# Conformally embedded spacetimes and the space of null geodesics

## TL;DR

This paper investigates conditions under which the space of null geodesics in causally simple Lorentzian manifolds is Hausdorff, revealing obstructions to conformal embeddings into globally hyperbolic spacetimes.

## Contribution

It establishes a criterion linking conformal embeddings and the Hausdorff property of null geodesic spaces, providing new insights into spacetime embedding limitations.

## Key findings

- Hausdorffness of null geodesic space implies conformal embedding into globally hyperbolic spacetime
- Obstructions to conformal embeddings are identified for causally simple spacetimes
- Examples of causally simple spacetimes not conformally embeddable are provided

## Abstract

It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.