Critical Coupling Surfaces in $\kappa(R,T)$ Gravity: Regularity, Gravitational Screening, and Phase Transitions

Abstract

We investigate the critical regime $\kappa(R,T)=0$ in $\kappa(R,T)$ gravity. While most studies assume a non-vanishing effective gravitational coupling, the existence of critical hypersurfaces where $\kappa$ vanishes is a generic feature of many admissible coupling functions. We show that the apparent singularity of the non-conservation equation is an artifact of a rewritten form of the conservation law and that the fundamental equations remain regular at $\kappa=0$. We further analyze the structure of critical hypersurfaces, derive the associated compatibility condition $(\nabla^\mu\kappa)T_{\mu\nu}=0$, and discuss their interpretation as gravitational screening surfaces separating attractive and repulsive gravitational phases. The existence of critical coupling hypersurfaces also obstructs a global Einstein-frame description, distinguishing $\kappa(R,T)$ gravity from theories based solely on algebraic redefinitions of the energy-momentum tensor. Possible cosmological and astrophysical consequences are briefly explored.