How Black Holes Emitting Hawking Radiation At Best Give Non-Trivial Information About Planckian Physics: Towards Entanglement Entropy.

The analogy between quantised sound waves in fluids and quantum fields in curved space-times facilitates an interdisciplinary knowhow transfer in both directions. On the one hand, one may use the microscopic structure of the fluid as a toy model for unknown high-energy (Planckian) effects in quantum gravity, for example, and investigate the influence of the corresponding cut-off. Examining the derivation of the Hawking effect for various dispersion relations, one reproduces Hawking radiation for a rather large class of scenarios, but there are also counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawkings result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics.

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On the other hand, the emergence of an effective geometry/metric allows us to apply the vast amount of universal tools and concepts developed for general relativity (such as horizons), which provide a unified description and better understanding of (classical and quantum) non-equilibrium phenomena (e.g., freezing and amplification of quantum fluctuations) in condensed matter systems. As an example for such a universal mechanism, the Kibble-Zurek effect describes the generation of topological effects due to the amplification of classical/thermal fluctuations in non-equilibrium thermal phase transitions. The loss of causal connection underlying the Kibble-Zurek mechanism can be understood in terms of an effective horizon – which clearly indicates the departure from equilibrium. The associated breakdown of adiabaticity leads to an amplification of thermal fluctuations (as in the Kibble-Zurek mechanism) as well as quantum fluctuations (at zero temperature). The zero-temperature version of this amplification mechanism is completely analogous to the early universe and becomes particularly important for the new and rapidly developing field of quantum phase transitions.

Furthermore, these analogue models might provide the exciting opportunity of measuring the analogues of these exotic effects – such as Hawking radiation or the generation of the seeds for structure formation during inflation – in actual laboratory experiments, i.e., experimental quantum simulations of black hole physics or the early universe. Even though the detection of these exotic quantum effects is partially very hard and requires ultra-low temperatures etc., there is no (known) principal objection against it. The analogue models range from black and/or white hole event horizons in flowing fluids and other laboratory systems over apparent horizons in expanding Bose–Einstein condensates, for example, to particle horizons in quantum phase transitions etc.

However, one should stress that the analogy reproduces the kinematics (quantum fields in curved space-times with horizons etc.) but not the dynamics, i.e., the effective geometry/metric is not described by the Einstein equations in general. An important and strongly related problem is the correct description of the back-reaction of the quantum fluctuations (e.g., phonons) onto the background (e.g., fluid flow). In gravity, the impact of the (classical or quantum) matter is usually incorporated by the (expectation value of) energy-momentum tensor. Since this quantity can be introduced at a purely kinematic level, one may use the same construction for phonons in flowing fluids, for example, the pseudo energy-momentum tensor. The relevant component of this tensor describing the energy density (which is conserved for stationary flows) may become negative as soon as the flow velocity exceeds the sound speed. These negative contributions explain the energy balance of the Hawking radiation in black hole analogues as well as super-radiant scattering. However, the (expectation value of the) pseudo energy-momentum tensor does not determine the quantum back-reaction correctly.

One should not neglect to mention the occurrence of a horizon in the laboratory – the Unruh effect. A uniformly accelerated observer cannot see half of the (1+1- dimensional) space-time, the two Rindler wedges are completely causally disconnected by the horizon(s). In each wedge, one may introduce a set of observables corresponding to the measurements made by the observers confined to this wedge – thereby obtaining two equivalent copies of observables in one wedge. In terms of these two copies, the Minkowski vacuum is an entangled state which yields the usual phenomena (thermo-field formalism) including the Unruh effect – i.e., the uniformly accelerated observer experiences the Minkowski vacuum as a thermal bath: For rather general quantum fields (Bisognano-Wichmann theorem), the quantum state ρ obtained by restricting the Minkowski vacuum to one of the Rindler wedges behaves as a mixed state ρ = exp{−2πHˆτ/κ}/Z, where Hˆτ corresponds to the Hamiltonian generating the proper (co-moving wristwatch) time τ measured by the accelerated observer and κ is the analogue to the surface gravity and determines the acceleration.

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Space-time diagram with a trajectory of a uniformly accelerated observer and the resulting particle horizons. The observer is confined to the right Rindler wedge (region x > |ct| between the two horizons) and cannot influence or be influenced by all events in the left Rindler wedge (x < |ct|), which is completely causally disconnected.

The thermal character of this restricted state ρ arises from the quantum correlations of the Minkowski vacuum in the two Rindler wedges, i.e., the Minkowski vacuum is a multi-mode squeezed state with respect the two equivalent copies of observables in each wedge. This is a quite general phenomenon associated with doubling the degrees of freedom and describes the underlying idea of the thermo-field formalism, for example. The entropy of the thermal radiation in the Unruh and the Hawking effect can be understood as an entanglement entropy: For the Unruh effect, it is caused by averaging over the quantum correlations between the two Rindler wedges. In the black hole case, each particle of the outgoing Hawking radiation has its infalling partner particle (with a negative energy with respect to spatial infinity) and the entanglement between the two generates the entropy flux of the Hawking radiation. Instead of accelerating a detector and measuring its excitations, one could replace the accelerated observer by an accelerated scatterer. This device would scatter (virtual) particles from the thermal bath and thereby create real particles – which can be interpreted as a signature of Unruh effect.