About Hawking radiation

Hawking radiation is the prediction made by Hawking that black holes will, if one takes into account quantum effects, not be completely black but radiate. This radiation will be a typical black body radiation, which has only one parameter, the temperature. The Hawking temperature will be extremely low, many orders below the CMB radiation, so that the effect will be anyway not visible in comparison with this background radiation. But from a purely theoretical point of view, this prediction is, of course, interesting.

Unfortunately, even if the derivation given by Hawking is in itself unproblematic, it makes no sense. The problem is that it is a prediction based on quantum field theory on a curved background, or semiclassical gravity, that means, an inconsistent approximation of a yet unknown quantum theory of gravity. It combines quantum field theory for the matter fields with a classical theory of gravity, general relativity. This mixture is not a self-consistent theory at all, it makes only sense as an approximation of a more fundamental quantum theory of gravity. And, as an approximation, it has only a limited range of applicability. This range can be approximately defined by the natural units of distance and time which can be obtained by combining Plancks constant, the speed of light, and the gravitational constant - the so-called Planck units: Planck length and Planck time. While the formulas of semiclassical gravity formally make sense even below the limit, it makes no sense at all to think that this approximation can give useful answers.

But this is the problem of the derivation of Hawking radiation. The derivation is fine and reasonable during the gravitational collapse itself. But already after a short time - for some star-sized black hole this would be less than a second after the moment relativistic time dilation on the surface of the collapsing star becomes important - we reach a region where surface time dilation becomes so large that the whole time of existence of the universe, for an outside observer, corresponds to less than Planck time on the surface. So, surface time dilation becomes so extremal that a photon of Hawkings temperature far away would have energies much greater than Planck mass. But for radiation with such a large energy it makes no sense at all to think that semiclassical theory could be applicable.

Moreover, the problem becomes much worse with time. After another second, the surface time dilation becomes many orders larger, and therefore the problem that to apply semiclassical theory makes no sense much much more serious.

This problem itself is well-known, and has even a name: The trans-Planckian problem.

Despite this, the problem is widely ignored, and Hawking radiation is accepted as a reasonable and certain prediction.

Why there cannot be a derivation independent of the trans-Planckian problem

If the trans-Planckian problem is mentioned at all, it is usually said that this is a problem of the original derivation, and that after this many other derivations have been found. The problem with this claim is that all these other derivations do not solve the trans-Planckian problem. One part of such derivations are simply derivations using the same semiclassical theory, but simply other methods of this same theory. So, they cannot solve the problem that the theory used is not applicable.

Another argument is an analogy with Unruh radiation. An acceleration of a particle detector in Minkowski space leads to a similar effect, namely that the particle detector seems to detect particles. Now, a particle detector at a fixed distance is also similar to an accelerated particle detector - that is sort of the equivalence principle. Thus, it is reasonable to conclude, by analogy, that it will also detect particles.

Unfortunately, it is possible to show that this analogy is wrong. The point is that stable stars do not Hawking-radiate. Such radiation is an effect of the change of the geometry, and a stable geometry does not give this effect. In particular, it can be (and has been) shown that the collapse stops at some fixed distance slightly greater than the Schwarzschild horizon size of the collapsing star, so that we do not obtain a black hole, then there will be no Hawking radiation, or, more accurate, the Hawking-like radiation which exists during the collapse stops once the final radius has been reached.

And this result can be used to show that no attempt to solve the trans-Planckian problem can be successful: Once a stable star with radius of say \(r_{Schwarzschild} + 10^{-100}l_{Pl}\) does not Hawking-radiate, Hawking radiation can be only a consequence of the difference between such a stable star and the GR solution. But this difference is completely in a region with that astronomical surface time dilation that the Planck time on the surface corresponds to the whole time of existence of the universe for an outside observer. Outside this clearly trans-Planckian region everything is equal.

How a trans-Planckian cutoff "solves" the problem

Beyond these attempts to "solve" the problem with other derivations based on essentially the same semiclassical theory there are also other ways to derive Hawking radiation: To modify the semiclassical theory in the trans-Planckian region, and to show that there will be, nonetheless, Hawking radiation. The point would be that, once such a modification of the theory does not change the final result, namely Hawking radiation, then the effect itself is stable: Even if every particular derivation depends on its particular assumptions for trans-Planckian theory, once the Hawking radiation does not depend on it, Hawking radiation is fine.

The problem with this is that this program fails. What we have already proposed - that the collapse stops at some \(r_{Schwarzschild} + 10^{-100}l_{Pl}\) - is an example of some trans-Planckian modification of the theory, and the result is no Hawking radiation.

On the other hand, there are indeed a lot of various modifications which lead to Hawking radiation. What is the cause of this? Very simple, whenever the physics changes in time, quantum field theory will predict some sort of Hawking-like radiation. So, if one introduces some cutoff, all what is necessary to obtain some Hawking-like radiation from the resulting theory is that one has to introduce a cutoff which somehow changes in time.

For example, let's replace the continous field theory with a theory using a spatial lattice. Then, if the lattice points move in time, then the whole quantum theory has an element of change in it, and the consequence is some Hawking-like radiation. If one, instead, uses a fixed lattice, there will be no Hawking radiation.

It remains to "justify" somehow that one has to prefer some cutoff method which has some elements which change in time. Which is done by introducing some preference for infalling observers or so.

To summarize, these methods to use a theory with some trans-Planckian cutoff to justify Hawking radiation use a modification which creates the resulting Hawking-radiation, because they introduce a time-dependence into the theory with cutoff which was not present in the original theory.