Emergent Horizons in the Laboratory

Transcription

1 Emergent Horizons in the Laboratory Ralf Schützhold Fachbereich Physik Universität Duisburg-Essen Emergent Horizons in the Laboratory p.1/26

2 Event Horizon Collapsing matter Singularity Light cones, light rays Event horizon Distortion of quantum fluctuations Hawking effect T Hawking = 1 c 3 8πM G N k B gedanken experiments Black hole entropy... 0> t r T Hawking Emergent Horizons in the Laboratory p.2/26

3 Perpetuum mobile (2 nd kind)? Emergent Horizons in the Laboratory p.3/26

4 Resolution: Quantum Effects W. G. Unruh, R. M. Wald, Phys. Rev. D 25, 942 (1982) Emergent Horizons in the Laboratory p.4/26

5 Transplanckian Problem Collapsing matter Singularity Light cones, light rays Event horizon Distortion of quantum fluctuations Hawking effect T Hawking = 1 c 3 8πM G N k B Problem: red-shift Planck scale E Pl = c 5 /G N J 0> t r T Hawking Emergent Horizons in the Laboratory p.5/26

6 Bill Unruh s Idea Sound waves in irrotational flow δv = φ ( ) ( ) t + v 0 0 c 2 s t + v 0 φ = ( 0 φ) Scalar field φ in curved space-time 1 ( eff φ = µ geff g µν eff νφ ) = 0 geff Painlevé-Gullstrand-Lemaître metric g µν eff = 1 ( 1 v0 0c s v 0 v 0 v 0 c 2 s1 Phonons (quantized) Quantum fields Fluid flow (classical) Gravitational field Euler equation Einstein equations W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981) Emergent Horizons in the Laboratory p.6/26 )

7 Generalizations General linearized low-energy effective action for scalar Goldstone-mode quasi-particles (e.g., phonons) L eff = 1 2 ( µφ)( ν φ)g µν (x) + O(φ 3 ) + O( 3 ) Analogy to quantum fields in curved space-times G µν g µν eff geff Universal properties e.g., phonons, ripplons, magnons... Similarly for (non-scalar) photons in certain media g µν eff = gµν M + (ε 1)uµ u ν Gordon metric [Barceló, Corley, Fischer, Jacobson, Liberati, Unruh, Visser, and many others] R. S., Class. Quantum Grav. 25, (2008) Emergent Horizons in the Laboratory p.7/26

8 De Laval Nozzle v>c v<c v=c Fluid flow, Wall, Sound waves, Event horizon T Hawking = 2π k B r (v 0 c s ) = O(nK...K) Toy model for underlying theory (including quantum gravity) Experimentally measurable?!? 0> Emergent Horizons in the Laboratory p.8/26 t r T Hawking

9 Impact of Dispersion Relation v>c v<c v=c Small k Euler equation linear dispersion ω = c s k Large k deviations: - sub-sonic - super-sonic W. G. Unruh and R. S., Phys. Rev. D 71, (2005) [Corley, Jacobson, Parentani etc.] ω OK (z.b.: Bose Einstein Kondensate) ω=kc S?? OK k Emergent Horizons in the Laboratory p.9/26

10 Supersonic Dispersion Initial wave-packet in its ground state is ripped apart into: Hawking radiation plus infalling partner Entanglement (squeezed state) thermality Caution: Bekenstein entropy does not apply! v>c v=c v<c Emergent Horizons in the Laboratory p.10/26

11 Eddington-Finkelstein Metric Metric ds 2 = (1 2M/r)dV 2 2dV dr [ (2 V r + r 1 2M ] ) r + f( 2 r) r Φ = 0, Dispersion relation f( r) 2 Hawking temperature T Hawking (ω) = v group(ω)v phase (ω) 8πM Ultra-violet catastrophe? ω 2 = c 2 k 2 (1 + l 2 Pk 2 ck ) vs ω = 1 l 2 P k 2 Cf. special relativity E = mc 2 / 1 v 2 /c 2 R. S. and W. G. Unruh, Phys. Rev. D 78, (R) (2008) Emergent Horizons in the Laboratory p.11/26

12 Miles Instability Lab system: Ω (observer at rest at r ) Local fluid system: ω (freely falling observer) (Ω + vk) 2 +iγω = ω 2 (k) Miles instability W. G. Unruh and R. S., Phys. Rev. D 71, (2005) [Corley, Jacobson, Parentani etc.] v>c v<c v=c Emergent Horizons in the Laboratory p.12/26

13 Lessons for Quantum Gravity? Hawking radiation is quite robust (i.e., independent of the microscopic structure) for a large class of systems and does not require the Einstein equations. [Unruh, Jacobson, Corley, RS, Parentani etc.] But there are also (physical) examples, which show deviations from the Hawking effect e.g., depending on whether the space-time foam is freely falling or prefers the static frame... v>c v<c ω OK (z.b.: Bose Einstein Kondensate) ω=kc S?? OK v=c k Impact of (non-linear) interactions not understood... Emergent Horizons in the Laboratory p.13/26

14 Kinematics Dynamics Kinematics of phonons: quantitative analogy g µν eff Dynamics of background: only qualitative analogy E.g., Bose-Einstein condensates Euler equation (classical) Einstein equations Phonons (quantum) Gravitons(?) Healing length Planck length? Many-body Hamiltonian Quantum gravity? Zero-point pressure Cosmological constant? R. S., M. Uhlmann, Y. Xu and U. R. Fischer, Phys. Rev. D 72, (2005) R. Balbinot et al, Phys. Rev. Lett. 94, (2005) Emergent Horizons in the Laboratory p.14/26

