Is energy conserved in general relativity? -- Noether’s 1st theorem with local symmetries

To answer the question in the title is rather non-trivial. In a flat   spacetime, as a consequence of Noether's 1st theorem, a time translation symmetry not only defines a corresponding energy density as the time   component of the Noether current, but also tells that a total energy given by a volume integral of the energy density is conserved. This construction of a conserved energy  does not work in general relativity, since a time translational symmetry is a part of local symmetries, general coordinate transformations, to which Noether's first theorem can not be directly applied. Recently we propose a general method to derive a conserved current associated with a global symmetry which is a part of local symmetries. We apply the method to general relativity and obtain the following answer to the question. “No, the matter energy is not conserved in general. However  there always exist a more general conserved charge associated with matters.” In my talk, I will explain how this conclusion is derived.