About the Gravitational Energy

In general relativity a correct definition of gravitational energy is a controversial topic. In the linear version of Einstein’s equations, the distribution of matter and energy determines the geometry of spacetime but the evolution of matter and energy is determined by conservation equations in which there is no contribution of gravitational field since the conservation laws preserve the same form that they have in special relativity. In other words, up the first order, there is no gravitational contribution to energy. This remark suggests that a form of gravitational energy must be sought in the nonlinear formulation of general relativity. In this chapter, we face the problem of defining a gravitational energy that, together with the momentum and energy of matter and fields, supplies expressive conservation laws for an isolated distribution S of matter and energy occupying a bounded region of an asymptotically Minkovskian spacetime. In other words, we cannot define in general the content of gravitational energy of an arbitrary region of spacetime but we can introduce some pseudo-tensors $$t^{\alpha \beta }$$ t α β that have reasonable physical properties and, together with the momentum-energy tensor $$T^{\alpha \beta }$$ T α β of S, lead to global conservation laws of momentum and energy of matter and gravitational field.