Self-gravitating field configurations: The role of the energy–momentum trace

Static spherically-symmetric matter distributions whose energy–momentum tensor is characterized by a non-negative trace are studied analytically within the framework of general relativity. We prove that such field configurations are necessarily highly relativistic objects. In particular, for matter fields with T≥α⋅ρ≥0T≥α⋅ρ≥0 (here T and ρ are respectively the trace of the energy–momentum tensor and the energy density of the fields, and α   is a non-negative constant), we obtain the lower bound maxr⁡{2m(r)/r}>(2+2α)/(3+2α)maxr⁡{2m(r)/r}>(2+2α)/(3+2α) on the compactness (mass-to-radius ratio) of regular field configurations. In addition, we prove that these compact objects necessarily possess (at least) two photon-spheres, one of which exhibits stable trapping of null geodesics. The presence of stable photon-spheres in the corresponding curved spacetimes indicates that these compact objects may be nonlinearly unstable. We therefore conjecture that a negative trace of the energy–momentum tensor is a necessary condition for the existence of stable, soliton-like (regular) field configurations in general relativity.