:evolutionAH). To overcome this problem, we consider the AH as a completely general parametric surface $(x(u),y(u),z(u))$, where $u$ is the parameter. We then solve the elliptic PDEs that arise from setting the expansion and a gauge condition for $u$ to zero. The technical details of this construction can be found in the Supplemental Material Sup and in Tunyasuvunakool (2016).

We solve the CCZ4 equations numerically with the GRChombo code Clough et al. (2015); Adams et al. (2015) using up to $22$ levels of refinement (each refined in a 2:1 ratio) with a coarsest grid spacing of $0.35\mu^{\frac{1}{3}}$. We discretize the equations using fourth-order finite differences and integrate in time using RK4. We choose our refinement levels such that the AH is covered by at least 57 points at all times. Convergence studies indicate that the order of convergence is $\approx 3$. Some relevant numerical tests are presented in the Supplemental Material Sup .

*Results.—*In Fig. LABEL:fig:evolutionAH Top we present different snapshots of the embedding of a constant rotational angle section of the AH into $\mathbb{R}^{4}$ at representative stages of the evolution 222Videos can be found at http://grchombo.github.io.. In the range of $a/\mu^{\frac{1}{3}}$ that we have explored, the ring mode grows fastest and governs the non-linear evolution. We find that initially only a large ring forms at the outermost edge of the horizon (second snapshot in Fig. LABEL:fig:evolutionAH Top), even if we perturb with a “saturn-shaped” perturbation by setting $k=2$ in (3). In Table 1, we summarize the growth rates of the first unstable mode for different values of $a/\mu^{\frac{1}{3}}$ as calculated from our simulations. To our knowledge, these are not currently available in the literature.

To estimate how much mass and angular momentum are contained within the outermost ring, we calculate the corresponding Komar integrals on the AH. The calculated mass is only accurate once the system has settled down to a steady state. Towards the end of our simulations, the Komar mass changes by less than $1\%$, thus indicating that the majority of the AH has settled down sufficiently. We find that the outermost ring accounts for 98–99% of the total mass and more than 99.99% of the angular momentum. The radiated mass is too small ($<2\%$) to be distinguishable from changes in the Komar mass due to the system not having settled down completely. Angular momentum is conserved because of our symmetry assumptions.

After the outermost large ring has formed, the region of the horizon connecting it to the rotation axis resembles a thin, locally boosted black membrane (see Fig. LABEL:fig:evolutionAH). The evolution of the ultraspinning instability takes place in the radial direction, while the local boost is along an orthogonal $U(1)$ direction. Therefore, the dynamics of the black membrane under this instability should be insensitive to the local boost and, since the transverse direction is flat, similar to the GL dynamics of a 5D black string. The portion of the AH that resembles a black membrane is indeed GL unstable and can accommodate many unstable modes (see Table 2). Its subsequent evolution leads to a sequence of ever thinner rings connected by segments of black membrane which are GL unstable.

As evidence that the horizon has the geometry of concentric rings connected by membrane sections, we evaluate the (suitably normalized) spacetime Kretschmann invariant, $K=R_{abcd}R^{abcd}\,Z_{\textrm{AH}}^{4}/12$, on the AH. Here $Z_{\textrm{AH}}$ is the radius of the transverse sphere, which measures the thickness of the AH. The normalization is such that $K=1$ for a black membrane and $K=6$ for a 6D black string. The results (Fig LABEL:fig:evolutionAH Bottom) are in close agreement with $K=1$ on the membrane sections and approach $K=6$ on most of the fully formed rings, suggesting that they are well approximated by stationary black strings.

There are three fundamental differences between the dynamics of unstable black strings and ultraspinning MP BHs. Firstly, the latter have compact horizons that do not wrap any topological direction in spacetime. Thus, any self-similarity is broken near the edges, and also in the early stages of the instability when the radial extent of the unstable membrane sections is comparable to the size of the whole BH. Secondly, MP BHs are rotating and the imposition of axisymmetry introduces a new constraint: the conservation of angular momentum. Furthermore, the rotation causes a centrifugal force which redistributes angular momentum outwards and leads to different membrane sections having different thicknesses (see third snapshot in Fig. LABEL:fig:evolutionAH). Hence, the local GL instabilities of each membrane section evolve on different timescales. Thirdly, throughout the whole evolution, the small concentric rings that form after the first generation move around, causing significant additional stretching over the time it takes to form a new generation. The non-zero boost velocity imparted upon the membrane delays the formation of the $i^{\text{th}}$ Hovdebo and Myers (2006) generation, while the stretching itself also causes the membrane to become thinner, resulting in the earlier formation of the $(i+1)^{\text{th}}$ generation.

The combination of these three effects implies that the evolution of the ultraspinning instability is not self-similar: while we do observe newly formed membrane sections all undergoing the GL instability, the time elapsed between the formation of successive generations does not decrease with a universal factor (c.f. Table 2), even for later generations. Instead, in the $a/\mu^{\frac{1}{3}}=1.7$ run, we observe factors between 0.07 and 0.42. Furthermore, they cause the pinch-off to happen sooner, mostly due to the quick drop in the formation times between generations at the beginning. The largest factor between generations that we observed was $X_{\max}=0.41$. Since $X_{\max}<1$, we can bound the pinch-off time by a geometric series

[TABLE]

While this upper bound is not sharp, it provides evidence that the BH pinches off in finite asymptotic time.

