Vitor Cardoso (The University of Mississippi) 16 th MidWest Relativity Meeting St. 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)
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Vitor Cardoso (The University of Mississippi) 16 th MidWest Relativity Meeting St. Louis, 17-18 November 2006 Black Holes and Strings in the Water Tap.
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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