Quantum Field Theory on compact stars near the Buchdahl limit

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.