Pullback-invariant matter couplings to degenerate tetrads

An algorithm is described for the construction of actions for scalar, spinor, and vector gauge fields that remains well defined when the metric is degenerate and that involves no contravariant tensor fields. These actions produce the standard matter dynamics and coupling to gravity when the tetrad is non-degenerate, but have the property that all fields on a spacetime manifold M that appear in them can be pulled back to any smooth manifold through an arbitrary map , and that this pullback leaves the action invariant when has degree one.