  We investigate a link between the energy-momentum dispersion relation and the
spectral distance in the context of a Lorentzian almost-commutative spectral
geometry, defined by the product of Minkowski spacetime and an internal
discrete noncommutative space. Using the causal structure, the
almost-commutative manifold can be identified with a pair of four-dimensional
Minkowski spacetimes embedded in a five-dimensional Minkowski geometry.
Considering fermions travelling within the light cone of the ambient
five-dimensional spacetime, we then derive the energy-momentum dispersion
relation.
