  While a natural ultraviolet cutoff, presumably at the Planck length, is
widely assumed to exist in nature, it has proven difficult to implement a
minimum length scale covariantly. A key reason is that the presence of a fixed
minimum length would seem to contradict the ability of Lorentz transformations
to contract lengths. In this paper, we implement a fully covariant Planck scale
cutoff by cutting off the spectrum of the d'Alembertian. In this scenario,
consistent with Lorentz contractions, wavelengths that are arbitrarily smaller
than the Planck length continue to exist. However, the dynamics of modes of
wavelengths that are significantly smaller than the Planck length possess a
very small bandwidth. This has the effect of freezing the dynamics of such
modes. While both, wavelengths and bandwidths, are frame dependent, Lorentz
contraction and time dilation conspire to make the freezing of modes of
transplanckian wavelengths covariant. In particular, we show that this
ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We
focus on Friedmann Robertson Walker (FRW) spacetimes and their much-discussed
transplanckian question: The physical wavelength of each comoving mode was
smaller than the Planck scale at sufficiently early times. What was the mode's
dynamics then? Here, we show that in the presence of the covariant UV cutoff,
the dynamical bandwidth of a comoving mode is essentially zero up until its
physical wavelength starts exceeding the Planck length. In particular, we show
that under general assumptions, the number of dynamical degrees of freedom of
each comoving mode all the way up to some arbitrary finite time is actually
finite. Our results also open the way to calculating the impact of this natural
UV cutoff on inflationary predictions for the CMB.
