Black Hole Analogue: Extreme Blue Shift Disturbance. Thought of the Day 141.0

One major contribution of the theoretical study of black hole analogues has been to help clarify the derivation of the Hawking effect, which leads to a study of Hawking radiation in a more general context, one that involves, among other features, two horizons. There is an apparent contradiction in Hawking’s semiclassical derivation of black hole evaporation, in that the radiated fields undergo arbitrarily large blue-shifting in the calculation, thus acquiring arbitrarily large masses, which contravenes the underlying assumption that the gravitational effects of the quantum fields may be ignored. This is known as the trans-Planckian problem. A similar issue arises in condensed matter analogues such as the sonic black hole.

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Sonic horizons in a moving fluid, in which the speed of sound is 1. The velocity profile of the fluid, v(z), attains the value −1 at two values of z; these are horizons for sound waves that are right-moving with respect to the fluid. At the right-hand horizon right-moving waves are trapped, with waves just to the left of the horizon being swept into the supersonic flow region v < −1; no sound can emerge from this region through the horizon, so it is reminiscent of a black hole. At the left-hand horizon right-moving waves become frozen and cannot enter the supersonic flow region; this is reminiscent of a white hole.

Considering the sonic horizons in one-dimensional fluid flow, the velocity profile of the fluid as depicted in the figure above, the two horizons are formed for sound waves that propagate to the right with respect to the fluid. The horizon on the right of the supersonic flow region v < −1 behaves like a black hole horizon for right-moving waves, while the horizon on the left of the supersonic flow region behaves like a white hole horizon for these waves. In such a system, the equation for a small perturbation φ of the velocity potential is

(∂t + ∂zv)(∂t + v∂z)φ − ∂z2φ = 0 —– (1)

In terms of a new coordinate τ defined by

dτ := dt + v/(1 – v2) dz

(1) is the equation φ = 0 of a scalar field in the black-hole-type metric

ds2 = (1 – v2)dτ2 – dz2/(1 – v2)

Each horizon will produce a thermal spectrum of phonons with a temperature determined by the quantity that corresponds to the surface gravity at the horizon, namely the absolute value of the slope of the velocity profile:

kBT = ħα/2π, α := |dv/dz|v=-1 —– (2)

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Hawking phonons in the fluid flow: Real phonons have positive frequency in the fluid-element frame and for right-moving phonons this frequency (ω − vk) is ω/(1 + v) = k. Thus in the subsonic-flow regions ω (conserved 1 + v for each ray) is positive, whereas in the supersonic-flow region it is negative; k is positive for all real phonons. The frequency in the fluid-element frame diverges at the horizons – the trans-Planckian problem.

The trajectories of the created phonons are formally deduced from the dispersion relation of the sound equation (1). Geometrical acoustics applied to (1) gives the dispersion relation

ω − vk = ±k —– (3)

and the Hamiltonians

dz/dt = ∂ω/∂k = v ± 1 —– (4)

dk/dt = -∂ω/∂z = − v′k —– (5)

The left-hand side of (3) is the frequency in the frame co-moving with a fluid element, whereas ω is the frequency in the laboratory frame; the latter is constant for a time-independent fluid flow (“time-independent Hamiltonian” dω/dt = ∂ω/∂t = 0). Since the Hawking radiation is right-moving with respect to the fluid, we clearly must choose the positive sign in (3) and hence in (4) also. By approximating v(z) as a linear function near the horizons we obtain from (4) and (5) the ray trajectories. The disturbing feature of the rays is the behavior of the wave vector k: at the horizons the radiation is exponentially blue-shifted, leading to a diverging frequency in the fluid-element frame. These runaway frequencies are unphysical since (1) asserts that sound in a fluid element obeys the ordinary wave equation at all wavelengths, in contradiction with the atomic nature of fluids. Moreover the conclusion that this Hawking radiation is actually present in the fluid also assumes that (1) holds at all wavelengths, as exponential blue-shifting of wave packets at the horizon is a feature of the derivation. Similarly, in the black-hole case the equation does not hold at arbitrarily high frequencies because it ignores the gravity of the fields. For the black hole, a complete resolution of this difficulty will require inputs from the gravitational physics of quantum fields, i.e. quantum gravity, but for the dumb hole the physics is available for a more realistic treatment.

 

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.

Black Hole Complementarity: The Case of the Infalling Observer

The four postulates of black hole complementarity are:

Postulate 1: The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.

Postulate 2: Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

Postulate 3: To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy S(M).

