Note on identifying four-dimensional vacuum solutions from Weyl invariants

TL;DR

Problem: identifying when two four-dimensional vacuum metrics describe the same solution under diffeomorphisms. Approach: compute Weyl invariants in the Newman-Penrose formalism for a Hopf-structured Kerr-Schild metric and for Kerr, and derive a coordinate map that shows their local equivalence. Contributions: provides explicit invariant-based evidence that the two metrics are locally diffeomorphic and furnishes an explicit transformation; demonstrates that a plane-boundary type-D solution is diffeomorphic to the Hopf metric. Significance: offers a practical, invariant-driven method to identify and connect exact GR solutions, reducing reliance on finding explicit coordinate transformations.

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.