Carter's case [D] admits the 2nd Canonical Form of the Killing Tensor

TL;DR

This paper explores the canonical forms of Killing tensors within vacuum solutions of Einstein's equations, revealing Carter's Case [D] as a key solution in the context of Petrov Type D spacetimes.

Contribution

It identifies all Petrov Types that admit the 2nd and 3rd canonical forms of Killing tensors in vacuum, highlighting Carter's Case [D] as a notable solution.

Findings

Carter's Case [D] admits the 2nd canonical form of the Killing tensor.

The study classifies Petrov Types compatible with these canonical forms.

Carter's solution is derived within the Newman-Penrose formalism.

Abstract

The study of the Canonical Forms of the Killing Tensor concerns the simultaneous resolving of the Integrability Conditions of the Killing Tensor along with the Einstein's Field Equations employing the framework of Newman-Penrose Formalism. We present all the Petrov Types admitting the 2nd and 3rd Canonical Forms of Killing Tensor in Vacuum in the frame of General Theory of Relativity. During the investigation of the Type D solution of 2nd Canonical form of the Killing Tensor the Carter's Case [D] solution in Vacuum emerged.