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General relativity in the radial gauge: Reduced phase space and canonical structure

Firstly, we present a reformulation of the standard canonical approach to spherically symmetric systems in which the radial gauge is imposed. This is done via the gauge unfixing techniques, which serves as their exposition in the context of the radial gauge. Secondly, we apply the same techniques to the full theory, without assuming spherical symmetry, resulting in a reduced phase space description of General Relativity. The canonical structure of the theory is analysed. In a companion paper a quantization of the reduced phase space is presented. The construction is well suited for the treatment of spherically symmetric situations and allows for a quantum definition thereof.