Covariant Charges in Chern-Simons AdS_3 Gravity

We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular reference to Chern-Simons AdS_3 Gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed in gr-qc/0110104 and gr-qc/0107074 to calculate the variation of conserved quantities. The problem to give a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black holes mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern-Simons AdS_3 Gravity theory and the variation of conserved quantities in General Relativity is analysed in detail.