The main aim of the present article is to prove that any globally hyperbolic space-
time M can be smoothly isometrically embedded in Lorentz-Minkowski LN, for some N, in
the spirit of Nash's theorem. This will be a consequence of the following two results, with
interest in its own right: (1) a Lorentzian manifold is isometrically embeddable in LN if and
only if it is a stably causal spacetime which admits a smooth time function ¿ with jr¿ j > 1,
and (2) any globally hyperbolic spacetime (M; g) admits a global orthogonal decomposition
M = R£S; g = ¡¯dt2 +gt with bounded function ¯. The role of the so-called problems
on smoothability" is stressed.