Black Hole White Hole 1.6 Black Hole

problems of horizon in effective gravity G. Volovik Helsinki U. Technology Landau Institute

v Valencia Feb. 4, 2009

Valencia Feb. 4, 2009 v < c v < c v > c horizon A-phase black hole white hole 1.6 black hole

1.4 ergoregion instability 1.2

1.0 brane

I = 4 A I = 3 A I valve = 2 A (rad/s) 0.8 valve valve

Ω horizon 0.6 T / T white hole 0.4 c 0.5 0.6 0.7 0.8 B-phase problems of horizon in effective gravity

* sources of effective gravity in condensed matter

* black and white hole horizons

* Hawking radiation as quantum tunneling

* analog of Zeldovich-Starobinsky radiation from rotating BH

* vacuum instability in the presence of horizons & ergoregions

* from condensed matter to quantum gravity: quantum vacuum as self-sustained Lorentz invariant medium * possible instability of astronomical black holes sources of effective gravity in condensed matter

* Fermi point gravity

* acoustic gravity

* ripplon 2+1 gravity

* magnon BEC gravity (application of Cornell idea to magnon BEC)

...

* optical space-times moving dielectric, optical solitons, slow light ... * elasticity theory of dislocations and disclinations 4D world crystal Spontaneous phase-coherent precession disordered precession spontaneously organized two-domain precession decays after pumping S−S two-domain precession since Sz is not conserved, z but coherence is preserved with the same total spin Sz

γΗ < ω

γΗ=ω

H γΗ > ω ∇H Ginzburg-Landau energy for magnon BEC

iα(t) phase of precession Ψ =|Ψ| e ≡ condensate phase deficit of S−S α n = |Ψ|2 = z spin projection h along field ≡ magnon number α(t) = ωt + α0

2 |∇Ψ| 2 2 FGL = + (ωL(r) − ω) |Ψ| + FD (|Ψ| ) 2m . local frequency of α = ω magnon Larmor coherent spin-orbit energy mass frequency precession ≡ . ≡ ≡ interaction α = µ external chemical between magnons potential potential spontaneous coherent precession = 2 magnon BEC |∇Ψ| 2 2 FGL = + (U(r) − µ) |Ψ| + F (|Ψ| ) 2m Sonic metric: effective metric in Landau two-fluid hydrodynamics

Doppler shifted phonon spectrum in moving superfluid and BEC review: Barcelo, Liberati & Visser, E = cp + p.vs c speed of sound Analogue Gravity Living Rev. Rel. 8 (2005) 12 vs superfluid velocity move p.vs to the left

E − p.vs= cp take square

2 2 2 µν p =(−E, p) (E − p.vs) − c p = 0 g pµpν = 0 ν 00 0 i i ij 2 ij i j g = −1 g = −vs g = c δ −vs vs effective metric

inverse metric gµν determines effective spacetime in which phonons move along geodesic curves

2 2 2 2 2 µ ν ds = − c dt + (dr − vsdt) ds = gµνdx dx reference frame for phonon is dragged by moving liquid geometry is emergent Superfluid 4He and BEC Universe

acoustic gravity metric theories of gravity general relativity

geometry of effective space time g geometry of space time for quasiparticles (phonons) µν for matter 2 µ ν geodesics for phonons ds = gµνdx dx = 0 geodesics for photons

Landau two-fluid equations Einstein equations of GR . Matter dynamic equations ρ + .(ρvs + P ) = 0 Matter for metric field gµυ 1 (Rµν - gµνR/2) = Tµν . 2 vs + (µ + vs /2) = 0 8πG equations equation equation for superfluid for normal µν T;ν Matter = 0 for matter 1/2 of GR component component message from: cond-mat to: quantum gravity

is gravity fundamental ? it may emerge as classical output of underlying quantum system

as hydrodynamics ?

underlying microscopic superfluid helium quantum system at high energy quantum vacuum

classical 2-fluid emergent classical general hydrodynamics low-energy relativity effective theory