From Table 2, we see that the typical ratio $R_{i}/L_{i}$ between the thickness and the length of a membrane section varies between 300 and 600. For the GL instability of black strings, this ratio is approximately 100 across generations Lehner and Pretorius (2010). Therefore, the membranes that form in the evolution of the ultraspinning instability are more unstable, indicating a faster pinch-off time.

Let us now explain the local dynamics which leads to the non-constant factors between generations. We calculate the radial velocity $dr/dt$ of null rays which co-rotate with the BH to estimate the local radial flow velocity of the AH. The results are shown in Fig. 2. They paint a very consistent picture: near each ring, the radial velocity either decreases or reverses completely, leading to a build up of mass. This also explains the numerous sign changes around the thinnest point of the membrane, where many higher generation rings are present.

The outermost ring very quickly settles down to an almost stationary state. However, as Fig. 2 Top shows, it is still rigidly expanding outwards in the rotation plane. Compared with the balanced 6D black rings Kleihaus et al. (2013); Dias et al. (2014a), we find that the area and angular velocity are still 7% below and 15% above their respective equilibrium values in the final frame of our simulation. These values are consistent with the fact that the ring still has to expand by an additional 7% in the rotation plane in order to reach the equilibrium $S^{1}$ radius while conserving angular momentum.

At late times, Fig. 2 shows that the flow in the $U<1$ region is unaffected by the pull from the outermost ring, and the dynamics is therefore determined by the higher generation rings. These differ from the first generation in that they carry far too little angular momentum to be balanced. Instead, they are held in place by the tension of the membrane sections surrounding them. The tension of a membrane in 6D is proportional to its thickness, and different parts of the membrane have different thickness due to the pull from the outermost ring from the outset. These differences are amplified as the GL instability develops on each of these local sections. As the thicknesses of the surrounding membranes change, higher generation rings develop a radial velocity towards the thicker sections. This is clearly visible in Fig. 2 Middle, and is large enough to significantly change the width of a membrane section during the development of a new generation, thus affecting its formation time.

To obtain a precise value for the pinch-off time, we track the global minimum thickness of the membrane. Even though the dynamics of higher generation rings prevents the formation of new generations from being self-similar, the minimum thickness closely follows the scaling law

[TABLE]

similar to black strings Lehner and Pretorius (2010, 2011) and the Rayleigh-Plateau instability of fluid columns (see Fig. 2 Bottom). This strongly supports our earlier conclusion that the BH will pinch off in finite asymptotic time, $t_{c}$, giving rise to a naked singularity. By performing a 2-parameter fit with (5), we can obtain values for the pinch-off time $t_{c}$ and the dimensionless constant $\alpha$. The value for the latter, $\alpha=(9.9\pm 0.2)\times 10^{-3}$, is universal in that it is the same for all of our runs and is independent of the rotation parameter and initial data.

We may finally speculate about the end point. Fig. 2 suggests that after pinch-off the outermost ring will settle down to its balanced configuration, absorbing the nearby ($U\gtrsim 1$) membrane section. Since the angular velocity in the central sections of the membrane is much too low to form balanced rings, sections closer to the centre will collapse into a spherical BH with negligible angular momentum. Therefore, the end point will be a black saturn in 6D. However, it will not be the saturn that maximizes the entropy for a given final mass and angular momentum, which consists of a central BH carrying all the mass surrounded by a thin ring that accounts for all the angular momentum Elvang et al. (2007). Nevertheless, we find that as $a/\mu^{\frac{1}{3}}$ is increased, the end point becomes more similar to this optimal configuration.

*Discussion.—*Our results provide evidence that the ultraspinning instability evolves into a naked singularity in finite asymptotic time, and thus can be interpreted as a potential counterexample to the WCC in higher-dimensional asymptotically flat spaces. In the approach to pinch-off, the minimum membrane thickness very closely follows a scaling law (5), with a universal constant $\alpha$. However, in $D\geq 6$ MP BHs are unstable to non-axisymmetric modes Shibata and Yoshino (2010b, a); Dias et al. (2014b). Therefore, to find a generic violation of the WCC in MP BH spacetimes, one has to consider the evolution under all these instabilities. Work in this direction is underway. Since the growth rate of the bar mode instability saturates Dias et al. (2014b), we expect that for sufficiently rapidly spinning MP BHs the ultraspinning instability will dominate. Once the first generation ring has formed, the membrane inside becomes thinner by a factor of 50. Thus, the ultraspinning instability in this paper is an order of magnitude faster than the axisymmetry-breaking GL instability of the outermost ring. Therefore, we expect that for sufficiently large values of $a/\mu^{\frac{1}{3}}$, modes that preserve the axisymmetry are dominant in all stages of the evolution, and that the violation of the WCC presented in this paper should be generic.