ABSTRACT
The existence of Killing vector fields ensures that one can define conserved quantities from geodesics and from the energy-momentum tensor of matter fields. This chapter discusses the notion of energy of the gravitational field and the challenges this conception poses. The existence of Killing vector fields not only allows us to identify the geometric directions of symmetry in a given spacetime, but also enables to construct quantities which are locally conserved. The chapter examines how Weyl rescalings affect diverse physical theories and under which conditions Weyl rescaling invariance can be achieved. It aims to define the energy density of the gravitational field within general relativity as the energy value per unit volume. The Newtonian gravitational energy density has always a negative value and the same holds for its integral over any spatial volume.
