The Conformal Pseudodistance and Null Geodesic Incompleteness

TL;DR

This paper explores the link between null geodesic completeness and conformal pseudodistance in Einstein Lorentz manifolds, showing that certain pseudodistance properties imply null geodesic incompleteness.

Contribution

It establishes a connection between conformal pseudodistance and null geodesic incompleteness in Einstein manifolds, providing conditions under which null geodesics are incomplete.

Findings

Nontrivial pseudodistance implies at least one incomplete null geodesic.

Nondegenerate pseudodistance implies all null geodesics are incomplete.

Presence of a 'physical metric' satisfying specific conditions leads to null geodesic incompleteness.

Abstract

We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its pseudodistancfe is nontrivial. If its pseudodistance is nondegenerate, all of its null geodesics must be incomplete. Thus an Einstein manifold (M,g) has no complete null geodesic if there is a "physical metric" in the conformal class of g satisfying the null convergence and null generic conditions.