The First Order Symmetry Operator on Gravitational Perturbations in the 5-dimensional Myers-Perry Spacetime with Equal Angular Momenta

It has been revealed that the first order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing-Yano 3-form. This might be used to construct all or part of solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers-Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.