Towards understanding the final fate of the black string

Luis LehnerUBC-CITA-PIMS

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

Black holes propertiesEvent horizon features

1. Hide singularities.2. Do not bifurcate (at least classically!)3. 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 wavesNot unique? grav. waves in Schwarzschild backgroundCan this happen?

• 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

Event Horizons can’t bifurcate

• Black Hole Event Horizon:– Region not accessible by ‘far’ observers

– Ruled by null geodesics

Violates pple of equivalence! (unlessNaked singularity!)

Not too many options!,

1. No hair theorems. Late soln parametrized by just (M,J,Q)

2. No ‘naked’ singularities. “Cosmic censorship conjecture”

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

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?

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

g g hab ab

B

ab

h A r e t i w ( )

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!

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)

I+

I-

EH

CH

ds fd drd d dz

e dz e d

2 2

2 2 2

2 2

Raychaudhuri /

; / ( )

2 2

2 2

2

3

2 2 3

6

Now

C

...

. .

( ) ( )

1 1 0 2 0

12 6

2

If expansion non-negative infinite time to bifurcate!

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:

2 4

G

L tt z, ,

2 0GM r L/ ;

L e L

M R e R

t i zo

R

2 2/ sinh( )

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

The way out?

• Treatment of the problem in fullNumerical simulation required

2 efforts: Vacuum and EKG system F. Pretorius, I. Olabarrieta, LL

H. Villegas, R. Petryk,

M. Choptuik, W. Unruh

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

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?

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

g RHS Hrr ADM

ds g dt g drdt

g dr g drdz g dz g drr

r rrr

rr rz zz

2 2 2

2 2 2

2

2

( )

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’CAVEAT: Boundaries ‘too close’

ara f r z ( , )

g g Gaussian r zrr rr ( , )

Stable case

Stable Case.(but not from linearized studies)

‘Unstable’ case

‘double mode’

Is it really unstable?• Checked convergence of R2 to a divergent

behavior– Apparent 1/(t+c)b behavior with b~0.5

R2

amax/amin

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

A different solution?

L < 15M 15M<L<~19Mamax/amin-1

amax/amin

L=15M

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?)

Warning for current use in num rel! -- Be careful with invariants….

Note: this only makes sense in the stationary regime!

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

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

Preliminary results…• Instabilities apparently present

• Not yet found the stationary solution!– ID dependent?

Case 1: L=12M

Case 2 : L=17M

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.

• 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)