Positiveness of mass and the strong energy condition

An important equestion in general relativity is, what conditions are sufficient to guarantee that the mass of a bounded system to positive. This problem is approached for static non-vacuum systems with the help of a formula for the gravitational mass (a generalization of one given earlier by Tolman) which separates the contributions of the singularities from those of the matter fields. For a singularity-free system, if the matter fields obey the strong energy condition familiar from the 'singularity theorems', then the mass will be positive. For systems with matter not obeying the strong energy condition, a much weakened energy condition is still sufficient to guarantee positive mass. Both of the cases are illustrated with concrete examples. (author)