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 $T\geq\alpha\cdot\rho\geq0$ (here $T$ and $\rho$ are respectively the trace of the energy-momentum tensor and the energy density of the fields, and $\alpha$ is a non-negative constant), we obtain the lower bound $\text{max}_r\{2m(r)/r\}>(2+2\alpha)/(3+2\alpha)$ on the compactness (mass-to-radius ratio) of regular field configurations. In addition, we prove that these compact objects necessarily possess (at least) {\it two} photon-spheres, one of which exhibits {\it 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 {\it necessary} condition for the existence of stable, soliton-like (regular) field configurations in general relativity.