ABSTRACT
The solution of the equations describing a physical problem can be considerably simplified if the latter has some symmetry. A physical quantity is symmetric if it is invariant with respect to some transformation. In classical mechanics energy is conserved when the Hamiltonian is independent of time; thus, the conservation of energy is associated to a symmetry with respect to time translations. This chapter shows that if a spacetime admits a timelike Killing vector field, with a suitable choice of coordinates the metric tensor can be made time independent; in this coordinate frame, the diffeomorphism along the Killing vector field is the time translation. The existence of a hypersurface-orthogonal vector field allows us to choose a coordinate frame such that the metric has a much simpler form.
