ACOUSTIC BLACK HOLES. MASSIMILIANO RINALDI Université de Genève

Transcription

1 ACOUSTIC BLACK HOLES MASSIMILIANO RINALDI Université de Genève

2 OUTLINE Prelude: GR vs QM Hawking Radiation: a primer Acoustic Black Holes Hawking Radiation in Acoustic Black Holes Acoustic Black Holes in BECs Trans-planckian problem in BECs

3 General Relativity is characterized by two fundamental constants: Newton Constant: Speed of Light: G = c = m s m3 Kg s 2 R µν 1 2 g µν = 8πG c 4 T µν

4 General Relativity is characterized by two fundamental constants: Newton Constant: Speed of Light: G = c = m s m3 Kg s 2 R µν 1 2 g µν = 8πG c 4 T µν Quantum Mechanics by another one: Kg Reduced Planck constant: = m2 s i t Ψ(x, t) = 2 2m 2 + ˆV Ψ(x, t)

5 There is only one combination of,,and that gives a length: c G p = G c 3 = m GeV These are the scales where gravitational effects are comparable to quantum ones!

6 There is only one combination of,,and that gives a length: c G p = G c 3 = m GeV These are the scales where gravitational effects are comparable to quantum ones! Hawking radiation from black holes: T = c 3 8πGMk b For a black hole with mass of the Sun: T 10 7 K

7 What is a black hole? - Collapse of a star - Gravity on the surface increases - Beyond a certain limit, nothing can escape - No need for GR: this happens in Newtonian gravity as well! A luminous star, of the same density as the Earth, and whose diameter should be two hundred and fifty times larger than the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the larges luminous bodies in the Universe may, through this cause, be invisible (P. Laplace, The system of the World, Book 5,Chapter VI, 1798)

8 Why black holes are not black? - Curved Spacetime - Event Horizon - Collapse (not essential) When spacetime is curved : ( + m 2 )φ(x) =0 φ = 1 det g µ ( det g g µν ν φ) g µν is the metric solution of R µν 1 2 g µνr =0

9 Why black holes are not black? - Curved Spacetime - Event Horizon - Collapse (not essential) When spacetime is curved : ( + m 2 )φ(x) =0 φ = 1 det g µ ( det g g µν ν φ) g µν is the metric solution of R µν 1 2 g µνr =0 No Poincaré group, no unique vacuum state: φ(x) = i â i u i (x)+â i u i (x) = j â i 0 a = ˆb j 0 b =0 but â i 0 b =0 ˆbj v j (x)+ˆb j v j (x)

10 Bogolubov transformations: v j = i (α ji u i + β ji u i ) â i = j (α jiˆbj + β jiˆb i ) ˆN i =â i âi Number operator 0 b ˆN i 0 b = j β ji 2 =0

11 Bogolubov transformations: v j = i (α ji u i + β ji u i ) â i = j (α jiˆbj + β jiˆb i ) ˆN i =â i âi Number operator 0 b ˆN i 0 b = j β ji 2 =0

12 Bogolubov transformations: v j = i (α ji u i + β ji u i ) â i = j (α jiˆbj + β jiˆb i ) ˆN i =â i âi Number operator 0 b ˆN i 0 b = j β ji 2 =0 Asymptotic solutions: u(t, r) (4πω) 1 exp( iωv), v = t + r ds 2 = C(r)dt 2 + C 1 (r)dr 2, C(r hor )=0 BH Metric v(t, r) (4πω) 1 exp( iωv) exp(iωe κu )

13 Surface Gravity: κ = 1 2 The β N ω = dc dr r=rh Bogolubov does not vanish so: 1 1 2πω ω exp exp 1 κ k b T Thermal emission with temperature: 1 k b T = κ 2π Hawking 1974

14 Surface Gravity: κ = 1 2 The β N ω = Negative energy particle flux dc dr r=rh Bogolubov does not vanish so: 1 1 2πω ω exp exp 1 κ k b T Thermal emission with temperature: 1 k b T = κ 2π Hawking 1974 The mass of an astrophysical black hole decreases at a faster and faster pace: Event horizon C =1 2M r, κ = 1 Hawking Flux 4M

15 Black hole thermodynamics

16 Black hole thermodynamics The size of an isolated black hole cannot change -> zeroth law

17 Black hole thermodynamics The size of an isolated black hole cannot change -> zeroth law The mass-energy of the black hole is conserved -> first law R H = 2GM c 2, A =4πR 2 H, dm = κ 8π da

18 Black hole thermodynamics The size of an isolated black hole cannot change -> zeroth law The mass-energy of the black hole is conserved -> first law R H = 2GM c 2, A =4πR 2 H, dm = κ 8π da The size of a black hole cannot decrease -> second law Hawking (1972) Bekenstein (1973) da 0 S = 1 4 k b A 4 2 p

