A duality between Lorentzian Twisted Spectral Triples and Spectral Triples

We present a duality between spectral triple and a Lorentzian version of twisted spectral triples, offering an original approach to Lorentzian geometry within the context of noncommutative geometry. This duality is used to define a spectral triple for which the fermionic action is Lorentz invariant, providing a new candidate for the almost-commutative manifold's spectral triple of the noncommutative standard model of particles physics.