Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship
aa r X i v : . [ h e p - t h ] O c t Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship
Luis Lehner , , and Frans Pretorius Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada Canadian Institute For Advanced Research (CIFAR), Cosmology and Gravity Program, Canada Department of Physics, Princeton University, Princeton, NJ 08544,USA.
We describe the behavior of 5-dimensional black strings, subject to the Gregory-Laflamme insta-bility. Beyond the linear level, the evolving strings exhibit a rich dynamics, where at intermediatestages the horizon can be described as a sequence of 3-dimensional spherical black holes joined byblack string segments. These segments are themselves subject to a Gregory-Laflamme instability,resulting in a self-similar cascade, where ever-smaller satellite black holes form connected by ever-thinner string segments. This behavior is akin to satellite formation in low-viscosity fluid streamssubject to the Rayleigh-Plateau instability. The simulation results imply that the string segmentswill reach zero radius in finite asymptotic time, whence the classical space-time terminates in anaked singularity. Since no fine-tuning is required to excite the instability, this constitutes a genericviolation of cosmic censorship.
PACS numbers: 04.50.Gh, 04.20.q, 04.25.D
Introduction:
While stationary black holes in 4 space-time dimensions (4D) are stable to perturbations, higherdimensional analogues are not. Indeed, as first illustratedby Gregory and Laflamme in the early 90s [1], blackstrings and p-branes are linearly unstable to long wave-length perturbations in 5 and higher dimensions. Sincethen, a number of interesting black objects in higher di-mensional gravity have been discovered, many of themexhibiting similar instabilities (see e.g. [2]).An open question for all unstable black objects is whatthe end-state of the perturbed system is. For blackstrings [1] conjectured that the instability would causethe horizon to pinch-off at periodic intervals, giving riseto a sequence of black holes. One reason for this con-jecture comes from entropic considerations: for a givenmass per unit length and periodic spacing above a criticalwavelength λ c , a sequence of hyperspherical black holeshas higher entropy than the corresponding black string.Classically, event horizons cannot bifurcate without theappearance of a naked singularity [3]. Thus, reachingthe conjectured end-state would constitute a violation ofcosmic censorship, without “unnatural” initial conditionsor fine-tuning, and be an example of a classical systemevolving to a regime where quantum gravity is required.This conjecture was essentially taken for granted un-til several years later when it was proved that the gen-erators of the horizon can not pinch-off in finite affinetime [4]. From this, it was conjectured that a new, non-uniform black string end-state would be reached [4]. Sub-sequently, stationary, non-uniform black string solutionswere found [5, 6], however, they had less entropy thanthe uniform string and so could not be the putative newend-state, at least for dimensions lower than 13 [7].A full numerical investigation studied the system be-yond the linear regime [8], though not far enough toelucidate the end-state before the code “crashed”. At that point the horizon resembled spherical black holesconnected by black strings, though no definitive trendscould be extracted, still allowing for both conjecturedpossibilities: (a) a pinch-off in infinite affine time, (b)evolving to a new, non-uniform state. If (a), a questionarises whether pinch-off happens in infinite asymptotic time; if so, any bifurcation would never be seen by out-side observers, and cosmic censorship would hold. Whilethis might be a natural conclusion, it was pointed outin [9, 10] that due to the exponentially diverging ratebetween affine time and a well-behaved asymptotic time,pinch-off could occur in finite asymptotic time.A further body of (anecdotal) evidence supporting theGL conjecture comes from the striking resemblance ofthe equations governing black hole horizons to those de-scribing fluid flows, the