Yet another proof of Hawking and Ellis's Lemma 8.5.5

TL;DR

This paper presents a simpler proof of a key lemma related to the null generators of future Cauchy horizons, building on previous proofs by Hawking, Ellis, Budzyński, Kondracki, Królak, and Minguzzi.

Contribution

It offers a more straightforward proof of Hawking and Ellis's Lemma 8.5.5, improving clarity and accessibility.

Findings

Proof confirms past completeness of null generators

Simplifies previous complex proofs

Supports the validity of the lemma in general relativity

Abstract

The fact that the null generators of a future Cauchy horizon are past complete was proved first by Hawking and Ellis [1]. Then Budzy\'nski, Kondracki, and Kr\'olak outlined a proof free from an error found in the original one [2]. Finally, a week ago Minguzzi published his version of proof [3] patching a previously unnoticed hole in the preceding two. I am not aware of any flaws in that last proof, but it is quite difficult. In this note I present a simpler one.