Towards understanding the final fate of the black string

1. Towards understanding the final fate of the black string Luis Lehner UBC-CITA-PIMS

2. Outline • Black holes/black strings features • Unique? • Gregory-Laflamme results (93) • Consequences • Present ‘knowledge’ • Horowitz-Maeda; Unruh-Wald results • NR to the ‘rescue’? • Particulars of the implementation • Results and preliminary results

3. Black holes propertiesEvent horizon features • Hide singularities. • Do not bifurcate (at least classically!) • In 4D unique spherical BH: Birkhof’s theom. “Only one static spherically symmetric solution to vacuum Einstein eqns” [Schwarzschild solution] Why? Degree of freedom in Einstein eqns: Gravity waves Not unique? grav. waves in Schwarzschild background Can this happen?

4. Grav. Wave? Transverse, Traceless • Polarization orthogonal to propagation • BUT: spherical symmetry (3D)  propagation in radial direction  polarization must be the same in all directions  not traceless! • Hence: No spherically sym. grav. waves in 3D

5. Violates pple of equivalence! (unless Naked singularity!) Event Horizons can’t bifurcate • Black Hole  Event Horizon: • Region not accessible by ‘far’ observers • Ruled by null geodesics • Not too many options!, • No hair theorems. Late soln parametrized by just (M,J,Q) • No ‘naked’ singularities. “Cosmic censorship conjecture”

6. Black String problem • Little intuition (in fact conflicting conjectures!) • problem only ‘guessed’ from ‘simple’ calculations • NR to tell us what happens! and use it to test ideas for 3D Num.Rel

7. Black strings… • BH Solutions in higher dimensions • Natural objects in string theories • In higher dimensions… electroweak interactions ~ gravity interactions • Some conjecture can be created at LHC! • Simplest : 4D spherical hole x S or 5D hypers. BH. • Solve Einstein eqns in 5D • Features? • Singularity inside (OK) • Bifurcation (OK) • Unique in spherical symmetry (NO!) • Why? Gravitational waves can exist! (r-z plane!) What are the possible solutions?, are those stable?

8. Stability • R. Gregory and R. Laflamme : NO! (94) • Linearized treatment around black string ds2=ds2Schw+dw2 • Found unstable modes allowed (contrary to the 4D case!) • Postulate • Insert into Einstein eqns with the anzats

9. Results from linear pert… • Instabilities for ‘long’ strings (L>L1 [=15M]) • Also, noted that: SBS<SBH(for a given total mass) • Conjecture: Instability leads to a series of BH’s….. which then form a single one… • Density of states from Ads/CFT correspondence • Discussions of BH on brane worlds. • Unstable brane configurations • BUT: this is just from linear analysis and entropy considerations!

10. Conjecture #1: ‘nothing happens’ • Horowitz-Maeda 2001. Just a ‘curvy’ solution • Assume horizon is smooth, • and that expansion must • be nonnegative infinite • time to shrink to zero • No naked singularity. • Observation: assumption rather strong! (totally deterministic)

11. CH I+ EH I- If expansion non-negative  infinite time to bifurcate!

12. Conjecture #2 (Unruh-Wald) • Newtonian Gravity • Set of spheres, radius R • No pressure in extra dimension (z) • Spheres can move in z-direction, not in r • Eqns: • Background: • Solns:

13. Conjecture #2: ‘all hell breaks loose’ • in preparation…: Just ‘Jean’s instability’. • Jean’s instability: ‘pressure and gravity fight’; (the cause we are here!) • Purely newtonian analogue (similar exponents found) whole spacetime collapses. • No naked singularity (Unruh-Wald; Geddes 01) • Observation: Newtonian analogue might not be good enough

14. The way out? • Treatment of the problem in full • Numerical simulation required 2 efforts: Vacuum and EKG system F. Pretorius, I. Olabarrieta, LL H. Villegas, R. Petryk, M. Choptuik, W. Unruh

15. Why do it ... • The obvious… • Ideal test bed for Num Rel. • 2D; not an axis problem (and not ‘trivial 1D’) • Dynamical horizon • Similar problems faced in full 3D • Need long evolutions • Careful boundary treatment • Initial data issues • Understanding physics from the solution

16. Full GR study • Numerical Implementation of the problem • Watch out for very different scenarios: • Initial data: perturbation of BS static soln • Test different ‘string lengths’ • Special care needed: • Distinguish num. instability from real ones: • Collapse can be really difficult to handle! • Inappropriate boundary conds.can drive things • Sufficient accuracy to distinguish other options?