latter which do exhibit insta-bilities that often result in break-up of the fluid. Thefluid/horizon connection harkens back to the membraneparadigm [11], and also in more recently developed cor-respondences [12, 13]. In [14] it was shown that the dis-persion relation of Rayleigh-Plateau unstable modes inhyper-cylindrical fluid flow with tension agreed well withthose of the GL modes of a black string. Similar behaviorwas found for instabilities of a self-gravitating cylinder offluid in Newtonian gravity [15]. In [16], using a pertur-bative expansion of the Einstein field equations [13] torelated the dynamics of the horizon to that of a viscousfluid, the GL dispersion relation was derived to good ap-proximation, thus going one step further than showinganalogous behavior between fluids and horizons.What is particularly intriguing about fluid analogies,and what they might imply about the black string case,is that break-up of an unstable flow is preceded by for-mation of spheres separated by thin necks. For high vis-cosity liquids, a single neck forms before break-up. Forlower viscosity fluids, smaller “satellite” spheres can formin the necks, with more generations forming the lowerthe viscosity (see [17] for a review). In the membraneparadigm, black holes have lower shear viscosity to en-tropy ratio than any known fluid [18].Here we revisit the evolution of 5D black strings usinga new code. This allows us to follow the evolution wellbeyond the earlier study [8]. We find that the dynamicsof the horizon unfolds as predicted by the low viscosityfluid analogues: the string initially evolves to a configu-ration resembling a hyperspherical black hole connectedby thin string segments; the string segments are them-selves unstable, and the pattern repeats in a self-similarmanner to ever smaller scales. Due to finite computa-tional resources, we cannot follow the dynamics indefi-nitely. If the self-similar cascade continues as suggestedby the simulations, arbitrarily small length scales, and inconsequence arbitrarily large curvatures will be revealedoutside the horizon in finite asymptotic time. Numerical approach:
We solve the vacuum Einsteinfield equations in a 5-dimensional (5D) asymptoticallyflat spacetime with an SO (3) symmetry. Since pertur-bations of 5D black strings violating this symmetry arestable and decay [1], we do not expect imposing this sym-metry qualitatively affects the results presented here.We use the generalized harmonic formulation of thefield equations [19], and adopt a Cartesian coordinatesystem related to spherical polar coordinates via ¯ x i =(¯ t, ¯ x, ¯ y, ¯ z, ¯ w ) = ( t, r cos φ sin θ, r sin φ sin θ, r cos θ, z ). Theblack string horizon has topology S × R ; ( θ, φ ) are co-ordinates on the 2-sphere, and z ( ¯ w ) is the coordinate inthe string direction, which we make periodic with length L . We impose a Cartesian Harmonic gauge condition,i.e. ∇ α ∇ α ¯ x i = 0, as empirically this seems to result inmore stable numerical evolution compared to sphericalharmonic coordinates. The SO (3) symmetry is enforcedusing the variant of the “cartoon” method [20] describedin [19], were we only evolve a ¯ y = ¯ z = 0 slice of the space-time. We further add constraint damping [21], which in-troduces two parameters κ and ρ ; we use ( κ, ρ = 1 , − . ρ is essential to damp an unstable zero-wavelength mode arising in the z direction.We discretize the equations using 4th order finite dif-ference approximations, and integrate in time using 4thorder Runge-Kutta. To resolve the small length scalesthat develop during evolution we use Berger and Oligeradaptive mesh refinement. Truncation error estimatesare used to dynamically generate the mesh hierarchy, andwe use a spatial and temporal refinement ratio of 2.At the outer boundary we impose Dirichlet conditions,with the metric set to that of the initial data. These con-ditions are not strictly physically correct at finite radius,though the outer boundary is placed sufficiently far thatit is causally disconnected from the horizon for the timeof the simulation. We use black hole excision on the innersurface; namely, we find the apparent horizon (AH) us-ing a flow method, and