The title should read: How polaritons produce Hawking radiation. On Fri, Apr 12, 2019 at 4:08 PM Axil Axil <[email protected]> wrote:
> https://www.mathpages.com/home/kmath591/kmath591.htm > > > *Hawking Radiation as Delayed Choice* > > > According to classical general relativity, mass-energy cannot depart from > the interior of a black hole, because nothing can cross an event horizon > “in reverse”. The event horizon is the surface at which the forward light > cones are tilted inward to the extent that they are entirely contained > inside the surface. Therefore, the passage of any mass-energy from inside > to outside the event horizon would require (according to classical general > relativity) superluminal propagation, and hence it is ruled out, and as a > consequence the mass-energy content of a black hole can never decrease. > However, applying quantum mechanical concepts, Hawking described a process > by which a quiescent black hole would radiate, so the mass-energy content > would decrease over time. The predicted rate of this radiation for > stellar-sized black holes is extremely small, but the prediction of any > outward radiation at all from an event horizon – no matter how weak – is > remarkable, since it appears to conflict with one of the foundational > principles of general relativity, namely, the principle of no super-luminal > propagation of mass-energy. > > > Several different explanations of Hawking radiation can be found in > popular descriptions. A few of these explanations are summarized below. > > > (1) Quantum fluctuations lead to the production of pairs of particles and > anti-particles just outside the horizon; one of these falls into the black > hole and the other escapes as radiation with positive mass-energy. The > in-falling particle has negative mass-energy, so its absorption results in > a reduction in the mass-energy of the black hole. > > > (2) The same as (1), except that both particles have positive mass-energy > (recognizing that anti-particles don’t have negative mass-energy). The > reduction in the mass-energy of the black hole is attributed to the > expenditure of work by tidal forces to separate the two particles. This > work exceeds the mass-energy of the absorbed particle by an amount equal to > the mass-energy of the radiated particle. > > > (3) Fluctuations occurring just *inside* the horizon of a black hole > produce pairs of particles, and one particle of such a pair may “tunnel” > out through the event horizon (exploiting the phenomenon of quantum > tunneling), becoming external radiation, leaving the other inside with > negative energy, diminishing the mass-energy of the black hole. > > > (4) Quantum vacuum fluctuations can be analyzed into positive and negative > frequencies. If the propagating fluctuations in the far future (far from > the black hole) are extrapolated back into the far past (prior to the > formation of the black hole), it is found that the future > positive-frequency fluctuations are consistent with the emission of > radiation from the black hole and the absorption of negative frequency > components by the black hole, diminishing its mass-energy. > > > The fourth explanation seems to be the closest to the actual derivation, > and it’s worth noting that it explicitly relies on an analysis beginning > prior to the formation of the black hole. Thus it would not apply (at least > not in any obvious way) to a black hole that has always existed. It might > be argued that such an object is not possible, since the universe itself > seems to have a definite beginning in the finite past, but even in this > context we might imagine primordial black holes, i.e., gravitationally > collapsed regions which have existed from the very beginning of the > universe. It’s unclear (to me) how the Hawking analysis (4) could be > applied to such an object, because there would be no time prior to the > formation of the black hole. > > > It’s interesting to compare this with the question > <https://www.mathpages.com/home/kmath528/kmath528.htm> of whether a > uniformly accelerating charged particle radiates. According to the > Lorentz-Dirac equations of classical electrodynamics, the answer seems to > be no, but this applies only to a particle that has *always* been > uniformly accelerating (and that will always continue to accelerate > uniformly), which is presumably not a realistic condition. For any > realistic particle, there must have been a time prior to when the particle > began its uniform acceleration, and this change in acceleration results in > radiation, in a sense because the Fourier transform of the particle’s > motion acquires impurities from the change in acceleration. This is > analogous to the fact that a purely monotonic signal must be eternal, > because any beginning or ending introduces other frequency components to > the function for the entire signal. > > > In the context of general relativity it isn’t too difficult to arrive at > an interpretation of Hawking radiation that is consistent with these ideas. > As discussed in another note <https://www.mathpages.com/rr/s7-03/7-03.htm>, > the worldline of every particle that falls into a black hole continues to > have a lightlike (null) connection to all external distances at all future > times. This is illustrated in the qualitative spacetime diagram below. > > > [image: image001.gif] > > > The red curve