Tensor gauge fields and dark matter in general relativity with fermions

Abstract

Abstract The action of general relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate transformations act on coordinates and tensor indices, while local Lorentz transformations act on Dirac and Lorentz indices, much like a noncompact internal symmetry. The internal-symmetry character of local Lorentz invariance suggests that it might be implemented by tensor gauge fields with their own Yang–Mills action rather than by the spin connection as in standard formulations. But because the Lorentz group is noncompact, their Yang–Mills action must be modified by a neutral vector field whose average value at low temperatures is timelike. This vector field and the tensor gauge fields are neutral and interact gravitationally, so they contribute to hot and cold dark matter. The two independent symmetries of the action are reduced to a single symmetry of the vacuum, local Lorentz invariance, by the nonzero average values of the tetrads c k a . The local Lorentz invariance of general relativity with fermions can be extended to local U(2, 2) invariance. If the contracted squares of the covariant derivatives of the tetrads multiplied by the square of a mass M are added to the action, then in the limit M 2 → ∞ , the equation of motion of the tensor gauge fields is the vanishing of the covariant derivatives of the tetrads, which is Cartan’s first equation of structure. In the same limit, the tensor gauge fields approach the spin connection.