Unconstrained degrees of freedom for gravitational waves, $\beta$--foliations and spherically symmetric initial data

A new parameterization of unconstrained degrees of freedom for gravitational field, used in Classical and Quantum Gravity {\bf 11}, 1055-1068 (1994), has been generalized to one-parameter family of such parameterizations, depending on a real parameter $\beta \in [0,2]$. The description introduced in CQG 11, 1055 corresponds to the special choice $\beta =0$. The method is closely related to the proof of the positivity of the energy presented in Phys. Rev. D {\bf 36}, 1041-1044 (1987) where $\beta$-foliations have been introduced (see also applications to black holes dynamics in Classical and Quantum Gravity {\bf 6}, 1535-1539 (1989) and Acta Physica Polonica B {\bf 25}, 1413-17 (1994)). Spherically symmetric initial data corresponding to trivial degrees of freedom is analyzed along these lines. In particular, the quasi-local energy content of the Schwarzschild initial data is analyzed for different choices of the $\beta$-gauge.