Lorentzian manifolds isometrically embeddable in 𝕃^{ℕ}

Abstract

Abstract. In this article, the Lorentzian manifolds isometrically embeddable in 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 τ with |∇τ | > 1. Then, we prove that any globally hyperbolic spacetime (M, g) admits such a function, and, even more, a global orthogonal decomposition M = R × S, g = −βdt 2 + g t with bounded function β 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 . The role of the so-called "folk problems on smoothability" in Clarke's approach is also discussed.

Recent progress on the notion of global hyperbolicity

“…The existence of a steep temporal function is interesting for a spacetime M , as it solves the problem of its isometric embeddability in some L N (in the spirit of Nash' theorem [36]), see [35]. In fact, it was noticed by Greene [25] and Clarke [14] that any semi-Riemannian (or even degenerate) manifold can be isometrically immersed in some semi-Euclidean space R N s of sufficiently big dimension N and index s. The problem is a bit subtler for s = 1, but a simple argument in [35] shows that a spacetime can be isometrically embedded in L N if and only if it admits a steep temporal function.…”

“…Nevertheless, in general one cannot expect that, for example, if t ± is only a continuous time function constructed from some admissible measure then, by changing this measure, the new functions t ± will be smooth. In fact, t ± are time functions if and only if the spacetime is causally continuous [35], that is, t ± are not continuous if the spacetime is only stably causal. Nevertheless, in such a spacetime, the existence of a time function is ensured thanks to an original argument by Hawking [28].…”

“…This embeddability had been already claimed by Clarke [14]; however, his proof was affected by the folk problems of smoothability. The proof in [35] is based in a constructive procedure of a steep Cauchy temporal function, which also starts at Geroch's Cauchy time function. This construction is independent and easier than the one in [6].…”

“…Finally, it is worth pointing out that, for a spacetime which admits a temporal function but it is not globally hyperbolic, the existence of a steep temporal function can be lost or gained by changing conformally the metric [35]. So, one has:…”

Symplectic 4-manifolds via Lorentzian geometry

“…Theorem 1 above is essentially a fourdimensional symplectic version of the construction in [1], made possible for the following reason: because a null vector field k uniquely satisfies k ⊂ k ⊥ , one can thus consider the two-dimensional quotient subbundle k ⊥ /k instead of the full three-dimensional subbundle k ⊥ -this is the crucial (and well known) fact that ultimately makes Theorem 1 possible. Regarding the existence of the function f , there is a well known class of Lorentzian 4-manifolds, namely, the globally hyperbolic ones, which possess Cauchy temporal functions f as defined in [3,14], which naturally satisfy the property that k(f ) is nowhere vanishing. These 4-manifolds split diffeomorphically as R × S. Finally, it is also worth noting that the existence of the function f is also satisfied in any stably causal spacetime, by choosing f to be merely a temporal function [3]; i.e., one whose level sets are not necessarily Cauchy hypersurfaces, as they are for Cauchy temporal functions (stably causal spacetimes comprise a strictly larger class of Lorentzian 4-manifolds than globally hyperbolic ones; see [13]).…”