Coframe geometry and gravity

The possible extensions of GR for description of fermions on a curved space, for supergravity and for loop quantum gravity require a richer set of 16 independent variables. These variables can be assembled in a coframe field, i.e., a local set of four linearly independent 1-forms. In this chapter we study the gravity field models based on a coframe variable alone. We give a short review of the coframe gravity. This model has the viable Schwarzschild solutions even being alternative to the standard GR. Moreover, the coframe model treating of the gravity energy may be preferable to the ordinary GR where the gravity energy cannot be defined at all. A principle problem that the coframe gravity does not have any connection to a specific geometry even being constructed from the geometrical meaningful objects. A geometrization of the coframe gravity is an aim of this chapter. We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an n dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Maxwell-like system of equations. In the framework of the coframe geometry construction, we propose a geometrical action for the coframe gravity. It is similar to the Einstein-Hilbert action of GR, but the scalar curvature is constructed from the general coframe connection. We show that this geometric Lagrangian is equivalent to the coframe Lagrangian up to a total derivative term. Moreover there is a family of coframe connections which Lagrangian does not include the higher order terms at all. In this case, the equivalence is complete. February 2, 2008