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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 c 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 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 gµν(p - eA - e .W )(p - eA - e .W ) = 0 µ µ τ µ ν ν τ ν 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.
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