19 Black hole thermodynamics The size of an isolated black hole cannot change -> zeroth law The mass-energy of the black hole is conserved -> first law R H = 2GM c 2, A =4πR 2 H, dm = κ 8π da The size of a black hole cannot decrease -> second law Hawking (1972) Bekenstein (1973) da 0 S = 1 4 k b A Extremal black holes : J 2 + e 2 M 2 = M 4 have κ =0 (but A = 0). These correspond to the lowest possible temperature, i.e. zero Kelvin. -> third law 4 2 p

20 Can we mimic gravitational systems in the lab? fluid flow sonic horizon supersonic subsonic

21 Can we mimic gravitational systems in the lab? fluid flow sonic horizon supersonic Acoustic waves propagate with speed: subsonic dx dt = c sn + v Impose: n 2 =1, 1 v2 c 2 s dt = dt +(c 2 s v 2 ) 1 v dx 1 dt v2 dx 2 =0 c 2 s c 2 s

22 Can we mimic gravitational systems in the lab? fluid flow sonic horizon supersonic Acoustic waves propagate with speed: subsonic dx dt = c sn + v Impose: n 2 =1, 1 v2 c 2 s dt = dt +(c 2 s v 2 ) 1 v dx 1 dt v2 dx 2 =0 c 2 s c 2 s Confront with the Schwarzschild case! 1 2GM rc 2 dt GM rc 2 1 dr 2 =0

23 Acoustic metric (Unruh 1981) Consider an irrotational and homentropic fluid: v = ψ P = P (ρ) Classical action: S = d 4 x ρ ψ ρ( ψ) 2 + u(ρ)

24 Acoustic metric (Unruh 1981) Consider an irrotational and homentropic fluid: v = ψ P = P (ρ) Classical action: S = d 4 x ρ ψ ρ( ψ) 2 + u(ρ) Variation wrt Variation wrt ψ -> continuity: ρ + (ρv) =0 ρ -> Bernoulli: ψ v2 + du dρ =0

25 Acoustic metric (Unruh 1981) Consider an irrotational and homentropic fluid: v = ψ P = P (ρ) Classical action: S = d 4 x ρ ψ ρ( ψ) 2 + u(ρ) Variation wrt Variation wrt ψ -> continuity: ρ + (ρv) =0 ρ -> Bernoulli: ψ v2 + du dρ =0 Linear fluctuations: ρ ρ 0 + ρ 1, ψ ψ 0 + ψ 1 1 S = S 0 + S 2, S 2 = d 4 x 2 ρ 0( ψ 1 ) 2 ρ 0 2c 2 ( ψ 1 + v ψ 1 ) 2 s Where the speed of sound is: c s = dp (ρ) dρ

26 Acoustic metric (Unruh 1981) Variation of S 2 with respect to : f µν = ρ 0 c 2 s 1 v i 0 v i 0 c 2 sδ ij v i 0v j 0 ψ 1 µ (f µν ν )ψ 1 =0 Set f µν = det(g) g µν and find: S 2 = 1 2 d 4 x det(g) g µν µ ψ 1 ν ψ 1

27 Acoustic metric (Unruh 1981) Variation of S 2 with respect to : f µν = ρ 0 c 2 s 1 v i 0 v i 0 c 2 sδ ij v i 0v j 0 ψ 1 µ (f µν ν )ψ 1 =0 Set f µν = det(g) g µν and find: S 2 = 1 2 d 4 x det(g) g µν µ ψ 1 ν ψ 1 Klein-Gordon equation for a massless scalar field: ψ 1 on the acoustic metric: ds 2 g µν dx µ dx ν = ρ 0 1 µ ( det(g) g µν ν ψ 1 )=0 det(g) c s [ (c 2 s v 2 0)dt 2 2δ ij v i 0dx j dt + δ ij dx i dx j ]

28 vdx In two dimensions, set: t = T c 2 s v 2 ds 2 = ρ 0 (c 2s v 20)dT 2 + c2 sdx 2 c s (c 2 s v0 2) Confront with two-dimensional Schwarzschild b.h.: ds 2 = 1 2GM c 2 r dt GM c 2 r 1 dr 2

29 vdx In two dimensions, set: t = T c 2 s v 2 ds 2 = ρ 0 (c 2s v 20)dT 2 + c2 sdx 2 c s (c 2 s v0 2) Confront with two-dimensional Schwarzschild b.h.: ds 2 = 1 2GM c 2 r dt GM c 2 r 1 dr 2 The temperature of the black hole depends on the surface gravity T = κ, κ = c2 d 1 2GM 2πck b 2 dr rc 2 = c4 4GM c 2 r=2gm For the acoustic black hole replace c with c s and: κ = 1 2 d(c 2 s v 2 ) dx v=cs water helium B.E.C o K 10 4 o K nk

30 H = BLACK HOLES IN BOSE-EINSTEIN CONDENSATES dx 2 2m ˆΨ (x) ˆΨ(x)+V (x) ˆΨ (x) g(x) ˆΨ(x)+ 2 ( ˆΨ ) 2 ( ˆΨ) 2 Atomic mass External Potential atom-atom interaction constant [ ˆΨ(x), ˆΨ (x )] = δ(x x ) Bose commutation rules

