Lorentzian manifolds isometrically embeddable in L^N

TL;DR

This paper characterizes which Lorentzian manifolds can be embedded into Lorentz-Minkowski space, showing that globally hyperbolic spacetimes can always be isometrically embedded, using a novel construction of a smooth time function.

Contribution

It provides a new characterization of embeddability for Lorentzian manifolds and introduces a method to construct smooth time functions with bounded gradient.

Findings

Globally hyperbolic spacetimes can be isometrically embedded in L^N.

A new construction of smooth time functions with bounded gradient is presented.

The role of smoothability in embedding problems is emphasized.

Abstract

We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bounded away from zero -and, thus, an orthogonal global splitting of the spacetime with bounded lapse. The role of the so-called "folk problems on smoothability" is stressed.