Lorentzian manifolds isometrically embeddable in $\mathbb {L}^N$

Abstract:

In this article, the Lorentzian manifolds isometrically embeddable in $\mathbb {L}^N$ (for some large $N$, 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 $\tau$ with $|\nabla \tau |>1$. Then, we prove that any globally hyperbolic spacetime $(M,g)$ admits such a function, and, even more, a global orthogonal decomposition $M=\mathbb {R} \times S, g=-\beta dt^2 + g_t$ with bounded function $\beta$ 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 $\mathbb {L}^N$. The role of the so-called “folk problems on smoothability” in Clarke’s approach is also discussed.