We take as implicit in postulate 2 that the semi-classical field equations are those of a low energy effective field theory with local Lorentz invariance. These postulates do not refer to the experience of an infalling observer, but states a ‘certainty,’ which for uniformity we label as a further postulate:

Postulate 4: A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

To be more specific, we will assume that postulate 4 means both that any low-energy dynamics this observer can probe near his worldline is well-described by familiar Lorentz-invariant effective field theory and also that the probability for an infalling observer to encounter a quantum with energy E ≫ 1/rs (measured in the infalling frame) is suppressed by an exponentially decreasing adiabatic factor as predicted by quantum field theory in curved spacetime. We will argue that postulates 1, 2, and 4 are not consistent with one another for a sufficiently old black hole.

Consider a black hole that forms from collapse of some pure state and subsequently decays. Dividing the Hawking radiation into an early part and a late part, postulate 1 implies that the state of the Hawking radiation is pure,

|Ψ⟩= ∑ii⟩E ⊗|i⟩L —– (1)

Here we have taken an arbitrary complete basis |i⟩L for the late radiation. We use postulates 1, 2, and 3 to make the division after the Page time when the black hole has emitted half of its initial Bekenstein-Hawking entropy; we will refer to this as an ‘old’ black hole. The number of states in the early subspace will then be much larger than that in the late subspace and, as a result, for typical states |Ψ⟩ the reduced density matrix describing the late-time radiation is close to the identity. We can therefore construct operators acting on the early radiation, whose action on |Ψ⟩ is equal to that of a projection operator onto any given subspace of the late radiation.

To simplify the discussion, we treat gray-body factors by taking the transmission coefficients T to have unit magnitude for a few low partial waves and to vanish for higher partial waves. Since the total radiated energy is finite, this allows us to think of the Hawking radiation as defining a finite-dimensional Hilbert space.

Now, consider an outgoing Hawking mode in the later part of the radiation. We take this mode to be a localized packet with width of order rs corresponding to a superposition of frequencies O(r−1s). Note that postulate 2 allows us to assign a unique observer-independent s lowering operator b to this mode. We can project onto eigenspaces of the number operator bb. In other words, an observer making measurements on the early radiation can know the number of photons that will be present in a given mode of the late radiation.

Following postulate 2, we can now relate this Hawking mode to one at earlier times, as long as we stay outside the stretched horizon. The earlier mode is blue-shifted, and so may have frequency ω* much larger than O(r−1s) though still sub-Planckian.

Next consider an infalling observer and the associated set of infalling modes with lowering operators a. Hawking radiation arises precisely because

b = ∫0 dω B(ω)aω + C(ω)aω —– (2)

so that the full state cannot be both an a-vacuum (a|Ψ⟩ = 0) and a bb eigenstate. Here again we have used our simplified gray-body factors.

The application of postulates 1 and 2 has thus led to the conclusion that the infalling observer will encounter high-energy modes. Note that the infalling observer need not have actually made the measurement on the early radiation: to guarantee the presence of the high energy quanta it is enough that it is possible, just as shining light on a two-slit experiment destroys the fringes even if we do not observe the scattered light. Here we make the implicit assumption that the measurements of the infalling observer can be described in terms of an effective quantum field theory. Instead we could simply suppose that if he chooses to measure bb he finds the expected eigenvalue, while if he measures the noncommuting operator aa instead he finds the expected vanishing value. But this would be an extreme modification of the quantum mechanics of the observer, and does not seem plausible.

Figure below gives a pictorial summary of our argument, using ingoing Eddington-Finkelstein coordinates. The support of the mode b is shaded. At large distance it is a well-defined Hawking photon, in a predicted eigenstate of bb by postulate 1. The observer encounters it when its wavelength is much shorter: the field must be in the ground state aωaω = 0, by postulate 4, and so cannot be in an eigenstate of bb. But by postulate 2, the evolution of the mode outside the horizon is essentially free, so this is a contradiction.

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Figure: Eddington-Finkelstein coordinates, showing the infalling observer encountering the outgoing Hawking mode (shaded) at a time when its size is ω−1* ≪ rs. If the observer’s measurements are given by an eigenstate of aa, postulate 1 is violated; if they are given by an eigenstate of bb, postulate 4 is violated; if the result depends on when the observer falls in, postulate 2 is violated.

To restate our paradox in brief, the purity of the Hawking radiation implies that the late radiation is fully entangled with the early radiation, and the absence of drama for the infalling observer implies that it is fully entangled with the modes behind the horizon. This is tantamount to cloning. For example, it violates strong subadditivity of the entropy,

SAB + SBC ≥ SB + SABC —– (3)

Let A be the early Hawking modes, B be outgoing Hawking mode, and C be its interior partner mode. For an old black hole, the entropy is decreasing and so SAB < SA. The absence of infalling drama means that SBC = 0 and so SABC = SA. Subadditivity then gives SA ≥ SB + SA, which fails substantially since the density matrix for system B by itself is thermal.

