Classification of Einstein spaces with Stackel metric of type (3.0)

TL;DR

The paper classifies Einstein spaces with Stackel metric type $(3.0)$ that admit a simply transitive abelian group of motions $G_3$, expressed in privileged coordinates where metric components depend on a single variable. It reduces the Einstein equations to two autonomous subsystems, solves the first to obtain three variants for the transverse metric coefficients, and then resolves the second to derive a master equation that yields four cases for the remaining sector. By combining these results, the authors enumerate ten explicit Einstein-space metrics (Variants A–C, items 1–10) and provide their closed forms, thereby completing the classification of vacuum and electrovacuum Stackel spaces of this type. The work demonstrates complete separability of the geodesic equations and advances the organization and understanding of Stackel spaces within general relativity, closing the gap on Einstein spaces of type $(3.0)$ in privileged coordinates.

Abstract

The classification of the Einstein spaces with the Stackel metric of the (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. Thus the classification of vacuum and electrovacuum Stackel spaces of all types is completed and the complete list of metrics of such spaces in privileged coordinate systems is given.