Solutions of the Einstein–Cartan–Dirac equations with vanishing energy-momentum tensor

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Abstract
It is shown that a frame can always be chosen with respect to which a Dirac spinor ψ has a particular simple form. This result is used to investigate the existence of solutions of the Einstein–Cartan–Dirac equations with vanishing energy-momentum tensor, but nonvanishing current (‘‘ghost solutions’’). We concentrate on solutions with a massive Dirac field, subject to the restriction (ψ̄ψ)2+(ψ̄γ5ψ)2>0. In this case, the field equations can be reduced to a set of (covariant) equations on the group SL(2, R). A family of new exact solutions is presented.

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