In black hole physics and inflationary cosmology, the trans-Planckian problem is the problem of the appearance of quantities beyond the Planck scale, which raise doubts on the physical validity of some results in these two areas, since one expects the physical laws to suffer radical modifications beyond the Planck scale. In black hole physics, the original derivation of Hawking radiation involved field modes that, near the black hole horizon, have arbitrarily high frequencies—in particular, higher than the inverse Planck time, although these do not appear in the final results. A number of different alternative derivations have been proposed in order to overcome this problem. The trans-Planckian problem can be conveniently considered in the framework of sonic black holes, condensed matter systems which can be described in a similar way as real black holes. In these systems, the analogue of the Planck scale is the interatomic scale, where the continuum description loses its validity. One can study whether in these systems the analogous process to Hawking radiation still occurs despite the short-scale cutoff represented by the interatomic distance. The trans-Planckian problem also appears in inflationary cosmology. The cosmological scales that we nowadays observe correspond to length scales smaller than the Planck length at the onset of inflation. Trans-Planckian problem in Hawking radiation The trans-Planckian problem is the issue that Hawking's original calculation includes quantum particles where the wavelength becomes shorter than the Planck length near the black hole's horizon. This is due to the peculiar behavior there, where time stops as measured from far away. A particle emitted from a black hole with a finite frequency, if traced back to the horizon, must have had an infinite frequency, and therefore a trans-Planckian wavelength. The Unruh effect and the Hawking effect both talk about field modes in the superficially stationary spacetime that change frequency relative to other coordinates which are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration which constantly Doppler shifts the modes. 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