On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

TL;DR

This paper discovers a new symmetry operator for the wave equation in Kerr black hole backgrounds, linked to a conserved quantity and a Killing tensor, enhancing understanding of wave solutions in curved spacetime.

Contribution

It introduces a novel symmetry operator S that commutes with the wave operator and the time evolution generator in Kerr spacetime, revealing a new invariance property.

Findings

Derived the constant of motion associated with a Killing tensor.

Found a new symmetry operator S for wave solutions.

Showed S commutes with the wave operator and the time evolution generator.

Abstract

This short paper derives the constant of motion of a scalar field in the gravitational field of a Kerr black hole which is associated to a Killing tensor of that space-time. In addition, there is found a related new symmetry operator S for the solutions of the wave equation in that background. That operator is a partial differential operator with a leading order time derivative of the first order that commutes with a normal form of the wave operator. That form is obtained by multiplication of the wave operator from the left with the reciprocal of the coefficient function of its second order time derivative. It is shown that S induces an operator that commutes with the generator of time evolution in a formulation of the initial value problem for the wave equation in the setting of strongly continuous semigroups.