31 H = BLACK HOLES IN BOSE-EINSTEIN CONDENSATES dx 2 2m ˆΨ (x) ˆΨ(x)+V (x) ˆΨ (x) g(x) ˆΨ(x)+ 2 ( ˆΨ ) 2 ( ˆΨ) 2 Atomic mass External Potential atom-atom interaction constant [ ˆΨ(x), ˆΨ (x )] = δ(x x ) Bose commutation rules i d ˆΨ dt = 2 2m d 2 ˆΨ dx 2 + ˆV ˆΨ + g ˆΨ 2 ˆΨ Gross-Pitaevski Equation

32 Madelung representation: ˆn ˆΨ = e iˆθ ˆθ = θ + ˆθ 1 ˆn = n +ˆn 1 θ 1 1 det(g) µ ( det(g) µ ˆθ1 )=0 satisfy the background G-P equation The fluctuation of the phase satisfies a Klein-Gordon equation!

33 Madelung representation: ˆn ˆΨ = e iˆθ ˆθ = θ + ˆθ 1 ˆn = n +ˆn 1 θ 1 1 det(g) µ ( det(g) µ ˆθ1 )=0 satisfy the background G-P equation The fluctuation of the phase satisfies a Klein-Gordon equation! The fluctuation θ 1 sees the acoustic metric: ds 2 = g µν dx µ dx ν = n mc c 2 dt 2 +(dx vdt) 2 We work in the hydrodynamic approximation, namely perturbation wavelength λ ξ mgn healing length Shorter wavelengths see the atomic structure: non-linear dispersion relations.

34 Detection of Hawing radiation in BECs through density-density correlations G 2 (x, x ) ˆn(x)ˆn(x ) ˆn = n +ˆn 1 t ˆθ1 + m ˆθ ˆθ 1 ˆn 1 = g Typical configuration: dc κ = dx x=0 x and x on opposite sides of the horizon

35 Detection of Hawing radiation in BECs through density-density correlations G 2 (x, x ) ˆn(x)ˆn(x ) ˆn = n +ˆn 1 t ˆθ1 + m ˆθ ˆθ 1 ˆn 1 = g Typical configuration: dc κ = dx x=0 x and x on opposite sides of the horizon G 2 (x, x ) 1 cosh 2 κ 2 ( x c r v + Balbinot et. al. (2008) x ) v c l

36 Analytical result for density-density correlations between Hawking partners Ab initio numerical simulation (Carusotto et al 2008)

37 Analytical result for density-density correlations between Hawking partners Ab initio numerical simulation (Carusotto et al 2008)

38 Beyond the hydrodynamical approximation: the trans-planckian problem. Infinite amount of time Modes with trans-planckian frequency Finite amount of time

39 Beyond the hydrodynamical approximation: the trans-planckian problem. Infinite amount of time No UV cutoff. Can we rely upon the theory beyond the Planck scale? Finite amount of time Modes with trans-planckian frequency In BECs there are effects at high frequency: modified dispersion relations ω 2 = k 2 + L 2 k 4 Interatomic distance

40 Dispersion and trans-planckian physics i d ˆΨ dt = 2 2m d 2 ˆΨ dx 2 + ˆV ˆΨ + g ˆΨ 2 ˆΨ At high frequency we cannot neglect this term! Let: i d ˆφ dt = ˆφ(t, x) = j ˆΨ ˆΨ0 (1 + ˆφ) 2 2m d 2 dx dψ 0 mψ 0 dx d dx â j φ j (t, x)+â j ϕ j (t, x) ˆφ + mc 2 ( ˆφ + ˆφ ) Mode decomposition

41 Dispersion and trans-planckian physics i d ˆΨ dt = 2 2m d 2 ˆΨ dx 2 + ˆV ˆΨ + g ˆΨ 2 ˆΨ At high frequency we cannot neglect this term! Let: i d ˆφ dt = ˆφ(t, x) = j ˆΨ ˆΨ0 (1 + ˆφ) 2 2m d 2 dx dψ 0 mψ 0 dx d dx â j φ j (t, x)+â j ϕ j (t, x) ˆφ + mc 2 ( ˆφ + ˆφ ) Mode decomposition i( t + v x )+ ξc 2 2 x c ξ i( t + v x )+ ξc 2 2 x c ξ φ j = c ξ ϕ j ϕ j = c ξ φ j Two coupled second order PDEs hence four modes!

42 Temporal formation of a step configration Spacetime diagram G 2 correlation function C. Mayoral, A. Fabbri, M. Rinaldi (2010)

43 Temporal formation of a step configration Spacetime diagram G 2 correlation function C. Mayoral, A. Fabbri, M. Rinaldi (2010) The main observation is that the Hawking thermal spectrum is robust against MDR

44 CONCLUSIONS Acoustic black holes offer a theoretical and experimental tool to investigate QFT on curved space They might show evidence of (acoustic)hawking radiation Hints towards the microscopic theory of black hole thermodynamics Possible developments towards analogue models of cosmology