Reversing the Null Limit of the Szekeres Metric

TL;DR

This paper develops an algorithm to reconstruct the timelike-dust metric from null-dust spacetimes, extending previous null limit results of the Szekeres metric to include a non-zero cosmological constant.

Contribution

It introduces a systematic procedure for reversing the null limit process to retrieve the timelike-dust metric from null-dust spacetimes, including the effects of a non-zero cosmological constant.

Findings

Elucidated the null limit process and the behavior of key quantities.

Highlighted the role of transformations and substitutions in the process.

Provided a reconstruction algorithm for the metric and matter tensor.

Abstract

The null limits of the Lemaitre-Tolman and Szekeres spacetimes are known to be the Vaidya and news-free Robinson-Trautman metrics. We generalise this result to the case of non-zero $\Lambda$, and then ask whether the reverse process is possible -- is there a systematic procedure to retrieve the timelike-dust metric from the null-dust case? We present such an algorithm for re-constructing both the metric and matter tensor components of the timelike-dust manifold. This undertaking has elucidated the null limit process, highlighted which quantities approach unity or zero, and necessitated a careful discussion of how the functional dependencies are managed by the transformations and substitutions used.