. Matter ρ + .(ρvs + P ) = 0 1 Matter . 2 (Rµν - gµνR/2) - Λgµν = Tµν vs + (µ + vs /2) = 0 8πG Tµν = 0 ν Matter Schwarzschild-Painleve-Gulstrand acoustics Superfluid 4He & BEC Gravity

speed of sound c speed of light phonon spectrum geometry gµνp p = 0 gµν g ds2 = g dxµdxν µ ν µν µν

ds2 = - dt2 c2 + (dr-vdt)2

Acoustic metric for phonons Painleve-Gulstrand metric propagating in radial flow v(r) outside gravitating body of mass M 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ g g 00 g0r after transformation 00 g0r dt = dt - vdr/(c2-v2)

Schwarzschild metric is obtained:

ds2 = - dt2 (c2-v2) + dr2 /(c2-v2) + r2dΩ2 v2(r) = ____2GM g r 00 grr Kinetic energy of superflow = potential of gravitational field Sonic Black Hole Liquids, BEC & superfluids Gravity acoustic horizon black hole horizon Painleve-Gulstrand metric (Unruh, 1981) 2 2 2 2 2 2 2 ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ g 00 g0r Schwarzschild metric ds2 = - dt2 (c2-v2) + dr2 /(c2-v2) + r2dΩ2

2 2 r g 2 2____GM 2 __rh v (r) = c __h 00 g v (r) = = c r rr r r

vs(r) gravity v(r) Information from interior region cannot be vs= c transferred black hole r = rh by light or sound v(rh)=c horizon at g00= 0 superfluid (or v(r ) = c) h vacuum Landau critical velocity = black hole horizon Painleve-Gulstrand metric BEC, superfluid 4He Gravity 2 2 2 2 2 2 2 ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ g 00 g0r acoustic horizon black hole horizon Schwarzschild metric 2 2 r 2 2____GM 2 __rh v (r) = c __h ds2 = - dt2 (c2-v2) + dr2 /(c2-v2) + r2dΩ2 v (r) = = c r r r gravity v(r) vs(r) g00 grr

horizon at g00= 0 vs= c where flow velocity black hole r = rh (velocity of local frame in GR) v(r )=c reaches Landau critical velocity h

superfluid v(rh) = vLandau = c vacuum (Unruh, 1981)

Hawking radiation is phonon/photon creation above Landau critical velocity gravity emerging near Fermi point Atiyah-Bott-Shapiro construction: linear expansion of Hamiltonian near the nodes in terms of Dirac γ-matrices

0 emergent relativity linear expansion near H = e k Γi .(p − p ) Fermi point i k k

emergent emergent fierbein gravity γ− matrices

pz µν g (pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0 py

effective metric: effective effective px emergent gravity SU(2) gauge isotopic spin effective field effective electromagnetic electric charge field e = + 1 or −1 hedgehog in p-space

all ingredients of Standard Model : gravity & gauge fields chiral fermions & gauge fields are collective modes emerge in low-energy corner of vacua with Fermi point together with spin, Dirac Γ−matrices, gravity & physical laws: Lorentz & gauge invariance, equivalence principle, etc crossover from hydrodynamics to Einstein general relativity they represent two different limits of hydrodynamic type equations

equations for gµν depend on hierarchy of ultraviolet cut-off's: Planck energy scale EPlanck vs Lorentz violating scale ELorentz

EPlanck >> ELorentz EPlanck << ELorentz emergent hydrodynamics emergent general relativity

3He-A with Fermi point Universe

ELorentz << EPlanck ELorentz >> EPlanck −3 9 ELorentz ~ 10 EPlanck ELorentz > 10 EPlanck Fermionic horizons: horizon in flowing 3He film & in moving soliton

