  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.
