  We set up a nonperturbative gravitational coarse graining flow and the
corresponding functional renormalization group equation on the as to yet
unexplored "tetrad only" theory space. It comprises action functionals which
depend on the tetrad field (along with the related background and ghost fields)
and are invariant under the semi-direct product of spacetime diffeomorphisms
and local Lorentz transformations. This theory space differs from that of
Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric
constitutes the fundamental variable and because of the additional symmetry
requirement of local Lorentz invariance. It also differs from "Quantum Einstein
Cartan Gravity" (QECG) investigated recently since the spin connection is not
an independent field variable now. We explicitly compute the renormalization
group flow on this theory space within the tetrad version of the
Einstein-Hilbert truncation. A detailed comparison with analog results in QEG
and QECG is performed in order to assess the impact the choice of a fundamental
field variable has on the renormalization behavior of the gravitational average
action, and the possibility of an asymptotically safe infinite cutoff limit is
investigated. Implications for nonperturbative studies of fermionic matter
coupled to quantum gravity are also discussed. It turns out that, in the
context of functional flow equations, the "hybrid calculations" proposed in the
literature (using the tetrad for fermionic diagrams only, and the metric in all
others) are unlikely to be quantitatively reliable. Moreover we find that,
unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related
to the local Lorentz transformations may not be discarded but rather contribute
quite significantly to the beta functions of Newton's constant and the
cosmological constant.