Quasiparticle energy spectrum in moving texture (Doppler shift) 2 2 2 2 2 2 2 (E-vpz) = cx (px - pF) +cz (z)pz + cy py

ds2 =- dt2 (1-v2/cz2(z)) -2(v/cz2(z))dzdt + cx-2dx2+ cy-2dy2 + cz-2(z)dz2

Soliton: speed of light cz(z) changes sign across the soliton

Hawking temperature TH = (h/2π) (dcz/dz)hor

Jacobson-GV, PRD 58, 064021 (1998) black hole in 3He-A film g < 0 g < 0 g > 0 00 00 00 grr > 0 grr > 0 grr < 0 |v(r)| < c

horizon |v(r)| < c horizon |v(r)| > c

rh rh 3He-A film superfluid 4He film ) covering by 4He film

black Painlevé-Gullstrand form of 2D black hole: postpones development of vacuum instability ds2 = - dt2 (c2-v2(r)) + 2v(r) dr dt + dr2 + r2dφ2

hole ( c = 3 cm/s If 3He-A film is moving to the hole v(r) = b / r black hole horizon |v| < c |v| < c horizon is at rh = b / c

|v| > c

4He film

3He-A film white hole horizon Horizon in moving soliton Jacobson-GV, PRD 58, 064021 (1998)

speed of light cz(z) changes sign across the soliton

cz(z) velocity of soliton v v Hawking radiation leads z |c (z)| > v |cz(z)| < v |cz(z)| > v to deceleration of soliton until horizons shrink z leaving bare singularity

-zh 0 zh white hole black hole g00 < 0 g00 > 0 g00 < 0 physical consequence gzz > 0 horizon gzz < 0 horizon gzz > 0 of Hawking radiation g00=0, gzz=0 -v is the decay of horizon

cz(z) this is what to be measured

effective metric ds2 =- dt2 (1-v2/cz2(z)) -2(v/cz2(z))dzdt + cx-2dx2+ cy-2dy2 + cz-2(z)dz2

horizons at cz(zh) = ± v g00=0 gzz=0 between horizons particles move in one direction

z=0 - curvature singularity : cz(z)=0 Hawking radiation in semiclassical description: quantum tunneling between classical trajectories GV: JETP Lett. 69,705 (1999) Parikh-Wilczek: PRL 85,5042 (2000)

g > 0 pr 00 empty pr (r) = Elab / (c+v(r)) grr < 0 positive energy |v(r)| > c states Elab = c|pr| + prv(r) > 0 Ecomoving = c|pr| >0 trajectories of outgoing particles horizon r r r=0 h tunneling trajectories of ingoing particles

pr (r) =− Elab / (c+v(r)) g00 < 0 grr > 0 occupied Elab = c|pr| + prv(r) > 0 positive energy |v(r)| < c Ecomoving = c|pr| >0 states

pr(r) = Elab / (c+v(r)) tunneling exponent and Hawking temperature

E = − c|p | + p v(r) > 0 lab r r S(E) = 2 Im ∫ dr pr (r) = 2E Im ∫ dr/(c+v (r))= E /TH Ecomoving = − c|pr| < 0 − − W = we S(E) = we E/TH TH = v'/2π problem of Hawking radiation in de Sitter universe GV: 0803.3367 2 2 2 2 2 2 2 2 2 2 2 2 ds = - dt c + (dr-vdt) ds = − dt (c -H r ) + 2 Hr dr dt + dr + r dΩ v (r) = Hr de Sitter expansion * tunneling approach

outside de Sitter horizon r > c/H inside de Sitter horizon r < c/H

pr (r) = E / (−c+v(r)) > 0 pr (r) = E / (−c+v(r)) < 0 tunneling from occupied E = − c|pr| + prv(r) > 0 positive energy states E = c|pr| + prv(r) > 0 Ecomoving = − c|pr| < 0 Ecomoving = c|pr| >0

S(E) = 2 Im ∫ dr pr (r) = 2E Im ∫ dr/(− c+Hr )= 2π E /H − − W = we S(E) = we E/TH TH = H /2π

* Does Hawking radiation really occur in de Sitter universe ? BH has preferred reference frame, de Sitter Universe does not

