Lorentz invariance of gravitational Lagrangians in the space of reference frames

The recently proposed theories of gravitation in the space of reference framesS are based on a Lagrangian invariant with respect to the homogeneous Lorentz group. However, in theories of this kind, the Lorentz invariance is not a necessary consequence of some physical principles, as in the theories formulated in space-time, but rather a purely esthetic request. In the present paper, we give a systematic method for the construction of gravitational theories in the spaceS, without assuming a priori the Lorentz invariance of the Lagrangian. The Einstein-Cartan equations of gravitation are obtained requiring only that the Lagrangian is invariant under proper rotations and has particular transformation properties under space reflections and space-time dilatations