Óscar Dias

Based on:

Vitor Cardoso, OD, hep-th/0602017,

OD, Troels Harmark, Rob Myers, Niels Obers (2007).

University of Barcelona

Multi-black holes from the

analogue fluid model

Einstein’s Gravity in Higher Dimensions,Hebrew University, Jerusalem, February 2007

Troels’ talk

Fluid analogue of the Gregory-Laflamme instability

Outline:

Motivation: The liquid drop model for nuclear fission

Multi-black holes in the fluid analogue

Rotating fluid drops and ultra-spinning black holes

A farewell tale

The Liquid Drop model for Nuclei.

Meitner, Frisch (Nature 1939): Coin the term Nuclear Fission

Bohr, Wheeler (Phys Rev 1939): The mechanism for Nuclear Fission

“”

Electric repulsive Forces Electric repulsive Forces

Excitation modes of nucleus Oscillation modes of liquid drop

The basic Map of the Analogy:

++++

Surface Tension of nuclear matter Surface Tension of fluid

The Liquid Drop model for Nuclei.

Perturbation:

Bohr, Wheeler (1939)

Volume: Surface Tension O , Uniformly charged with Ze

Energy perturbed drop:

Quadratic term in α 2:

Critical value :

Incompressible fluid with:

A first estimation :

Membrane Paradigm (Thorne, MacDonald, Price...)

Gregory-Laflammeinstability

Rayleigh-Plateau instability

Surface gravity

Surface tension

Vitor Cardoso, OD, hep-th/0602017See also, Cardoso, Gualtieri, hep-th/0610004

For given E, the black object prefers the configuration with more S.

For a given V, the fluid picks the configuration with less surface area.

dM = T dA

λ

z

R0 r(z)

= 2π /k

22

10 )cos( )( RkzRRzr εε ++=

)]([ 212

120

220 2 RRRRzV ++= επ

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−+= 1

22 2

02

0

21

20 Rk

RRRzA επ

0

21

2const

Mass .Conserv4

R

RR −=⎯⎯⎯⎯⎯⎯ →⎯=ρ

( ) 1 2

20

2

0

21/)(

0 TRkRRPzA-ATP −=⎯⎯⎯⎯⎯ →⎯ = π

yInstabilit 1for decreases energy Potential 0 ⇒<kR

The threshold mode:

Threshold mode.

Perturbation:

GL:

∫ ⎟⎠⎞

⎜⎝⎛+=

21

dzdrrdrdA θ

ρ = constPlateau, 1849

Gregory-Laflamme, 1993

RP:

λ

z

R0 r(z)

= 2π /k

22

10 )cos()cos( )( RmkzRRzr εφε ++=

Non-axisymmetric perturbations are stable

( ) TmRkRRP 1

2 22

02

0

21

+−=⎯→⎯π

Stability of non-axisymmetric modes

Perturbation:

RP:

GL: Non-axisymmetric perturbations are stable:

Kudoh, hep-th/0602001 Hovdebo, Myers, hep-th/0601079

Plateau, 1849

Rayleigh-Plateau threshold mode:

)dimensions spatial ( 20 DDRkc −=

) large ( ~0 DDRkc

Gregory-Laflamme threshold mode:

Kol, Sorkin, gr-qc/0407058

GL:

Dspatial dim.

4 5 6 7 8 9 49 99

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 9.75GL:

Threshold mode for higher dimensional spaces

RP:

RP:

max ↑Ω⇒↑D max ↑⇒↑ kD

Instability timescale

- Euler eq.- Continuity eq.- Const. density

Rayleigh (1878), D=3

For D spatial dimensions

As D increases:Ω

k

RP:

( T, ρ of D-dim Scharzschild BH )

Gregory-Laflamme, hep-th/9404071

RPGLRPmaxlower wouldredshift included, effects gravity & viscosity If max 10~max ΩΩΩ ⎯→⎯

Instability timescale

GL:

( Myers )

Ω

k

Ω

k

RP:

Critical Dimension

: 2

0 2RP length RP theat breakingcylinder aConsider

−=

D

Rπλ

)('' '

000

sphcyl

0 )(2sph

10 )(1cyl

RP

RRRRCV

RLCV

L

VV

DD

DD=⎯⎯⎯⎯⎯ →⎯

⎪⎭

⎪⎬⎫

=

=

≡

=−

λ

V fixedfor , 11for 1 10for 1

cyl

sph

⎩⎨⎧

≥>≤<

→DD

AA

⎪⎩

⎪⎨⎧

→>

→>

→ unstable lyentropical string Black

unstable GL string Black bhbs

min

GLmin

LL

LL

13 12,~for bh bsmin

GLmin

DLL →=

Sorkin, hep-th/0402216GL:

Hovdebo, Myers, hep-th/0601079

:13 :12

bh bs

min

GL

min

bh bs

min

GL

min

⎩⎨⎧

<≥>≤

→

→

LLDLLD

RP:

Kudoh, Miyamoto, hep-th/0506019

Photograph of a fluid jet.

Evolution of the Instability

Photographs of a Liquid Bridge.

From detailed review:Eggers, Rev. Mod. Phys. 69 (1997) 865.

RP:

Evolution of the Instability

Choptuik, Lehner, Olabarrieta, Petryk, Pretorius, Villegas, gr-qc/0304085

GL:

Tim

e e

volu

tion

RP:

When the classical theory doesn’t apply...

GL:When Pinch-off region reaches the Planck scale:

General Relativity

Quantum Gravity

RP:

When Radius liquid bridge ~ molecules size:

(continuum) hydrodynamic theory

Molecular dynamics

There is always an anoying but ...

Two different systems with different dynamics / physicswhich however exhibit similar features.

