Lorentzian fermionic action by twisting Euclidean spectral triples

We show how the twisting of spectral triples induces a transition from a Euclidean to a Lorentzian noncommutative geometry at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed Euclidean manifold, then that of a two-sheet Euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in Euclidean signature. We obtain the Weyl and the Dirac equations in Lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This permits us to interpret the field of 1-form that parametrises the twisted fluctuation of a manifold as the (dual) of the energy-momentum 4-vector.