15 Lessons for Quantum Gravity? we can quantize (linearized) phonons for small k beyond linear order: UV divergences (non-ren.) sum of zero-point fluctuations of phonon modes (extrapolating Euler equation to large k) up to cut-off does not yield correct result [R. S., Proceedings of Science (QG-Ph) 036 (2007)] quantization of vorticity from Euler equation?? Bose-Einstein condensates ξ m Γ superfluid Helium ξ roton m Γ several cut-off scales ( Planck scale?) breakdown of Euler ω 2 = c 2 k 2 (1 ± k 2 ξ 2 ) circulation quantum Γ of one vortex UV cut-off ξ Emergent Horizons in the Laboratory p.15/26

16 Summary Analogy: gravity condensed matter Robustness of Hawking radiation exceptions: friction, UV-catastrophe... Black-hole information paradox? Experiments: Hawking radiation in the lab? Kinematics dynamics: quantum fluids quantum back reaction, quantization of vorticity Outlook: cosmic horizons expanding fluids, phase transitions inflation as a quantum phase transition? Emergent Horizons in the Laboratory p.16/26

17 Apparent Horizon Expanding condensate Sound waves Apparent horizon at v Fluid = c Sound Analogous to expanding universe v=c Phonon modes: + oscillation (initially) + horizon crossing freezing φ k (t) v<c v>c φ frozen k t t Emergent Horizons in the Laboratory p.17/26

18 Quantum Fluctuations Oscillation horizon crossing freezing φ k (t) t φ frozen k t Amplification of quantum fluctuations (squeezing) (in complete analogy to early universe, e.g., WMAP) Quantum limit (accuracy) of time-of-flight imaging δˆ (r)δˆ (r ) = O(1%) M. Uhlmann, Y. Xu, and R. S., New J. Phys. 7, 248 (2005) Emergent Horizons in the Laboratory p.18/26

19 Dynamical Phase Transition Temperature T = 0 External parameter g = g(t) g c Two competing ground states Ψ < and Ψ > Quasi-particle excitations χ and Φ (unstable) Actual quantum state Ψ(t) E Φ Ψ < Ψ > Sweep through (1 st -order) transition (E n E 0 ) 0 response time diverges at critical point g c non-equilibrium dynamics g(t) g c χ Φ Ψ(t) Emergent Horizons in the Laboratory p.19/26 g(t)

20 Particle Horizon E Φ χ Φ Ψ(t) c(t) Ψ < Ψ > g c g(t) Effective Hamiltonian H = 1 2 [απ2 + β( Φ) 2 ] Speed of sound c 2 s = αβ 0 for g g c Homogeneous medium Particle trajectory Sound cones/waves Particle horizon analogy to expanding universe Emergent Horizons in the Laboratory p.20/26 t

21 Fluctuations Phonon mode + oscillation (initially) + horizon crossing + freezing φ k (t) φ frozen k t c(t) t universal behavior R. S., Phys. Rev. Lett. 95, (2005) Emergent Horizons in the Laboratory p.21/26

22 Bose-Einstein condensate L eff = 1 2 ( ) 1 g(t) Φ 2 0 m ( Φ)2 Phase fluctuations Φ, density fluctuations δ E Φ Ψ < Ψ > Mass of atoms/molecules m, background density 0 Coupling g(t) (e.g., Feshbach resonance) Critical point at g c = 0 (repulsive attractive) Linear sweep g(t) t frozen spectra: σ(φ) = k 4/3, σ(δ ) = k 4/3 ω 3 ω 2 ω 1 g c Ω 1 Ω 2 χ Φ Ψ(t) g(t) δ R. S., Phys. Rev. Lett. 95, (2005) Emergent Horizons in the Laboratory p.22/26

23 Similarities to Cosmic Inflation Release of energy (p)re-heating Robust against initial (small-scale) perturbations Universality (no fine-tuning) E Φ Ψ < Ψ > Amplification of quantum fluctuations g c χ Φ Ψ(t) g(t) But: different spectrum in general, e.g., σ(φ) = k 4/3 Preferred frame (rest frame of medium) No unique/constant propagation speed Neglect of (quantum) back-reaction Emergent Horizons in the Laboratory p.23/26

24 Speculations... Postulate: No (locally) preferred frame Unique/constant propagation speed A eff = 1 dt d 3 r Φ 2 ( Φ) 2 2 t 2 scale-invariance A[λt, λr] = A[t, r] Dominated by (quantum) back-reaction? correct 1/k 3 -spectrum (conformal de Sitter metric) Was cosmic inflation just a phase transition? R. S., Phys. Rev. Lett. 95, (2005) Emergent Horizons in the Laboratory p.24/26

25 Ion Trap Expansion or Contraction of Ion chain Expanding Universe Phonon pair creation Hawking??? probability for n thermal state <n>= motional state n R. S. et al., Phys. Rev. Lett. 99, (2007) squeezed state 0.6 <n>= motional state n Emergent Horizons in the Laboratory p.25/26

26 Acknowledgments German Research Foundation (DFG) Emmy-Noether programme Alexander von Humboldt foundation ESF-Programme Cosmology in the Laboratory Pacific Institute of Theoretical Physics EU-IHP ULTI, CIAR, NSERC many interesting discussions... Emergent Horizons in the Laboratory p.26/26