dynamically adjust this boundary t/M A / A low res.med reshigh res. FIG. 1: (Normalized) apparent horizon area vs. time. (the excision surface ) to be some distance within the AH.Due to the causal nature of spacetime inside the AH, noboundary conditions are placed on the excision surface.We adopt initial data describing a perturbed blackstring of mass per unit length M and length L = 20 M ≈ . L c ( L c is the critical length above which all pertur-bations are unstable). This data was used in [8] and werefer the reader to that work for further details.We evaluate the following curvature scalars on the AH: K = IR AH / , S = 27 (cid:0) J I − − (cid:1) + 1 , (1)where I = R abcd R abcd , J = R abcd R cdef R ef ab and R AH is the areal radius of the AH at the corresponding point(though note that I and J are usually defined in termsof the Weyl tensor, though here this equals the Riemanntensor as we are in vacuum). K and S have been scaledto evaluate to { , } for the hyperspherical black hole andblack string respectively. Results:
The results described here are from sim-ulations where the computational domain is ( r, z ) ∈ ([0 , M ] × [0 , M ]). The coarsest grid covering theentire domain has a resolution of ( N r , N z ) = (1025 , τ : [“low”,“medium”,“high”] resolution have τ =[ τ , τ / , τ /
64] respectively. This leads to an initial hier-archy where the horizon of the black string is covered by4, 5 and 6 additional refined levels for the low to high res-olutions, respectively. Each simulation was stopped whenthe estimated computational resources required for con-tinued evolution was prohibitively high (which naturallyoccurred later in physical time for the lower resolutions);by then the hierarchies were as deep as 17 levels.Fig. 1 shows the integrated AH area A within z ∈ [0 , L ] versus time. At the end of the lowest resolutionrun the total area is A = (1 . ± . A [27], where A is the initial area; interestingly, this almost reachesthe value of 1 . A that an exact 5D black hole of thesame total mass would have. Fig. 2 shows snapshots ofembedding diagrams of the AH, and Fig. 3 shows thecurvature invariants (1) evaluated on the AH at the lasttime step, both from the medium resolution run.The shape of the AH, and that the invariants are tend-ing to the limits associated with pure black strings orblack holes at corresponding locations on the AH, sug-gests it is reasonable to describe the local geometry as -4-2024 R / M t = 0 - 200M R / M t = 200 - 220 M Z/M -202 R / M t= 220 - 226 M FIG. 2: Embedding diagram of the apparent horizon at sev-eral instances in the evolution of the perturbed black string,from the medium resolution run. R is areal radius, and theembedding coordinate Z is defined so that the proper lengthof the horizon in the space-time z direction (for a fixed t, θ, φ )is exactly equal to the Euclidean length of R ( Z ) in the abovefigure. For visual aid copies of the diagrams reflected about R = 0 have also been drawn in. The light (dark) lines denotethe first (last) time from the time-segment depicted in thecorresponding panel. The computational domain is periodicin z with period δz = 20 M ; at the initial (final) time of thesimulation δZ = 20 M ( δZ = 27 . M ). z/M KSR AH /M FIG. 3: Curvature invariants evaluated on the apparent hori-zon at the last time of the simulation depicted in Fig. 2. Theinvariant K evaluates to 1 for an exact black string, and 6 foran exact spherical black hole; similarly for S (1). being similar to a sequence of black holes connected byblack strings. This also strongly suggests that satelliteformation will continue self-similarly, as each string seg-ment resembles a uniform black string that is sufficientlylong to be unstable. Even if at some point in the cas-cade shorter segments were to form, this would not be astable configuration as generically the satellites will havesome non-zero z -velocity, causing adjacent satellites tomerge and effectively lengthening the connecting stringsegments. With this interpretation, we summarize keyfeatures of the AH dynamics in Table I.We estimate when this self-similar cascade will end. Gen. t i /M R s,i /M L s,i /R s,i n s R h,f /M . ± . .
