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In this paper, we show that any globally hyperbolic space-time admits atleast one globally defined distance-like function, which is a viscositysolution to the Lorentzian eikonal equation. According to whether the timeorientation is changed, we divide the set of viscosity solutions into somesubclasses. We show if the time orientation is consistent, then a viscositysolution has a variational representation locally. As a result, such aviscosity solution is locally semiconcave, as the one in the Riemannian case.Also, if the time orientation of a viscosity solution is non-consistent, weanalyse its peculiar properties which make this kind of viscosity solutions aretotally different from the ones where the Hamiltonians are convex.

Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times