Black holes as open quantum systems Claus Kiefer Institut für Theoretische Physik Universität zu Köln
Hawking radiation 1 1 singularity II γ H γ γ H collapsing 111 star 1 1 I - future event horizon + i o S. W. Hawking (1975) T BH = 8πGk B M 6. 1 8M M K n k = 1 e 8πωGM 1 : (average number of particles) with ω = k k
Black holes have a finite lifetime: ( ) 3 ( ) 3 M τ BH 8895 t P 1.159 1 67 M yr m P M from the emission of gravitons and photons (D. Page 1976) The semiclassical approimation breaks down if the black hole approaches the Planck mass m P. If the black hole left only thermal radiation behind, a pure state for a closed system would evolve into a mied system information-loss problem This would be in contradiction to ordinary quantum theory where the entropy S = k B Tr(ρlnρ) is conserved for a closed system (unitary evolution).
Information is lost during the evaporation, ρ $ρ SρS The full evolution is unitary, but this cannot be seen in the semiclassical approimation The black hole leaves a remnant carrying all the information Final answer only within quantum gravity! Focus here on the second option
At no point in the calculation by Hawking (and others) is an eact mied (canonical) state used in the formalism. The coherent superposition used by Hawking is indistinguishable from a local thermal miture (L. Parker 1975). The reduced state of each mode in a two-mode squeezed state is a thermal state (canonical ensemble); in the special case of a black hole, the temperature is independent of k (universality). We can consider a spatial hypersurface that enters the horizon or a hypersurface that is locked at the bifurcation point (used in the following).
1 1 singularity II γ H γ γ H collapsing 111 star 1 1 I - future event horizon + i o For a massless scalar field φ on this background, an initial vacuum state evolves into Ψ = k ψ k with ψ k ep [ k coth(πkgm +ikt) φ(k) ]. This is a pure state, but ψ k n k ψ k = 1 e 8πωGM 1 two-mode squeezed state J.-G. Demers and C.K. (1996)
The above pure state can be written as [ ] ψ k ep k 1+eiϕ k tanhr k 1 e iϕ φ(k) k tanhrk ep [ (Ω R +iω I ) φ(k) ] with the squeezing parameter tanhr k = ep( 4πωGM) and the squeezing angle ϕ k = kt. Thus, r k for k r k for k. At maimum of Planck spectrum: r.5 kt = : kt = π/: Squeezing in φ Squeezing in p φ Ratio of corresponding widths is tanh (πkgm) (.37 at maimum of Planck spectrum)
Decoherence S A E Now, the superposition principle leads to ( ) t c n n Φ n E n n c n n Φ n E n All local observations follow from the reduced density matri of system plus apparatus: ρ SA n c n n n Φ n Φ n if E n E m δ nm The interferences eist, but they are not there.
(a) (b) From: E. Joos et al., Decoherence and the appearance of a classical world in quantum theory (3)
Wigner function W(,p) = 1 dy e ipy/ ρ( y,+y) π p (a) p (b)
Left: Decoherence through particle collisions. Right: Decoherence through heating of fullerenes. From: M. Arndt and K. Hornberger, Quantum interferometry with comple molecules, arxiv:93.1614v1
Squeezed states are very sensitive to decoherence. This sensitivity is responsible, for eample, for the quantum-to-classical transition for the primordial fluctuations in the early Universe. (During inflation, the Wigner ellipse becomes frozen.) The degree of decoherence can conveniently be studied with the Wigner function.