00 10 . . ± . . ± . . ±
1% 105 ±
1% 1 0 . ± ± . ± ≈ > . − . ≈ ≈ . ≈ > interpreted as proceeding through several self-similar generations, where each local string segment tem-porarily reaches a near-steady state before the onset of thenext GL instability. t i is the time when the instability hasgrown to where the nascent spherical region reaches an arealradius 1 . R s,i ,which has an estimated proper length L s,i (the critical L/R is ≈ . n s is the number of satellites that form persegment, that each attain a radius R h,f measured at the endof the simulation. Errors, where appropriate, come from con-vergence tests. After the second generation the number anddistribution of satellites that form depend sensitively on gridparameters, and perhaps the only “convergent” result we havethen is at roughly t = 223 a third generation does develop. Wesurmise the reason for this is the long parent string segmentscould have multiple unstable modes with similar growth rates,and which is first excited is significantly affected by truncationerror. We have only had the resources to run the lowest res-olution simulation for sufficiently long to see the onset of the4th generation, hence the lack of error estimates and presenceof question marks in the corresponding row. -5 -4 -3 -2 -ln(231-t/M) -4-3-2-1012 l n ( R AH ( z = z c ) / M ) z=15.0z=5.0z=6.5z=4.06 FIG. 4: Logarithm of the areal radius vs. logarithm of timefor select points on the apparent horizon from the simulationdepicted in Fig. 2. We have shifted the time axis assuming self-similar behavior; the putative naked singularity forms atasymptotic time t/M ≈ z = 15 , .
06 correspond to the maxima of the areal radii of thefirst and second generation satellites, and one of the thirdgeneration satellites at the time the simulation stopped. Thevalue z = 6 . The time when the first satellite appears is controlled bythe perturbation imparted by the initial data; here thatis T /M ≈ after that sourcedby the initial data is T /M ≈
80. Beyond that, withthe caveats that we have a small number of points andpoor control over errors at late times, each subsequentinstability unfolds on a time-scale X ≈ / t of the end-state is then t ≈ T + P ∞ i =0 T X i = T + T / (1 − X ). For thedata here, t /M ≈ r ∝ ( t − t ), or, d ln r/d ( − ln( t − t )) = −
1, where r is the stream radius(see [22], and [23] for extensions to higher dimensions);to within 10 −
20% this is the average slope we see (e.g.Fig. 4) at string segments of the AH at late times.
Conclusions:
We have studied the dynamics of a per-turbed, unstable 5D black string. The horizon behavessimilarly to the surface of a stream of low viscosity fluidsubject to the Rayleigh-Plateau instability. Multiple gen-erations of spherical satellites, connected by ever thin-ner string segments, form. Curvature invariants on thehorizon suggest this is a self-similar process, where ateach stage the local string/spherical segments resemblethe corresponding exact solutions. Furthermore, the timescale for the formation of the next generation is propor-tional to the local string radius, implying the cascade willterminate in finite asymptotic time. Since local curvaturescalars grow with inverse powers of the string radius, thisend-state will thus be a naked, curvature singularity. Ifquantum gravity resolves these singularities, a series ofspherical black holes will emerge. However, small mo-mentum perturbations in the extra dimension would in-duce the merger of these black holes, thus for a compactextra dimension the end state of the GL instability willbe a single black hole with spherical topology.The kind of singularity reached here via a self-similarprocess is akin to that formed in critical gravitational col-lapse [24]; however, here no fine-tuning is required. Thus,5 (and presumably higher) dimensional Einstein gravityallows solutions that generically violate cosmic censor-ship. Angular momentum will likely not alter this con-clusion, since as argued in [13], and shown in [25], rota-tion does not suppress the unstable modes, and moreoverinduces super-radiant and gyrating instabilities [26].
Acknowledgments:
We thank V. Cardoso, M. Chop-tuik, R. Emparan, D. Garfinkle, K. Lake, S. Gubser, G.Horowitz, D. Marolf, R. Myers, W. Unruh and R. Waldfor stimulating discussions. This work was supported byNSERC (LL), CIFAR (LL), the Alfred P. Sloan Foun-dation (FP), and NSF grant PHY-0745779 (FP). Simu- lations were run on the
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