Formal Solutions of Any-Order Mass, Angular-Momentum, and Dipole Perturbations on the Schwarzschild Background Spacetime

Abstract

Formal solutions of any-order mass, angular-momentum, dipole perturbations on the Schwarzschild background spacetime are derived in a gauge-invariant manner. Once we accept the proposal in [K. Nakamura, Class. Quantum Grav. \({\bf 38}\) (2021), 145010.], we can extend the gauge-invariant linear perturbation theory on the Schwarzschild background spacetime including the monopole (\(l=0\)) and dipole (\(l=1\)) modes to any-order perturbations of the same background spacetime through the arguments in [K. Nakamura, Class. Quantum Grav. \({\bf 31}\) (2014), 135013.]. As a result of this resolution, we reached to a simple derivation of the above formal solutions of any order.

Keywords

• general relativity
• Schwarzschild black hole
• any-order gauge-invariant perturbation
• monopole mode
• dipole mode