Spacetime with prescribed hidden symmetry

TL;DR

This paper explores spacetimes with specific hidden symmetries defined by Killing tensors, solving PDEs to find such metrics and analyzing their implications for geodesics and wave equations.

Contribution

It introduces a method to construct spacetimes with prescribed hidden symmetries by solving PDEs for the inverse metric involving Killing tensors.

Findings

Derived explicit metrics with higher-rank Killing tensors.

Analyzed null geodesics and photon regions in these spacetimes.

Demonstrated the feasibility of finding hidden symmetries more easily than previously thought.

Abstract

In this paper, we investigate spacetime characterized by a hidden symmetry defined by a given Killing tensor. To exhibit this hidden symmetry, the inverse metric must commute with the Killing tensor under the Schouten-Nijenhuis bracket, which translates into a system of partial differential equations (PDEs) for the inverse metric. For some significant examples, we solve these PDEs directly, deriving spacetimes with prescribed hidden symmetries, including those specified by higher-rank Killing tensors. Utilizing the hidden symmetries, we study related problems such as null geodesics, photon region, and separation of variables of wave equations. Through this work, we aim to demonstrate that hidden symmetry is more accessible than previously believed.