Conservation of Energy in Black Holes and in Cosmology

We first review the various definition of the total energy in the gravitational system. The naive definition has some defects, and we review how to modify the definition of the total energy. Then we explicitly demonstrate how to calculate the total energy of the system. Our example is the total energy of a black hole in the expanding closed de Sitter universe in (2+1) dimension. In general, we find that the contribution to the total energy comes only from the singularity. Then we can calculate the total energy by evaluating the contribution around the singularity