Ht * cond-mat simulation ds2 = − dt2 + dr2 /c2 c(t)=e− Schutzhold: PRL 95 (2005) 135703; GV: J.Low.Temp.Phys. 113 (1998) 667 activation in de Sitter space-time GV: 0803.3367 2 2 2 2 2 2 2 2 2 2 2 2 ds = - dt c + (dr-vdt) ds = − dt (c -H r ) + 2 Hr dr dt + dr + r dΩ v (r) = Hr de Sitter expansion

* ionization rate of an atom from electron at atomic level −ε0 2 p /2m − pHr = −ε0 0 r

/m 1/2 H −ε0 r0 = (2ε0 ) << c/H electron tunneling well inside de Sitter horizon from bound state

r0 effective activation temperature T = H /π S(−ε0) = 2 Im ∫ dr pr (r) = π ε0 /H 0 is twice the Hawking temperature W = we −S(-ε ) = we −ε /T 0 0 TH = H /2π suggestion for Hawking type radiation in de Sitter space-time GV: 0803.3367 2 2 2 2 2 2 2 2 2 2 2 2 ds = - dt c + (dr-vdt) ds = − dt (c -H r ) + 2 Hr dr dt + dr + r dΩ v (r) = Hr de Sitter expansion

pure de Sitter vacuum does not radiate

but observer views thermal bath with twice the Hawking temperature because he/she violates de Sitter symmetry

effective activation temperature T = H /π

is twice the Hawking temperature TH = H /2π Relativistic ripplons living on brane between two shallow superfluids

3He-A shallow h2 ripplon vA 3He-A brane brane (interface) h1 vB shallow 3He-B 3He-B

µν Spectrum of ripplons g pµpν = 0 Effective metric for ripplons 2 2 2 2 α1(ω-k.vA) +α2(ω-k.vB) =c k 2 µ ν ds = gµνdx dx Speed of light c2-W2-U2 1 ds2 = -dt2 + dr2 +r2 d 2 2 2 2 φ c =(F/ρs)h1h2/(h1+h2) c -U c2-W2-U2

F is restoring force: 2 2 either gravity or gradient of magnetic field U =α1α2(vA-vB) W =α1vA+α2vB black hole for ripplons at AB-brane c2-W2-U2 1 ds2 = -dt2 + dr2 +r2 dφ2 c2-U2 c2-W2-U2

signature +−+ signature −++ grr < 0 g00 > 0 g00 > 0 grr < 0

physical singularity within black hole horizon

g00=∞ g00=0 U2=c2 W2+U2=c2 Kelvin- Landau Helmholtz criterion instability

grr < 0 g00 < 0 signature −−+ horizon as ergosurface c2-W2-U2 1 ds2 = -dt2 + dr2 +r2 dφ2 c2-U2 c2-W2-U2

g00=∞ g00>0 physical ergoregion: ripplon energy <0 singularity within black hole U2=c2 horizon ergosurface: ripplon energy E=0 Kelvin- Helmholtz Landau criterion: instability ripplon energy E=0

E spectrum of ripplons

r horizon ergoregion: U2=c2 ripplon energy E < 0 ripplon energy E = 0 instability in ergosurface Schutzhold-Unruh: PRD 66, 044019 (2002) GV: JETP Lett. 75, 418 (2002)

g00=∞ g00>0 physical ergoregion: ripplon energy E = Re ω <0 singularity within black hole ergoregion: ripplon energy E = Re ω =0 U2=c2 horizon 2 2 2 Kelvin- W +U =c Helmholtz Landau instability criterion g00=0 lesson from AB-brane: Black hole Im ω may collapse due to vacuum instability spectrum of ripplons in ergoregion Re ω Im ω & Re ω cross 0 simultaneously: r behind horizon attenuation transforms W2+U2=c2 singularity horizon U2=c2 to amplification ergoregion: ripplon energy E =Re ω < 0 also Im ω < 0 which means instability Ergoregion instability 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) boundary of the AB interface at side wall A phase of cylindrical ergoregion: container AB interface region where the energy v>vc of some quasiparticles B phase is negative v0 non-relativistic analog of g00=0 experimental Landau criterion of ripplon ergregion instability

experimental set-up

H, Tesla p = 29 bar 0.6 45 NMR pick-up A phase 0.5 supercooled A phase 0.4 (false vacuum)

