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 the 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.

中文翻译：

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具有相等角动量的 5D Myers-Perry 时空中引力扰动的一阶对称算子

已经揭示了真空时空上线性化爱因斯坦方程的一阶对称算子可以由 Killing-Yano 3-形式构建。这可用于构造场方程的全部或部分解。在本文中，我们对 Schwarzschild 时空和 Myers-Perry 时空的 5 维角动量相等的度量扰动进行了模态分解，并具体研究了对称算子对特定模态的作用。我们表明，在这样的时空中，在对称算子的作用下，度量微扰的模式之间没有过渡，它最终是等距的无穷小变换的线性组合。