  Very compact stars seem to be forbidden in General Relativity. While
Buchdahl's theorem sets an upper bound on compactness, further no-go results
rely on the existence of two light rings, the inner of which has been
associated to gravitational instabilities. However, little is known about the
role of quantum fields in these strong gravity regimes. Here, we consider the
particularly simple model of a constant density star and we work in the probe
approximation where the backreaction is ignored. We show that the trapping of
modes inside the star leads the renormalized stress tensor of Conformal Field
Theories to diverge faster than the classical source in the Buchdahl limit.
This leads to the violation of the Null Energy Condition around the inner light
ring. The backreaction of quantum fields in this regime therefore cannot be
ignored. This happens as the star's surface approaches the Buchdahl radius
$9GM/4$ rather than the Schwarzschild radius. The results are independent of
the details of the interactions, but contain an ambiguity associated to the
renormalization scheme.
