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.
Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity
这系列的第二部分处理自旋$\pm2$组件（或极端组件），满足Teukolsky主方程，在一个缓慢旋转的克尔黑洞外部的线性化的引力。对于这两个分量中的每一个，在执行一次和两次一阶微分算子之后，所得到的方程与Teukolsky主方程本身一起构成线性自旋加权波系统。通过对该系统的处理，证明了自旋$\pm 2$分量的能量和Morawetz估计.这是联合工作的第一步（Andersson et al. in Stability for linearized gravity on the Kerr spacetime，arXiv：1903.03859，2019），旨在解决缓慢旋转Kerr度规的线性稳定性。
