Black-hole & white-hole horizons for capillary-gravity waves in superfluids

Black-hole & white-hole horizons for capillary-gravity waves in superfluids G. Volovik Helsinki University of Technology & Landau Institute Cosmology COSLAB Particle Particle physics Condensed matter Warwick December 9, 2005 1.6 A-phase 1.4 Ω c (rad/s) 1.2 1.0 0.8 I valve = 4 A I valve = 3 A I valve = 2 A black hole horizon 0.6 T / T c 0.4 0.5 0.6 0.7 0.8 B-phase 01

Effective Lorentzian metric is typical for condensed matter L= (-g) 1/2 g µν µ φ µ φ * Phonons in moving superluids & sound waves in moving liquid * Spin waves in inhomogeneous medium * Light in moving dielectric * Quasiparticles in superconductors with nodes Landau-Khalatnikov two-fluid equations are analogs of Einstein equations for gravity and matter * Elasticity theory with dislocations and disclinations T µν ;µ = 0 Gap (θ) * Quasiparticles in 3 He-A This system is most instructive since gravity appears together with chiral fermions, gauge fields,lorentz invariance gauge invariance, the same speed of light for fermions & bosons 02 * Ripplons on the surface of liquid or interface between liquids best system for simulating event horizon of black hole

Acoustic metric in liquids (Unruh, 1981) & superfluids Doppler shifted spectrum in moving liquid review: Barcelo, Liberati & Visser, Analogue Gravity gr-qc/0505065 E = cp + p. v c speed of sound E p. v= cp (E p. v) 2 + c 2 p 2 = 0 g µυ p µ p ν = 0 p ν =( E, p) g 00 = 1 g 0 i = v i g ij = c 2 δ ij v i v j inverse metric g µυ determines effective spacetime in which phonons move along geodesic curves 03 ds 2 = - dt 2 c 2 + (dr-vdt) 2 reference frame for phonon is dragged by moving liquid ds 2 = g µυ dx µ dx ν

Superfluids Acoustic gravity speed of sound c speed of light Gravity Doppler shifted spectrum in moving liquid E = cp + p. v g µυ g µυ p µ p ν = 0 acoustic metric g µυ geometry ds 2 = - dt 2 c 2 + (dr-vdt) 2 reference frame for phonon is dragged by moving liquid Effective metric for phonons propagating in radial superflow v(r) ds 2 = g µυ dx µ dx ν Painleve-Gulstrand metric after time transformation dt = dt - vdr/(c 2 -v 2 ) Schwarzschild metric: ds 2 = - dt 2 (c 2 -v 2 ) + 2 v dr dt + dr 2 + r 2 dω 2 g 00 g 0r ds 2 = - dt 2 (c 2 -v 2 ) + dr 2 /(c 2 -v 2 ) + r 2 dω 2 g 00 g rr 03a v 2 (r) Kinetic energy of superflow = potential of gravitational field v 2 (r) = 2GM r

Liquids & superfluids Acoustic Black Hole Gravity acoustic horizon (Unruh, 1981) Painleve-Gulstrand metric ds 2 = - dt 2 (c 2 -v 2 ) + 2 v dr dt + dr 2 + r 2 dω 2 black hole horizon v 2 (r) = c 2 r h r g 00 g 0r Schwarzschild metric ds 2 = - dt 2 (c 2 -v 2 ) + dr 2 /(c 2 -v 2 ) + r 2 dω 2 g 00 g rr v 2 (r) = 2GM = c 2 r h r r 03b v s (r) superfluid v s = c Information from interior region cannot be transferred by light or sound horizon at g 00 = 0 (or v(r h ) = c) gravity v(r) vacuum black hole r = r h v(r h )=c

Vacuum resistance to formation of horizon v s (r) Hydrodynamic instability of spherical black hole along the stream line of stationary flow: v s = c d(ρv) = ρ(1-v 2 (r)/c 2 ) dv continuity equation: ρv =Const / r 2 ρv unstable region behind horizon superfluid horizon cannot be achieved because continuity equation requires v=c(=s) v d(ρv) >0 dv Such instability is absent if speed of "light" c < s speed of sound in Fermi superfluids c < < s 04