0.3 0 z (T) (mm) 0 H (T) 0.2 AB z Normal phase 0.1 stable B phase stable A phase

B phase (T) (true vacuum) 0 -45 AB 0 0.5 1 1.5 2 2.5 3 H H T, mK 0 0.2 0.4 Tesla Landau criterion

22 ρsB(vsB -vn) +ρsA(vsA -vn) =2 √ Fσ

critical value vsB(T) depends on T via ρsB(T), F(T), σ(T) experimental Landaucriterionofripplonergregioninstability

Ωc (rad/s) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 lines -theoreticalvalues withoutfitting ρ . . . 0.8 0.7 0.6 0.5 sB T AB ( (H v sB max I

valve - )

v = 4 A = 4 n ) + 22 T AB ρ sA (H I T max (

valve v / sA

) T = 3 A = 3 c - v n )

= 2

√

F σ T

AB (H min )

Proposal for black hole for ripplons at AB-brane (azimuthal flow) c2-W2-U2 r2 dφ2 ds2 = -dt2 + +dr2 c2-U2 c2-W2-U2

black hole horizon U2+W2 =c2 v brane g00=0

g00=∞ physical g00<0 singularities g =∞ U =c 00

g00=0 v

white hole horizon U2+W2 =c2 Relativistic ripplons in shallow water

ripplon

surface v h liquid wall

µν Spectrum of ripplons g pµpν = 0 k << kc = √ ρg/σ (ω-k.v)2 =c2k2 +c2k4(σ/ρg − h2/3) kh << 1

Effective metric for ripplons 2 µ ν ds = gµνdx dx 1 ds2 = -dt2 (1-v2/c2) + dr2 +r2 dφ2 Speed of "light" c2-v2 c2=gh ripplonic white hole circular hydraulic jump

Courtesy Piotr Pieranski 1 Hydraulic jump as white hole ds2 = -dt2 (1-v2/c2) + dr2 +r2 dφ2 c2-v2

v2c1 Observation of instability in the ergoregion in superfluid 4He

E. Rolley et al. AIP Conf. Proc. 850, 141 (2006) physics/0508200 v2c1 horizon

ergoregion

Im ω (kc) Re ω (kc) in the ergoregion standing waves attenuation of ripplon are excited transforms to amplification in the ergoregion with Re ω (kc)=0 ergoregion instability inside the "white hole" in shallow water

standing waves inside the jump

Courtesy Piotr Pieranski Zeldovich-Starobinsky effect, rotational Unruh effect & ergorgion instability

Radiation by object rotating in quantum vacuum & by circulating superflow

Analog to Kerr Black Hole

Hawking radiation Unruh effect: looks as thermal with radiation looks as thermal with TH ~ hg/c TU ~ ha/c where g is gravity where a is acceleration of a body

metric in rotating frame ds2 = - dt2 c2 + (dr-vdt)2 H. Takeuchi, M. Tsubota and GV Zel’dovich-Starobinsky Effect v (r) = Ω x r in Atomic Bose-Einstein Condensates: Analogy to Kerr Black Hole J. Low Temp. Phys. 150, 624 (2008)

2 2 2 2 2 2 2 2 2 2 ds = − dt (c -Ω ρ ) + 2Ωρ dφ dt + ρ dφ + dρ + dz tunneling approach 2 2 2 2 2 2 2 2 2 2 ds = − dt (c -Ω ρ ) + 2Ωρ dφ dt + ρ dφ + dρ + dz

normal region ρ < c/Ω ergoregion ρ > c/Ω

body surface R < c/Ω ergosurface ρ = c/Ω tunneling from zero energy state at the body to zero energy state at the ergosurface

