The vierbein field of General Relativity contains components which can be
shifted by local Lorentz transformations and therefore do not show up in
the spacetime metric. In my talk I will show that these components can be
made physical in a consistent manner. They then appear in the form of an
antisymmetric tensor field which makes the action reminiscent of the
bosonic sector of supergravity. Both the metric and the antisymmetric
tensor can be made massive by adding a potential term for the vierbein to
the Einstein-Hilbert action. The form of this mass potential is inspired by
ghost-free massive gravity. I will discuss the features of the resulting
theory.