Analog Black Holes Painleve-Gulstrand metric ds 2 = - dt 2 (c 2 -v 2 ) + 2 v dr dt + dr 2 + r 2 dω 2 Acoustic horizon in Laval nozzle black hole v s v < c v > c BEC Horizons for quasiparticles in moving superfluids black hole white hole v s v < c v > c v < c 3 He, cold fermi atoms Horizons for quasiparticles in soliton c > v white hole horizon c < v black hole horizon c > v x velocity of soliton v 05 c(x)

black hole white hole c(x) v s v s c < v v < c v > c BEC v < c v > c v < c white hole horizon black hole horizon there are many principle difficulties with above schemes Let us try surface waves Schutzhold & Unruh 2002: horizon for gravity waves in shallow water Helsinki experiments 2002: ergoregion instability for surface waves at the interface between superfluids 05a ENS experiments 2005 (Rolley et al. physics/0508200): hydraulic jump in superfluids as white hole (physics/0508215)

* Kelvin-Helmholtz criterion (dynamic instability of interface under shear flow) ρ(v 1 -v 2 ) 2 = 4 Fσ k c = F/σ ρ = 2ρ 2 ρ 1 /(ρ 2 +ρ 1 ) gas v 1 =v G ρ 1 = ρ G A-phase v 1 =v sa ρ 1 =ρ sa liquid v 2 =v L ρ 2 = ρ L B-phase v 2 =v sb ρ 2 =ρ sb F =g(ρ 2 -ρ 1 ) F = (χ B H2 -χ A H 2 ) gravity in liquids magnetic forces in 3 He 06

3 criteria for interface instability at T=0 interface * Kelvin-Helmholtz criterion (dynamic instability of interface under shear flow) ripplon v sa =0 v sb =v A-phase B-phase ρ sb = ρ ρ sa = ρ ρv 2 = 4 Fσ k c = F/σ * Landau criterion (excitation of quasiparticles -- ripplons = capillary-gravity waves) ρv 2 = Fσ v = min k k c = F/σ ω(k) k kf+k 3 σ ω(k)= 2ρ ripplon spectrum in deep water * Thermodynamic (ergoregion) instability criterion (negative free energy = ergoregion at T=0) 07 ρv 2 = 2 Fσ k c = F/σ

* Thermodynamic instability criterion (negative free energy of perturbations) interface v sa A-phase v n =0 v sb B-phase 2 2 ρ sb (v sb -v n ) +ρ sa (v sa -v n ) =2 Fσ instability is caused by winds of superfluid components with respect to the normal component (or with respect to the container wall at T=0). It occurs even if v sb =v sa flapping of sails and flags (Rayleigh) flag pole flag wind velocity with respect to flag pole is the same on both sides of flexible membrane 08

Landau criterion (T=0) * Conventional Landau criterion (applicable for single superfluid liquid only) ω(k,v s )=ω(k)+ k.v s <0 interface v n =0 v sa v sb v = min * Generalized Landau criterion for two superfluid liquids ω(k) k A-phase B-phase ρ sa = ρ ρ sb = ρ ω(k,v sa,v sb ) <0 ripplon spectrum ω(k,v sa,v sb ) = k.(v sa +v sb )/2 + kf+k 3 σ - k 2 ρ(v sa -v sb ) 2 /2 2ρ 2 2 ρ(v sb -v n ) +ρ(v sa -v n ) =2 Fσ 09 Coincides with thermodynamic instability criterion

Landau-Ergoregion instability vs Kelvin-Helmholtz instability (effect of environment) interface v n =0 v sa =0 v sb amplitude a(t) ~ exp( t Im ω (k c ) ) increment of growth of critical ripplon k c = F/σ KH analysis taking into account interaction of interface with environment (normal component) Γ interaction parameter Re ω (k c ) slope ~ interaction with bulk Γ Im ω (k c ) ergoregion Attenuation of ripplon due to environment transforms to amplification v in the ergoregion sb Thermodynamic & Landau ergoregion instability condition 2 ρv sb =2 Fσ 2 KH instability condition at Γ=0 ρv sb =4 Fσ 10 Γ T 3 (Kopnin, 1987)

Ergoregion for ripplons living at the AB-brane in Helsinki experiments interface between static B-phase and A phase circulating with solid-body velocity v=ωr (velocity is shown by arrows) ergoregion: region where the energy of some quasiparticles is negative in rotating frame non-relativistic analog of g 00 >0 v>v c v<v c boundary of the AB interface at side wall of cylindrical container v=v c ergosurface A phase AB interface B phase non-relativistic analog of g 00 =0 11

