  We investigate the causal structure of two-sheeted space-times using the
tools of Lorentzian spectral triples. We show that the noncommutative geometry
of these spaces allows for causal relations between the two sheets. The
computation is given in details when the sheet is a 2- or 4-dimensional
globally hyperbolic spin manifold. The conclusions are then generalised to a
point-dependent distance between the two sheets resulting from the fluctuations
of the Dirac operator.
