Trans-Planckian problem
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In
In black hole physics, the original derivation of
The trans-Planckian problem can be conveniently considered in the framework of
The trans-Planckian problem also appears in inflationary cosmology. The cosmological scales that we nowadays observe correspond to length scales smaller than the
Trans-Planckian problem in Hawking radiation
The trans-Planckian problem is the issue that Hawking's original calculation includes
The
An outgoing Hawking radiated photon, if the mode is traced back in time, has a frequency which diverges from that which it has at great distance, as it gets closer to the horizon, which requires the wavelength of the photon to "scrunch up" infinitely at the horizon of the black hole. In a maximally extended external Schwarzschild solution, that photon's frequency stays regular only if the mode is extended back into the past region where no observer can go. That region seems to be unobservable and is physically suspect, so Hawking used a black hole solution without a past region which forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified: a microscopic point right at the moment that the black hole first formed.[citation needed]
The quantum fluctuations at that tiny point, in Hawking's original calculation, contain all the outgoing radiation. The modes that eventually contain the outgoing radiation at long times are redshifted by such a huge amount by their long sojourn next to the event horizon, that they start off as modes with a wavelength much shorter than the Planck length. Since the laws of physics at such short distances are unknown, some find Hawking's original calculation unconvincing.[2][3][4][5]
The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations. The same effect occurs for regular matter falling onto a white hole solution. Matter which falls on the white hole accumulates on it, but has no future region into which it can go. Tracing the future of this matter, it is compressed onto the final singular endpoint of the white hole evolution, into a trans-Planckian region. The reason for these types of divergences is that modes which end at the horizon from the point of view of outside coordinates are singular in frequency there. The only way to determine what happens classically is to extend in some other coordinates that cross the horizon.
There exist alternative physical pictures which give the Hawking radiation in which the trans-Planckian problem is addressed.[6] The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time.[7] In the Unruh effect, the magnitude of the temperature can be calculated from ordinary Minkowski field theory, and is not controversial.
Notes
- ^ Bibcode:2011arXiv1103.2271B.
- S2CID 16668175.
- .
- PMID 10014053.
- S2CID 26432764.
- S2CID 119250586.
- ISBN 0-521-29928-4.