Relativistic ripplons -- quasiparticles living on brane between two shallow superfluids (future experiments, see "The Universe in a Helium Droplet" Oxford 2003) A-phase v sa =v 2 ripplon brane (interface) v sb =v 1 B-phase v na =0 v nb =0 h 2 h 1 shallow A-phase brane shallow B-phase ripplon Spectrum of ripplons α 1 (ω-k. vsa ) 2 +α 2 (ω-k. vsb ) 2 =c 2 k 2 12 Speed of light c 2 =(F/ρ s )h 1 h 2 /(h 1 +h 2 ) ds 2 = -dt 2 c 2 -W 2 -U 2 + dr c 2 2 -U 2 Effective metric for ripplons 1 c 2 -W 2 -U 2 +r2 dφ 2 U 2 =α 1 α 2 (v 1 -v 2 ) 2 W =α 1 v 1 +α 2 v 2

Artificial black hole for ripplons at AB-brane (radial flow) g 00 = physical singularity within black hole 13 Kelvin- Helmholtz instability U 2 =c 2 W 2 +U 2 =c 2 Landau instability ds 2 = -dt 2 c 2 -W 2 -U 2 + dr c 2 -U 2 2 lesson from AB-brane: horizon g 00 =0 1 c 2 -W 2 -U 2 +r2 dφ 2 Black hole may collapse due to vacuum instability ergoregion g 00 >0 bulk A-vacuum bulk B-vacuum Im ω singularity slope is proportional to friction caused by interaction of brane with 3D environment ergoregion g 00 >0 W 2 +U 2 =c 2 U 2 =c 2 horizon Re ω Im ω & Re ω cross 0 simultaneously: behind horizon attenuation transforms to amplification r

Artificial black hole for ripplons within AB-brane (azimuthal flow) ds 2 = -dt 2 c 2 -W 2 -U 2 +r 2 dφ 2 c 2 -U 2 1 c 2 -W 2 -U 2 +dr 2 W 2 =α 2 1 v 2 U 2 =α 1 α 2 v 2 black hole horizon U 2 +W 2 =c 2 g 00 =0 v brane physical singularities U =c g 00 = g 00 = g 00 =0 g 00 <0 v 14 white hole horizon U 2 +W 2 =c 2

Hawking radiation as tunneling p r r=0 g 00 > 0 g rr < 0 v s (r) > c horizon r h tunneling empty positive energy states p r = E lab / (c+v s (r)) E lab = c p r + p r v s (r) > 0 E comoving = c p r >0 trajectories of outgoing particles r trajectories of ingoing particles p r = E lab / (c+v s (r)) E lab = -c p r + p r v s (r) > 0 E comoving = -c p r < 0 occupied positive energy states g 00 < 0 g rr > 0 v s (r) < c p r =- E lab / (c+v s (r)) E lab = c p r + p r v s (r) > 0 E comoving = c p r >0 15 W = we -S= we -E/T H S = 2 Im dr p r (E lab ) = E lab / T H tunneling exponent and Hawking temperature T H = v'/2π

Hydraulic jump 1. Relativistic ripplons in shallow water ripplon surface liquid wall v h Spectrum of ripplons (ω-k. v) 2 =c 2 k 2 +c 2 k 2 (k 2 /k c 2 k 2 h 2 /3) Effective metric for ripplons ds 2 = -dt 2 (1-v 2 /c 2 ) + dr 2 1 c 2 -v 2 +r2 dφ 2 k << k c = F/σ kh << 1 Speed of "light" c 2 =(F/ρ s )h 16

circular hydraulic jump 17a Courtesy Piotr Pieranski

2. Hydraulic jump as white hole h 1 ergoregion v 1 >c 1 white hole horizon v 2 <c 2 v h 2 ds 2 = -dt 2 (1-v 2 /c 2 ) + dr 2 1 c 2 -v 2 +r2 dφ 2 17b

3. Observation of instability in the ergoregion E. Rolley et al. physics/0508200 standing waves are excited in the ergoregion h 1 Re ω (k c )=0 v 1 >c 1 white hole horizon v 2 <c 2 v h 2 ergoregion Re ω (k c ) Im ω (k c ) in the ergoregion attenuation of ripplon transforms to amplification 18 ds 2 = dt 2 (1 v 2 /c 2 ) + dr 2 1 c 2 v 2 +r2 dφ 2

* Lesson for gravity: Conclusion * Ripplons on the surface of liquid or interface between liquids: best system for simulating event horizon of black & white holes * Thermodynamic instability of interface: analog of vacuum instability in the ergoregion * Kelvin-Helmholtz instability of interface: analog of black-hole singularity vacuum instability in the ergoregion may be the main mechanism of decay of black hole * Hydraulic jump in superfluids: first realization of instability of relativistic ergoregion 19