Dimension of the isometry group in spacetimes with an invariant frame

TL;DR

This paper provides a complete set of conditions for determining the dimension of the isometry group in spacetimes with an invariant frame, using the connection tensor and curvature tensors, enabling an algorithmic approach.

Contribution

It introduces an intrinsic, deductive, explicit, and algorithmic characterization of spacetimes with invariant frames based on the connection tensor and curvature tensors.

Findings

Derived necessary and sufficient conditions for isometry group dimension

Expressed the connection tensor in terms of Weyl and Ricci tensors for specific Petrov types

Developed an implementable algorithm using xAct Mathematica packages

Abstract

The necessary and sufficient conditions for a spacetime with an invariant frame to admit a group of isometries of dimension $r$ are given in terms of the connection tensor $H$ associated with this frame. In Petrov-Bel types I, II and III, and in other spacetimes where an invariant frame algebraically defined by the curvature tensor exists, the connection tensor $H$ is given in terms of the Weyl and Ricci tensors without an explicit determination of the frame. Thus, an IDEAL (intrinsic, deductive, explicit and algorithmic) characterization of these spacetimes follows. Some examples show that this algorithm can be easily implemented on the xAct Mathematica suite of packages.