Hawking radiation from decoherence Wigner ellipse rotates with a timescale 14GM typical observation times; thus, a coarse-graining with respect to the squeezing angle ϕ is justified This yields the entropy S k = (1+n k )ln(1+n k ) n k lnn k r k 1 r k Integration over all modes gives S = (π /45)TBH 3 V, the entropy of Hawking radiation The pure squeezed state is thus indistinguishable from a canonical ensemble with Hawking temperature T BH (although the hypersurface remains outside the horizon), ecept for times smaller than 14GM which is about 1.7 1 3 s for a primordial black hole with mass 5 1 14 g Thermal nature of Hawking radiation thus arises from decoherence These arguments may not apply after the Page time (when the Bekenstein Hawking entropy equals the semiclassical radiation entropy)
Decoherence of black holes by Hawking radiation The total state of black holes and Hawking radiation can be approimated at the semiclassical level (of canonical quantum gravity) by Ψ e is ψ If one considers a superposition of the form Ψ e is ψ +e is ψ (superposition of black hole with white hole) or a superposition of the form Ψ e is ψ +e is() ψvac (superposition of black hole with no hole), one can show by tracing out the Hawking radiation that decoherence occurs, i.e., that the superposition cannot be distinguished from a corresponding miture. J.-G. Demers and C.K. (1996)
Decoherence for string black holes For BPS states ( mass=charge ): increasing g: D-branes black holes; S strings = S BH Myers (1997); Amati (1999): averaging over all string ecitations of mass M leads to information loss by decoherence; thus, black holes eist only as macrostates Environment? Many non-metric coupled fields of ecited string modes
Quasi-normal modes For the Schwarzschild black hole, one has for fied l (angular momentum of perturbation) a countable infinity of modes (labelled by integers n), with the spectrum ω n = i(n+ 1 ) 4GM + ln3 ( 8πGM +O n 1/) ( = iκ n+ 1 ) + κ ( π ln3+o n 1/) The real part has the universal value Reω n ω QNM = ln3 8πGM ln3 T BH Connection to quantum properties of black hole? 8.85 M M khz
Hawking temperature as minimal noise temperature Idea: Treat the black hole as a harmonic oscillator with frequency ω QNM. In quantum optics, the measurement of an oscillator position (here, the field amplitude) possesses a fundamental limitation beyond the uncertainty relation if an amplifier is coupled to the system (Caves 198). In this process, a minimal noise temperature is added to the system. It leads to the ultimate position resolution QL = ln3mω 1.35 This is relevant for eperiments that attempt to reach the quantum limit (e.g. coupling a nano-mechanical oscillator to a single-electron transistor).
By the coupling of the system to an amplifier, a minimal noise temperature is added according to T n = ω k B ln 1 ( 3±G 1 1±G 1 where G denotes the gain of the amplifier. ), In the limit of infinite gain (needed because the signal is strongly damped in the limit n ), this leads in the black-hole case to T n = ω QNM k B ln3 T BH. The minimal noise temperature equals the Hawking temperature! C.K. (4)
Black-hole entropy from quasi-normal modes? Universality of the Bekenstein Hawking entropy S BH = k BA 4lP? The spectrum of quasi-normal modes is also universal. Can one recover the Bekenstein Hawking entropy from the entanglement between these modes and the black hole?
Insight from canonical quantum gravity Model black hole and Hawking radiation by a set of coupled oscillators (the coupling strength being a measure for the back action of the radiation on the black hole) Without back action, the black hole assumes a strongly squeezed state. Including back action, the squeezing is suppressed; the reduced density matrices for black hole and Hawking radiation become similar in the late evaporation phase, i.e. the difference between black hole and radiation vanishes; there is one strongly entangled quantum state. C.K., J. Marto, and P.V. Moniz (9)
Density matri for the black hole Restrict to diagonal elements ( probabilities ) ρ = tr y ρ =,y,y dy 3 3 3 3.5.5.5.5 t 1.5 t 1.5 t 1.5 t 1.5 1 1 1 1.5.5.5.5 4 4 4 4 4 4 4 4 3 3 3 3.5.5.5.5 t 1.5 t 1.5 t 1.5 t 1.5 1 1 1 1.5.5.5.5 4 4 4 4 4 4 4 4 Figure: Time evolution of ρ, with m P = = ω = = 1; t = and p = 1 for simplicity, and µ (graphics from left to right and top to bottom) assuming the values of the set {,.5,1,5,1,,5,1}, ω y = ω 1 5/, m y = m P 1 5.
Density matri for the Hawking radiation Restrict again to diagonal elements ( probabilities ) ρ yy = tr ρ =,y,y d 1 1 1 1 8 8 8 8 t 6 4 t 6 4 t 6 4 t 6 4 4 4 y 4 4 y 4 4 y 4 4 y Figure: Time evolution of ρ yy, with m P = = ω = = 1 and p = 1; t = for simplicity, and µ (graphics from left to right) assuming the values of the set {,1,5,1}, ω y = ω 1 5/, m y = m P 1 5. For large values of µ the results look qualitatively similar to the results for ρ. If the back reaction is large, the difference between the black hole and the Hawking radiation begins to disappear.
Summary and outlook Black holes arise from decoherence ( classical irreversible object). This is analogous to the emergence of particle trajectories in the classical limit of quantum mechanics. A black hole is thus an open quantum system and should be treated in this way. The information-loss problem can only refer to the total system of black hole plus Hawking radiation plus other interacting degrees of freedom; it can be solved only by a full quantum theory of gravity (fate of singularity is crucial). The thermal nature of Hawking radiation can be understood to emerge from decoherence. If decoherence is inefficient, one can have a superposition of black and white holes. The quasi-normal modes of the black hole may play an important role for understanding black-hole temperature and entropy.