Actually, assuming the Page argument, the inequality is violated even more strongly: for an old black hole the entropy decrease is maximal, SAB = SA − SB, so that we get from subadditivity that SA ≥ 2SB + SA.

Note that the measurement of Nb takes place entirely outside the horizon, while the measurement of Na (real excitations above the infalling vacuum) must involve a region that extends over both sides of the horizon. These are noncommuting measurements, but by measuring Nb the observer can infer something about what would have happened if Na had been measured instead. For an analogy, consider a set of identically prepared spins. If each is measured along the x-axis and found to be +1/2, we can infer that a measurement along the z-axis would have had equal probability to return +1/2 and −1/2. The multiple spins are needed to reduce statistical variance; similarly in our case the observer would need to measure several modes Nb to have confidence that he was actually entangled with the early radiation. One might ask if there could be a possible loophole in the argument: A physical observer will have a nonzero mass, and so the mass and entropy of the black hole will increase after he falls in. However, we may choose to consider a particular Hawking wavepacket which is already separated from the streched horizon by a finite amount when it is encountered by the infalling observer. Thus by postulate 2 the further evolution of this mode is semiclassical and not affected by the subsequent merging of the observer with the black hole. In making this argument we are also assuming that the dynamics of the stretched horizon is causal.

Thus far the asymptotically flat discussion applies to a black hole that is older than the Page time; we needed this in order to frame a sharp paradox using the entanglement with the Hawking radiation. However, we are discussing what should be intrinsic properties of the black hole, not dependent on its entanglement with some external system. After the black hole scrambling time, almost every small subsystem of the black hole is in an almost maximally mixed state. So if the degrees of freedom sampled by the infalling observer can be considered typical, then they are ‘old’ in an intrinsic sense. Our conclusions should then hold. If the black hole is a fast scrambler the scrambling time is rs ln(rs/lP), after which we have to expect either drama for the infalling observer or novel physics outside the black hole.

We note that the three postulates that are in conflict – purity of the Hawking radiation, absence of infalling drama, and semiclassical behavior outside the horizon — are widely held even by those who do not explicitly label them as ‘black hole complementarity.’ For example, one might imagine that if some tunneling process were to cause a shell of branes to appear at the horizon, an infalling observer would just go ‘splat,’ and of course Postulate 4 would not hold.

Unruh Radiation, Black Holes and Partial Waves. Note Quote.

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It is well known that Hawking radiation from an asymptotically flat Schwarzschild black hole is dominated by low angular momentum modes. This is a consequence of the fact that a black hole of Hawking temperaure TH and Schwarzschild radius rs has TH rs ∼ 1, so that high angular momentum modes of energy TH are trapped behind a large barrier in the effective radial potential. Since a local observer is unlikely to encounter such quanta, one might then conclude that a (much-weakened) version of postulate “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” might still hold in which the suppression is replaced by a fixed (1/area) power law. In addition, one would need to propose a mechanism through which these quanta would arise from the infalling perspective. This would appear to require that the infalling observer experience violations of local quantum field theory at this (power-law-suppressed) level.

This would already be a striking result: these quanta must appear quite close to the horizon and so violate the standard wisdom that the horizon is not a distinguished location. And they are not rare in the sense that their number is of the same order as the number of actual Hawking quanta.

As noted long ago by Unruh and Wald, it is possible to ‘mine’ energy from the modes trapped behind the effective potential. The basic procedure is to lower some object below the potential barrier, let the object absorb the trapped modes, and then raise the object back above the barrier. Unruh and Wald thought of the object as a box that could be opened to collect ambient radiation and then closed to keep the radiation from escaping. One may also visualize the object as a particle detector, though the two are equivalent at the level discussed here.

In the context of such a mining operation, one need only consider the internal state of the mining equipment to be part of the late-time Hawking radiation. In particular, postulate “outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations”, can be used to evolve the mode to be mined backward in time and to conclude for an old black hole that, even before the mining process takes place, the mode must be fully entangled with the early-time radiation. “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” is then violated for these modes as well, suggesting that the infalling observer encounters a Planck density of Planck scale radiation and burns up. One might say that the black hole is protected by a Planck-scale firewall.

Note that this firewall need not be visible to any observer that remains outside the horizon. All that we have argued is that the infalling observer does not experience a pure state. There remains considerable freedom in the possible reduced density matrices that could describe a few localized degrees of freedom outside the black hole, so that this matrix might still agree perfectly with that predicted by Hawking. In this case any local signal that an external observer might hope to ascribe to the firewall at distance 1/ω cannot be disentangled from the Unruh radiation that results from probing this scale without falling into the black hole.