Lorentzian manifolds isometrically embeddable in 𝕃^{ℕ}

Abstract

In this article, the Lorentzian manifolds isometrically embeddable in L N \mathbb {L}^N (for some large N N , in the spirit of Nash’s theorem) are characterized as a subclass of the set of all stably causal spacetimes; concretely, those which admit a smooth time function τ \tau with | ∇ τ | > 1 |abla \tau |>1 . Then, we prove that any globally hyperbolic spacetime ( M , g ) (M,g) admits such a function, and, even more, a global orthogonal decomposition M = R × S , g = − β d t 2 + g t M=\mathbb {R} \times S, g=-\beta dt^2 + g_t with bounded function β \beta and Cauchy slices. In particular, a proof of a result stated by C.J.S. Clarke is obtained: any globally hyperbolic spacetime can be isometrically embedded in Minkowski spacetime L N \mathbb {L}^N . The role of the so-called “folk problems on smoothability” in Clarke’s approach is also discussed.