17. Set up – 1st phase • Coordinate conditions: • Read off the BS solution in KS coordinates • Evolution equations? • Different options considered!, • Hyperbolic formulations tricky to use in the unstable case • Settled for ‘adjusted’ ADM formulation (Kelly et.al.,LL 01) • Variables re-normalized to have zero truncation error if dealing with the black string solution • Outer boundary: Dirichlet to BS or ‘Sommerfeld’-type

18. Set-up 1st phase (cont) • Initial Data • Option 1 • Linear ‘gauge’ induced initial data. • Set • Let ‘truncation’ error be the source • Option 2 • Perturbation by hand • Set • Adjust the source as ‘strong’ as wanted • Singularity treatment. • Excised • Apparent horizon located (through flow method) • Finite difference 2nd order code. CAVEAT: Boundaries ‘too close’

19. Stable case

20. Stable Case.(but not from linearized studies)

21. ‘Unstable’ case

22. ‘double mode’

23. Is it really unstable? • Checked convergence of R2 to a divergent behavior • Apparent 1/(t+c)b behavior with b~0.5 amax/amin R2

24. Other diagnostics • Followed null geodesics from the outer boundary to ‘pretty’ close to inner boundary • Which can’t be done in ‘stable’ cases • Verified behavior for different ID and boundary conditions

25. A different solution? amax/amin amax/amin-1 L < 15M 15M<L<~19M L=15M

26. New stable solutions? • Stationarity: d(V)/dt ~ 10(-13) (down from ~ 10(-1)) • Are they different? Gauge can fool us! • - Evaluate invariant quantities in and invariant way to decide • - if BS; S:= f(Riem2/[Riem Riem Riem]) identically 1 • (caveat: how much not 1 is not 1?)

27. Warning for current use in num rel! -- Be careful with invariants…. Note: this only makes sense in the stationary regime!

28. Results for ‘close’ boundaries • Instability range more restricted than pert.theory results. (L >~ L2= 19M instead of 15M) • Final fate? • Instability apparently present, no slowing down observed so far for L>L2 • No apparent collapse of 5th dimension • For L>L2; naked singularity at throat (!!!) • ‘new’ stationary soln for L1<L<L2

29. Towards the ‘real’ final fate • Boundaries ‘uncomfortably’ close for firm claims… • Answer: send the boundaries to…. io • Compactify slices. x(1-1/r) • Note: This is ‘almost heretic’! • Popular belief: ‘lack of resolution will kill the run’ • True when waves are significant/important, otherwise ‘filter’ them out! • Eg. This case and Garfinkle’s singularity studies

30. Preliminary results… • Instabilities apparently present • Not yet found the stationary solution! • ID dependent?

31. Case 1: L=12M

32. Case 2 : L=17M

33. Where are we?… • For ‘close’ boundaries, new solution found and bifurcation for L>LGL . • For the full problem, still working on… • Bifurcation?… so far apparently there • WHEN DOES IT HAPPEN?! • New solution? … need to study different ID • BUT, if it’s there… one ‘could’ find it as an initial data problem (see Gubser 01)! • Note: stationarity  only 4 variables left; 3 equations to solve! • Need some ‘feeling’ for correct boundary conditions at inner boundary. ‘Regularity’ might not be enough.

34. Should have the results in the short term! Keep tuned… What next? • Behavior at the bifurcation (critical?) • Studies of more generic black branes (see Gubser hep-th/0110193, section 4)