Classical Hawking radiation and particle pairs

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Classical Hawking radiation and particle pairs

Hawking radiation was originally derived using quantum mechanics and presented as an exclusively "quantum" effect. With the growing realisation that statistical behaviour of classical [[acoustic metric|acoustic systems] obeyed the same basic laws, Hawking radiation started to be presented as a semiclassical rather than a non-classical effect. However, these classical counterparts were sometimes described as not being literal Hawking radiation, but only analogues – "real" Hawking radiation was still an inherently non-classical effect associated with particle-pair production and could not be physically equivalent to any classical mechanism.

In the exercise below, we show how a classical description, accidentally treated to an over-mathematical SR-style approach, can be accidentally converted into one in which the radiation is created discontinuously as particle-pairs outside the horizon.

Pseudo particle-pair production

Step 1: Tachyonic behaviour

Although travel at more than lightspeed is generally considered impossible, some mathematical treatments appear to allow a description of hypothetical particles referred to as tachyons, that move at more than the speed of light. Since a particle's rate of timeflow under special relativity is usually described as dropping towards zero as its velocity approaches v=c 9creating the famous lightspeed barrier, the rate of timeflow for tachyons is described as being negative.

If a particle approached us at more than background lightspeed, and its existence didn't change the way that light propagated in the region, we'd expect not to see the particle until it hit us, after which we'd start to see the slower signals that it had generated during its journey starting to arrive at our location, in reverse order.

If the particle was emitted at location A, and arrived at our location B, then what we would see would be the particle starting at B and travelling towards A.

The time-reversal would also correspond to a reversal of the apparent "handedness" of the particle, so if it was an electron that had been deflected away from a negatively-charged plate during its journey, we'd see the time-reversed sequence apparently showing it to be attracted towards the plate, so that the time-reversal made it appear to have also flipped its electrical polarity. although we started out by assuming that the "tachyonic" particle was an electron travelling towards us at more then lightspeed, what we'd see (the observerspace physics) would be an "anti-electron" (or positron) travelling away from us at less than lightspeed.

=Step2: False paths

Under GR1960 or other SR-based models, a particle cannot escape outwards through a horizon if it is initially traveling at less than lightspeed.. Under systems that allow indirect radiation, there is no ballistic escape path though the horizon, but particles can escape along accelerated trajectories.

If our physics supports classical indirect radiation and one such particle is able to escape through a horizon by a sequence of accelerations, and we register its existence on our sensors, then if we do not take the necessary accelerations into account, and make a simple ballistic extrapolation of the arriving particle's supposed earlier trajectory, we will get a false trajectory for the particle which describes it starting out travelling at more than the local speed of light, before being slowed by the journey uphill across the gravitational differential.

Step 3: Particle-pairs

Our new (false) trajectory can then be considered as having two parts - the initial "superluminal" part where it escaped through the horizon, and the subluminal part, starting some region outside the horizon, at the point where the velocity had dropped to less than lightspeed.

Since the first part of this artificial trajectory (before the "lightspeed point") is "tachyonic", if the escaped particle is an electron we can describe the first stage of its path as being time-reversed. The description is then of an electron and positron both apparently being created at the same point outside the horizon, with the electron escaping and the positron travelling back into the horizon to be captured.

We then have an (somewhat artificial) description of Hawking radiation apparently being caused by particle-pair production outside the horizon.


One criticism of this approach is that the conversion has a degree of observer-dependence – observers can disagree as to which point along the trajectory should be considered the lightspeed point, and can therefore disagree about the height above the horizon at which the particle-pair is "really" created. However, this same behaviour also appears in standard QM descriptions of Hawking radiation.