Revisiting the Possibility of a Sharp Phase Transition in Cold Neutron Stars

First-order phase transitions (FOPTs) in cold neutron stars (NSs) have been extensively studied and have provided valuable insights into the behavior of the densest matter visible in our Universe, although a strong consensus has yet to emerge. Revisiting the possibility of a hadron-quark FOPT from a new perspective, we examine the interplay between the coupled nature of gravity and microscopic interactions in Tolman–Oppenheimer–Volkoff (TOV) equations and the fundamental requirements of thermodynamic consistency in NSs. We demonstrate that a sharp FOPT manifested as a plateau in the equation of state (EOS) $P(\epsilon )$, i.e., pressure $P$ versus energy density $\epsilon$, is intrinsically incompatible with the regularity conditions of the TOV solutions. Although numerical integrations of the TOV equations with EOSs incorporating FOPTs may yield seemingly reasonable mass-radius relations consistent with current observations, such results can mask underlying inconsistencies. Our analysis thus establishes a structural consistency criterion for constraining dense-matter EOSs using NS observables, complementing existing studies of possible phase transitions in NS interiors.