A Class of Integrable Metrics

TL;DR

This paper explores the structure of four-dimensional spacetimes with specific symmetries, revealing their connection to the Carter class of solutions through integrable metrics and the Goldberg-Sachs theorem.

Contribution

It establishes a link between the most general integrable metrics with Killing vectors and the Carter family of spacetimes in Einstein manifolds.

Findings

Metrics with two Killing vectors and a rank-two Killing tensor can be parameterized by ten functions.

Special vierbein choices reduce the system, leading to principal null congruences.

In Einstein manifolds, these metrics are Petrov type D and belong to the Carter class.

Abstract

In four dimensions, the most general metric admitting two Killing vectors and a rank-two Killing tensor can be parameterized by ten arbitrary functions of a single variable. We show that picking a special vierbien, reducing the system to eight functions, implies the existence of two geodesic and share-free, null congruences, generated by two principal null directions of the Weyl tensor. Thus, if the spacetime is an Einstein manifold, the Goldberg-Sachs theorem implies it is Petrov type D, and by explicit construction, is in the Carter class. Hence, our analysis provide an straightforward connection between the most general integrable structure and the Carter family of spacetimes.