Note on identifying four-dimensional vacuum solutions from Weyl invariants

TL;DR

This paper demonstrates how Weyl invariants can be used to identify and relate different coordinate representations of the same four-dimensional vacuum solutions in General Relativity, aiding in solution classification.

Contribution

It provides a complementary method using Weyl invariants to verify when different metric forms represent the same Einstein vacuum solution, including explicit coordinate transformations.

Findings

Weyl invariants can confirm equivalence of different metric forms.

Explicit coordinate transformations linking the metrics are constructed.

The method aids in solution identification in General Relativity.

Abstract

The diffeomorphism covariance is a fundamental property of General Relativity which leads to the fact that the same solution of Einstein equation can be given in completely distinct forms in different coordinate systems. Distinguishing or identifying two metrics as solutions of Einstein equation is particularly challenging. In a recent paper arXiv:2503.14586 [hep-th], it is proposed to apply the relations of different Weyl invariants to distinguish solutions. In this note, we present a complementary application of the Weyl invariants. We verify from Weyl invariants that two metrics with completely different forms are the same solution. We also present the coordinates transformation that connects the two metrics.