Energy and angular momentum of the weak gravitational waves on the Schwarzschild background - Quasilocal gauge-invariant formulation

A four-dimensional spherically covariant gauge-invariant quasilocal framework for the perturbation of the Schwarzschild metric is given. An important ingredient of the analysis is the concept of quasilocality, which does duty for the separation of angular variables in the usual approach. A precise and full analysis for the "mono-dipole" part of the theory is presented. Direct construction (from the constraints) of the reduced canonical structure for the initial data and explicit formulae for the gauge-invariants are proposed. The reduced symplectic structure explains the origin of the axial and polar invariants. This enables one to introduce an energy and angular momentum for the gravitational waves, which is invariant with respect to the gauge transformations. An explicit expression for the energy and new proposition for angular momentum is introduced, in particular, compatibility of the Christodoulou-Klainerman S.A.F. condition with well-possedness of our functionals is checked. Both generators (energy and angular momentum) represent quadratic approximation of the ADM nonlinear formulae in terms of the perturbations of the Schwarzschild metric. The previously known results are presented in a new geometric and self-consistent way. Both degrees of freedom fulfill the generalized scalar wave equation. For the axial degree of freedom the radial part of the equation corresponds to the Regge-Wheeler result and for the polar one we get the Zerilli result.