The gravity in 2+1 dimensions can be mentioned as efficient model for better understanding of 3+1 dimensional gravity. Due to this, first we review Anti-de Sitter space which is a exact solution of Einstein's equations in three dimensions with a negative cosmological constant. BTZ black hole can be obtained by identifying some directions of AdS space. In this thesis we investigate the quantum fields theory in the BTZ black hole background. By identifications and image method , one can get the green functions in BTZ space. Finally, by use of green functions, we calculate the expectation value of energy tensor of static black holes for vacuum state.