Quantum fields close to black hole horizons

Transcription

1 Quantum fields close to black hole horizons Kinematics Accelerated scalar fields and inertial forces Photons in Rindler space vs thermal photons Interactions Static interactions of scalar, electric and gravitational charges Instability of matter Symmetry breakdown close to horizons? F. Lenz, K. Ohta, K. Yazaki, P. R. D 83, (20)

2 Uniformly accelerated observer in Minkowski space - Kinematics Transformation to observer s rest frame t, x, x τ, ξ,x t(τ, ξ) = a eaξ sinh aτ, x(τ, ξ) = a eaξ cosh aτ 0 II 5 t ξ = τ = ξ = const. τ = const. (H) x2 t 2 = a 2 e2aξ x t t x = tanh aτ III SH x I < τ <, < ξ < 5 0 IV ξ = τ = (H) ds 2 = R r dt 2 dr2 R r r 2 dω 2, R =2MG ds 2 R, e 2aξ = r R R Stretched horizon (SH) close to mathematical horizon (H)

3 Scalar Fields in Rindler Spaces Rindler metric S = 2 ds 2 = dt 2 dx 2 dx 2 = e 2aξ (dτ 2 dξ 2 ) dx 2 Action dτ dξ d 2 x ( τ φ) 2 ( ξ φ) 2 (m 2 φ 2 +( φ) 2 ) e 2aξ Wave equation 2 τ 2 ξ +(m 2 2 ) e 2aξ φ =0, φ = e iωτ e ik x ϕ(ξ) d2 dξ 2 + m2 e 2aξ ω 2 ϕ(ξ) =0, ϕ(ξ) =K iω a m e aξ, m 2 =(m 2 + k 2 )/a 2 The inertial force: Exponentially growing potential K i ω a.5 m 2 e 2aξ d2 dξ 2 + m2 e 2aξ ω 2 K iω m e aξ =0 a ξ m =, 2. ω = ω k m =0.5

4 Hamiltonian H a = d 2 k Degeneracy: consequence of generalized scale invariance 0 dωωa (ω, k )a(ω, k ) τ = a (x t + t x ) Boosts and Dilations commute Distance of two spcace-time points in Rindler and Minkowski coordinates (x x ) 2 = 2ea(ξ+ξ ) 2-point function Transition to imaginary Rindler time Periodic time dependence Unruh Temperature a 2 cosh a(τ τ ) cosh η cosh η =+ i0 M T φ(τ, ξ, x )φ(0, ξ, 0 ) 0M = D (x x ) 2 = D cosh a(τ τ ) cosh η a, 2e a(ξ+ξ ) a 2 β = 2π a = T τ iτ D (x E x E )2 = D Partition function With acceleration temperature T and Hamiltonian change H a 2e a(ξ+ξ ) a 2 e aξ e aξ 2 + a 2 x x 2 2e a(ξ+ξ ) cos a(τ τ ) cosh η Z = tr e βh a β=2π/a

5 Modified Planck s formula Accelerated and thermal photons T = a 2π g(ξ, x ) 0 M : H E (ξ, x )+H B (ξ, x ): 0 M = π 2 e 4aξ ωdω ω2 +4π 2 T 2 e ω T T ξ = e aξ T = π2 5 T 4 ξ Density of states ω T T 3 /ω, T ω ω T ω 2 e ω/t, ω 2 e ω/t thermal acceleration ω/a Energy density varies in space Tolman s law T ξ g00 = const. is satisfied. Temperature T ξ diverges when approaching horizon

6 Free field Propagators Propagator of massless scalar particles Propagators in Rindler space D(x, x )=D(τ, ξ, ξ, x ) = i0 M T φ(τ, ξ, x )φ(0, ξ, 0 ) 0M = D(ξ, ξ, x )= = a2 e a(ξ+ξ ) 8iπ 2 cosh aτ cosh η iδ dτd(τ, ξ, ξ, x )= ae a(ξ+ξ ) 4π 2 ξ + e 2aξ 2 D(ξ, ξ, x )=δ(ξ ξ )δ(x ) 4iπ 2 (x x ) 2 iδ i + sinh η π η D satisfies Poisson equation, imaginary part (homogeneous) Laplace equation Im D = superposition of zero energy modes: ω =0, k 2,k 3 Imaginary part of self-energy of a static charge Bremsstrahlung in Minkowski space Im Σ = κ2 2 Im D(ξ, ξ, 0) = κ2 8π 2 a is given by the total rate of Required by energy conservation in Rindler space, Bremsstrahlungs photons observed in Minkowski space appear as zero energy, transverse photons (with finite transverse momentum) in Rindler space

7 Interaction of static charges Scalar interaction energy V s (s) = κ 2 e a(ξ+ξ ) D(ξ, ξ, x ) D 00 (τ, ξ, ξ, x )=( τ t τ t τ x τ x ) D(τ, ξ, ξ, x ) Electrostatic interaction energy V v (s) = e 2 D(ξ, ξ, x ) V s = κ2 4π v(η), V v = e2 4π V t = GM M 2 4π v(η) cosh η, v(η)(2 cosh 2 η ) + i π τ max cosh η V s, V v,v t η v(η) = cosh η =+ a sinh η + iη e aξ e aξ 2 + a 2 x x 2 π 2e a(ξ+ξ ) With cosh η, V s,v,t invariant under scale transformations 5 ξ, ξ ξ + ξ 0, ξ + ξ 0, x, x eaξ 0 x,e aξ 0 x

8 The instability of atoms in Rindler space Electron coupled to a static charge located in Rindler space at ξ = ξ 0, x = x 0 Weak acceleration H ξ m a B e 2aξ 0 a = e 2aξ 0 a me 2 + ma(ξ ξ 0 ) e2 4π (ξ ξ0 ) 2 +(x x 0 ) 2 In the weak acceleration limit inertial force acts like an external electric field ξ/a B t H 0 0 y aa B a g Ionization probability of hydrogen d H = a eaξ 0 d H [0 3 m]

9 Spontaneous symmetry breaking close to horizons? H a = dξd 2 x π 2 +( ξ φ) 2 + e 2aξ ( φ) 2 + V (φ) 2 Hamiltonian depends on acceleration a φ 4 - model, discrete symmetry φ φ V (φ) = λ 8 φ2 (φ 2 2φ 2 0) λ 8 φ4 0 + m2 2 ϕ2, m 2 = λφ 2 0 Symmetry restoration if energy density of fluctuations (ξ) = e 2aξ 0 M : π 2 +( ξ ϕ) 2 + e 2aξ ( ϕ) 2 + m 2 ϕ : 0 M 2 of the order of the energy gain by symmetry breakdown (ξ) λ 8 φ4 0 (d H ) 480π 2 d 4 H (md H 0.2) d H 4 60 π 2 λ φ 0 Higgs model parameters d H T Tolman 90 GeV

10 Kinematics Summary On safe grounds - rewriting Minkowski space propagators in terms of Rindler coordinates Rindler particles = Minkowski particles accelerating = heating Dynamics Indications of significant differences in Minkowski and Rindler space dynamics - new type of interactions mediated by zero energy excitations Hints for symmetry restoration by acceleration deconfinement? Influence of the zero-mode radiation field on the structure of stretched horizons? ImD(ξ, ξ, x )