Radiation by object rotating in quantum vacuum surface of optical potential made by rotating body ρ = R blue or red detuned laser Tunneling region moving in static BEC ΩR < Ω ρ < c or vortex cluster rotating in static BEC Ergosurface potential or a vortex moves with Ω ρe = c linear velocity ΩR

body surface R < c/Ω ergosurface ρ = c/Ω tunneling from zero energy state at the body to the zero energy state at the ergosurface

c/Ω 2 2 2 2 2 1/2 S(E=0) = 2 Im ∫ dρ pρ (ρ) = 2 Im ∫ dρ(L /ρ − L Ω /c ) = 2L ln (c/ΩR) R

W = we −S(E=0) = w (ΩR/c)2L= w (ωR/cL)2L

ω = ΩL Zeldovich resonance condition

Calogeracos-GV: JETP Lett. 69 (1999) 281 z Vacuum instability in ergoregion instability of ergoregion in vortex core with N>>1: vs no stable vacuum if vs > c r

c(r) reconstruction of vacuum in the vortex core with N>>1: vs ergoregion disappears, vs < c everywhere, shell of Planck size and Planck energy separates vacua with different gµν -re ergoregion re r c < vs g00 > 0 c(r) c(r)

v s vs=0 vs=0 vs

-rc rc r Vacuum resistance to formation of horizons vs(r) spherical acoustic black or white holes are not solutions of hydrodynamic equations v = c equation along the stream line s of stationary flow: ρv d( v) ____ρ = ρ(1-v2(r)/c2) dv d(____ρv) > 0 d(____ρv) < 0 superfluid dv dv

horizon cannot be achieved v v=c(=s) because continuity equation unstable region behind horizon ρv =Const / r2 requires Hydrodynamic instability is absent d( v) ____ρ > 0 if speed of "light" c < s speed of sound dv everywhere in Fermi superfluids c < < s possible horizons

Painleve-Gulstrand metric

2 2 2 2 2 2 2 ds = - dt (c -v ) + 2 v dr dt + dr + r dΩ

Acoustic horizon in Laval nozzle Horizons for fermionic quasiparticles

black hole black hole white hole vs vs v < c v > c v < c v < c v > c BEC 3He, cold fermi atoms horizon can be exactly in the middle speed of "light" < speed of sound no hydrodynamic instability Quasiequilibrium state |vs(r)| > c |vs(r)| < c behind horizon r Fischer-GV: IJMP D10,57 (2001 horizon rh g00 > 0 g00 < 0 grr < 0 grr > 0 vs(r)

-c

Physical cut-off

2 2 1/2 2 2 1/2 Teff(r) = T∞/(vs /c −1) Teff(r) = T∞/(1−vs /c ) unusual Tolman's law usual Tolman's law

horizon r

* messages from cond-mat to GR:

black hole is radiating, de Sitter is not GV: "On dS radiation via quantum tunneling" 0803.3367 but detector embedded in de Sitter sees radiation

vacuum resists to formation of horizon

vacuum may be unstable behind the horizon

* we need relativistic theory of quantum vacuum black holes in dynamics of quantum vacuum

F.R. Klinkhamer and GV "Self-tuning vacuum variable and cosmological constant", Phys. Rev. D 77, 085015 (2008) "Dynamic vacuum variable and equilibrium approach in cosmology", Phys. Rev. D; arXiv:0806.2805 "f(R) cosmology from q-theory", Pis'ma ZhETF 88, 339 (2008)

* quantum vacuum as self-sustained Lorentz invariant medium

* conserved vacuum charge

* thermodynamics of relativistic quantum vacuum

* dynamics of relativistic quantum vacuum

* application to cosmology: relaxation of cosmological constant from Planck scale to present value

* future application to black hole physics of BH singularity action

4 1/2 S = d x (-g) [ ε (q) + K(q)R ] + Smatter

gravitational coupling K(q) is determined by vacuum and thus depends on vacuum variable q dynamic equations

equation dε/dq + R dK/dq = µ integration for q constant Einstein K(Rg − 2R ) + (ε − µq)g − 2( ∇ ∇ − g ∇λ∇ ) K = T equations µν µν µν µ ν µν λ µν

gravitational Einstein cosmological matter coupling tensor term µν ∇µ T = 0