signifies the worldline of an in-falling particle, and the > gold curve represents a lightlike (null) locus from the in-falling particle > to the exterior region in the future. The important point is that the > in-falling worldline passes through every one of the external Schwarzschild > times (to future infinity) prior to actually crossing the event horizon. As > it passes through each of these times, there still null paths from the > particle to the future exterior, so the in-falling particle has (in a > sense) infinite opportunities to radiate to the exterior region, out to > arbitrarily late times, even though we would normally consider the particle > to have passed inside the event horizon by approximately the Schwarzschild > time denoted by t1 on the figure. According to the causal structure of > spacetime in classical general relativity, we can *never* (at any > external time) conclude for certain that the particle has emitted its last > photon prior to crossing the event horizon. > > > In fact, carrying this further, we might ask if there is ever a time when > we (in the external region) can conclude for certain that the particle > itself is going to cross the event horizon. From the classical standpoint > we might be able to compute a “point of no return”, but given the existence > of quantum fluctuations it may not be straightforward to determine this > point with absolute certainty. Also, the superposition of outcomes might be > resolved only at later times, as in “delayed choice” EPR experiments > involving quantum entanglement. As evaluated at some external time, the > state of an in-falling particle in the past may be uncertain, or rather in > a superposition of states, one falling through the event horizon without > emitting any more radiation, and the other emitting some additional > radiation or (perhaps) being ejected itself, and never falling through the > horizon. The irreversible outcome may be determined only at some time in > the distant future. In the mean time, the particle’s condition is analogous > to that of Schrödinger’s cat, neither alive nor dead, but in a > superposition of alive and dead. Every particle that has (presumptively) > crossed an event horizon and contributes to the mass-energy of a black hole > is arguably in this condition. If so, then a black hole itself is > essentially a quantum mechanical apparition (or fluctuation), whose > eventual “evaporation” is therefore less surprising. We could regard > Hawking radiation as the very gradual completion of a series of > delayed-choice experiments, eventually yielding the final irreversible > result that nothing ever actually fell through the event horizon. > > > The most obvious objection to this interpretation is that any radiation > emitted from a particle very close to the event horizon will be extremely > red-shifted to an external observer, and the intensity will drop > exponentially with time for that observer. But this assumes the classical > continuous model of radiation, propagating strictly along null intervals. > In the context of quantum field theory a photon has a non-zero amplitude to > be exchanged even along spacelike intervals. Such photons would be virtual > with respect to the local free-falling coordinates at the point of > emission, but may be real with respect to the stationary external > coordinates. This is closely related to the usual description of Hawking > radiation, except that it makes use of delayed-choice interactions with the > in-falling particles comprising the black hole, rather than regarding the > radiation as arising from vacuum field fluctuations. > > > Whether or not this approach leads to a consistent account of Hawking > radiation, it would not be surprising if there exists *some* description > of Hawking radiation in terms of direct actions between particles. Any > phenomena that can be described in terms of mediating fields can also be > described in terms of direct actions between particles, dispensing with the > fields entirely. For example, although electrodynamics is most commonly > described in terms of Maxwell’s field theory, it can also be developed > along the lines of Ampere and Weber as a distant-action theory with no > fields at all. Of course, the opposite is also true, i.e., quantum field > theory can be expressed entirely in terms of fields, without reference to > particles. > > > > > > *Discussion* > > > If a polariton Bose condensate based black hole is the root of the black > EVO, the light that it produces via Hawking radiation may be based on a > delayed mechanism for virtual photon release. Hawking's radiation theory is > based on cause #4 above, the photon black hole captures the virtual > particle at the moment of its creation and releases it to the far field > when the black hole is dissolved. A static long lived black hole produces > little if any hawking radiation. > > > A polariton has a lifetime of 1 to 10 picoseconds. A polariton lives and > dies a trillion times in one second. Each time that a polariton comes > into existence, the polariton captures virtual photons and when it dies, it > releases those photons to the far field. In a polariton condensate, a > single member polariton can generate 2 trillion virtual photons per second > because of its short lived nature. This is how a polariton condensate can > extract so much photonic power from the vacuum. >
