One gravitational function across matter phases: compatibility and interface constraints in static $f(R,T)$ gravity

Abstract

Conservative $f(R,T)=R+h(T)$ gravity fixes $h$ by demanding that the matter stress tensor be conserved, and the reconstruction is normally carried out for one equation of state at a time. We ask whether one function $h(T)$ can serve two matter phases separated by a sharp transition. The analysis is static and local: spherically symmetric backgrounds, the ${\mathcal{L}}_m=\rho$ closure, cold barotropic perfect fluids, and a Maxwell interface without a surface layer. A common $h$ must satisfy two conditions that single-phase reconstructions do not impose. It must solve both reconstruction equations at every trace value occupied by both phases, and it must make the effective normal pressure continuous across the interface; equality of material pressure and chemical potential does not secure the latter. For a hadronic polytrope ($K>0$, $\Gamma >1$) and an affine quark equation of state, an open overlap of the occupied trace ranges forces $h$ to be constant on both. When the ranges are separated, both conditions can be met by a nonconstant $C^2$ function, which we construct explicitly at a single interface. Two further results bound this local construction. If the same polytrope extends to zero density, regularity of $h$ at $T=0$ removes its nonconstant part there; and wherever $h^{\prime }(T)\ne 0$, separate conservation forbids any change in the density of a fluid element.

arXiv comment18 pages, 3 figures. Prepared for submission to JCAP