Vitor Cardoso (The University of Mississippi)

16th MidWest Relativity MeetingSt. Louis, 17-18 November 2006

Black Holes and Strings in the Water Tap

VC & Ó. Dias, Phys. Rev. Lett. 96, 181601 (2006) VC & L. Gualtieri, Class. Quantum Grav. 23, 7151 (2006)

Black Holes and Membranes

• Why not? – Liquid drops as model for planets and stars. Bohr and Wheeler’s model of nuclear forces as surface tension.

• Membrane paradigm (Thorne et al ’86; Parikh & Wilczek ‘98) – Event horizon behaves as a stretched membrane, with electrical conductivity and viscosity.

• The first law of BH Mechanics dE=TdA (Smarr ‘73) –

Fluids held by surface tension T.

hgg bs

)(~

rheeh ikzt

•Black string:

•Perturb the BS:

•S-wave perturbations

(Gregory-Laflamme, 1993)

z

2Schwbs dzgg

Gregory-Laflamme instability

(D-1)-dimensional

)(~

rheeh ikzt

c0 k k There are solutions with >0:

(Gregory-Laflamme, 1993)

Gregory-Laflamme instability

(From Kudoh, 2006)

(Plateau, 1849; Rayleigh, 1878)

Rayleigh-Plateau instability

Water in my kitchen faucet Rain (with high speed camera)

22

10 )cos( )( RkzRRzr

)]([ 212

120

220 2 RRRRzV

1

22 2

02

0

21

2

0 RkR

RRzA

0

21

2const

Mass .Conserv

4

R

RR

1 2

20

2

0

21/)(

0 TRkR

RPzA-ATP

yInstabilit 1for decreasesenergy Potential 0 kR

The threshold mode:

Perturbation:

(Plateau, 1849; Rayleigh, 1878)

Rayleigh-Plateau instability

Rayleigh-Plateau threshold mode:

20 DRkc

) large ( 3~0 DDRkc Gregory-Laflamme threshold mode:

Kol & Sorkin, 2004

GL:

RP:

D 5 6 7 8 9 10 50 100

kR0

1.41 1.73 2.00 2.24 2.45 2.66 6.78 9.80

kR0 0.87 1.27 1.58 1.85 2.09 2.30 6.72 9.75

Cardoso & Dias, 2006

Threshold Mode in Higher Dimensions

RP:

GL:

22

10 )cos()cos( )( RmkzRRzr

Non-axisymmetric perturbations are stable:

TmRkR

RP 1

2 22

02

0

21

Non-axisymmetric perturbations are stable:Kudoh, 2006

General Perturbations

RP:

GL:

,3 )(

))(')(( 22 kDkI

kkIkIT

max D

ckD

RPGLRPmaxmaxmax lower wouldredshift included, effectsgravity If 5~

(Rayleigh, 1878)

(Myers)

tR e ~

Instability Timescale

=D/2-2

max D

ckD

RP:

GL:

rF 2centrif ~

Rotation: de-stabilizes(Johns & Narayanan, 2002)

Rotation and Charge Effects

Hint: Take an ultra-rotating black string:

They are unstable (Cardoso and Gualtieri, 2006)

Charge: stabilizes(Chandrasekhar, 1953)

Rotation: ? Charge: stabilizes(Gregory and Laflame, 1994)

RP:

GL:

“Very Nice!”

Extensions

USML2 space-lab flight 1995

Conclusions

• Membranes can mimic horizons

• Black strings are unstable: water dripping from faucet

• Understand gravity and black objects with simple analogies

Thank you!