Restrictions on possible forms of classical matter fields carrying no energy

It is postulated in general relativity that the matter energy-momentum tensor vanishes if and only if all the matter fields vanish. In classical Lagrangian field theory the energy and momentum density are described by the variational (symmetric) energy-momentum tensor (named the stress tensor) and a priori it might occur that for some systems the tensor is identically to zero for all field configurations whereas evolution of the system is subject to deterministic Lagrange equations of motion. Such a system would not generate its own gravitational field. To check if these systems can exist in the framework of classical field theory we find a relationship between the stress tensor and the Euler operator (i.e. the Lagrange field equations). We prove that if a system of interacting scalar fields (the number of fields cannot exceed the spacetime dimension d) or a single vector field (in spacetimes with d even) has the stress tensor such that its divergence is identically zero (i.e. “on and off shell”), then the Lagrange equations of motion hold identically too. These systems have then no propagation equations at all and should be regarded as unphysical. Thus nontrivial field equations require the stress tensor be nontrivial too. This relationship between vanishing (of divergence) of the stress tensor and of the Euler operator breaks down if the number of fields is greater than d. We show on concrete examples that a system of n>d interacting scalars or two interacting vector fields can have the stress tensor equal identically to zero while their propagation equations are nontrivial. This means that non-self-gravitating (and yet detectable) field systems are in principle admissible. Their equations of motion are, however, in some sense degenerate. We also show, that for a system of arbitrary number of interacting scalar fields or for a single vector field (in some specific spacetimes in the latter case), if the stress tensor is not identically zero, then it cannot vanish for all solutions. There do exist solutions with nonzero energy density and the system back-reacts on the spacetime.