In the fluid we don’t have a non-local forcethat plays the role of

gravitational force between large spaced regions …

So should not expect analogy at the level of dynamics

Details of the fluid Instability: Linear theory

Linear analysis of inviscid liquid bridge (Rayleigh) :

Initial single sinusoidal perturbation keeps its single wavelength,

Minimum radius at midpoint decreases monotonically in time

Complete pinch-off occurs at midpoint

RP:

Details of the fluid Instability: addition of rotation

rF 2centrif ~ ω

In the fluid, the addition rotation increases the RP instability:

GL:

( Johns, Narayanan, 2002 )

Competition: Inertia + Surface Tension + Rotation

Kleihaus, Kunz, Radu, hep-th/0702053

Rotating black strings are unstable:

RP:

Successfulprediction

Details of the fluid Instability: addition of viscosity

Effect of viscosity on a liquid bridge :

Addition of viscosity required both for real fluids andby the membrane paradigm in the gravity side.

the most unstable mode becomes longer

strength of the instability becomes weaker :

breakup time increases

Viscous linear theory�

Inviscid linear theory�

Ω

k�

RP:

Eggers, 1997;Shi, Brener, Nagel,

Science 2004

GL:

Pinch-off in FINITE time

Horowitz, Maeda, hep-th/0105111 :

Pinch-off in affine INFINITE time?

R0=10 μm

L=2 cm

τ = 230 μs

Pinch-off in “Infinite” time ?

R=1.5 mm

RP:

Δt ~ 4000 μs >> τ

Pinch-off in “Infinite” time ...

Details of the fluid Instability: non-linear theory

Non-linear analysis on a liquid bridge :

What about non-linear effects ?

threshold wavenumber increases (so perturbations that are linearly stable may became unstable),

non-linear interaction between modes generates increasingly higher harmonics (starting w/ single sin. perturb)

RP:

Shookooh, Elrod (1987)

Experimental Evolution of the RP Instability. Multi-black holes

t =14.4 s

t =16.9 s

t =10.6 s

R ~ 0.4 mmL ~ 5 mm λ ~ L

Conical phase.Topology changing transition

( Kol )

( Choptuik et al )

( OD, Harmark, Myers, Obers )

Multi-black holes

(now available!)Tjahjadi, Stone, Ottino (1992)

( OD, Harmark, Myers, Obers )

Successfulprediction

Static (unstable) endpoint analogues

Both solutions are in “unstable” equilibrium

t

* Breakup point(Linear theory)

α : 1st pinch-off( experiment )

β : Final pinch-off( experiment )

Detail of the fluid evolution

linear

Details of the fluid Instability:

Non-universal behaviour:

Kol

Details of the fluid Instability:

“Double cone” structure at the pinch-off region:

Self-similar processes after 1st breakup:

Kol, Wiseman,Asmin, Smolkin

?

The story so far: known phases of static solutions

Asnin,Choptuik, Chu, Fox,Frolov,Gorbonos, Goldberger, Gubser,

Harmark, Horowitz, Karasik, Kastor,Kil,Kleihaus, Hovdebo,

Kol, Kudoh, Kunz, Lehner, Maeda,Miyamoto, Myers, Niarchos, Obers,

Olabarrieta, Oren, Piran, Petryk, Pretorius, Radu, Rothstein, Sahabandu,

Smolkin, Sorkin,Suranyi,Traschen, Villegas, Wijewarhana, Wiseman

...

Expect new static lumpy NUS

What’s next ?

?Merger line or merger point(s)

Dynamical evolution static solutions=

( ... )If lumpy NUS where do they join US ?

?Expect merger singular line

?Expect static bumpy BHs

Chandraseckar (1965), Brown,Scriven (1980):a liquid drop develops instabilities before zero Gaussian curvature is reached

occurs 0Curv. Gaussian polesat flattens Surface =→⇒↑a

Rotating Liquid Drop Instabilities

Rings

Emparan, Myers (2003):Ultra-spinning BHs become “pancaked”

Rotating liquid drops Rotating black holes

Smarr (1973): Instabilities appear in Kerr BH when J /M is high ?No ! Teukolsky: Kerr BHs are stable.

But Myers-Perry BHs have no Kerr bound on rotation for D >5...

Expect ultra-spinning BHs to be unstable?

Black branes

Gregory-Laflamme Instability

Liquid Drop Instabilities and Ultra-spinning instability of Myers-Perry BH

Smarr (1973)Cardoso, OD, hep-th/0602017See detailed study: Cardoso, Gualtieri, hep-th/0610004

...well, also no black btrings, no black rings...

Colliders, Black Saturn, Black Rings, Black Strings, Instability, Multi-black holes

Fluid:

Gueyffier and Zaleski, 1997

Frontal collision of two drops (Proton beams).

First a toroidal structure is formed (Black Saturn),

which then collapses and forms a cylinder (Black String).

This cylinder (Black String)breaks up as required by the RP instability (GL instability).

We endup with an array of drops and satellite drops (Multi-BHs)

A farewell tale:

In the context of the Membrane Paradigm:

The key known features of GL instability reproduced by the RP instability

Gregory-Laflamme Rayleigh-Plateau

The open questions of GL have been addressed in the RP side and have a definite answer in this side of the correspondence

Is RP giving clues for the GL unknown properties?

Conclusion

Multi-black holes Array of drops

Thanks !

Moral:“Black strings pinch-off ‘just as’

water from a faucet breaks up into small droplets”

?Expect new static lumpy NUS

What’s next ?

?Merger line or merger point(s)

Dynamical evolution static solutions=

?

( ... )

If lumpy NUS where do they join US

?Expect merger singular

?

line

Expect static bumpy BHs

?