Noether currents for the Teukolsky Master Equation

TL;DR

This paper constructs conserved currents for the Teukolsky Master Equation in Kerr spacetime using three approaches, extending Noether's theorem to nonvariational equations and boundary conditions, revealing deep symmetry properties.

Contribution

It introduces three methods to derive conserved currents for the TME, including an extension of Noether's theorem to nonvariational equations and boundary conditions, enhancing understanding of symmetries in Kerr spacetime.

Findings

Conserved currents are constructed for TME using three approaches.

An extension of Noether's theorem to nonvariational equations is demonstrated.

Currents related to boundary conditions like Sommerfeld are also derived.

Abstract

Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime and with scaling symmetry are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are taken, of which the third one applies only to the spacetime symmetries. The results yielded by the three approaches, which correspond to three variants of Noether's theorem, are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spin weight. The currents thus involve two independent solutions of the TME with opposite spin weights. The first approach provides an example of the application of an extension of Noether's theorem to nonvariational differential equations. This extension is also reviewed in general form. The variant of Noether's theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained by the contraction of the symmetric energy-momentum tensor with the relevant Killing vector fields. Symmetries and conserved currents related to boundary conditions are introduced as well, and Noether's theorem and its variant for nonvariational differential equations are extended to them. The extension of the latter variant is used to construct conserved currents related to the Sommerfeld boundary condition.