GENERAL RELATIVITY, GRAVITATIONAL ENERGY AND SPIN–TWO FIELD

In my lectures I will deal with three seemingly unrelated problems: i) to what extent is general relativity exceptional among metric gravity theories? ii) is it possible to define gravitational energy density applying field–theory approach to gravity? and iii) can a consistent theory of a gravitationally interacting spin–two field be developed at all? The connecting link to them is the concept of a fundamental classical spin–2 field. A linear spin–2 field introduced as a small perturbation of a Ricci–flat spacetime metric, is gauge invariant while its energy–momentum is gauge dependent. Furthermore, when coupled to gravity, the field reveals insurmountable inconsistencies in the resulting equations of motion. After discussing the inconsistencies of any coupling of the linear spin–2 field to gravity, I exhibit the origin of the fact that a gauge invariant field has the variational metric stress tensor which is gauge dependent. I give a general theorem explaining under what conditions a symmetry of a field Lagrangian becomes also the symmetry of the variational stress tensor. It is a conclusion of the theorem that any attempt to define gravitational energy density in the framework of a field theory of gravity must fail. Finally I make a very brief introduction to basic concepts of how a certain kind of a necessarily nonlinear spin–2 field arises in a natural way from vacuum nonlinear metric gravity theories (Lagrangian being any scalar function of Ricci tensor). This specific spin–2 field consistently interacts gravitationally and the theory of the field is promising.