Energy conditions in modified gravity are commonly imposed on an “effective” stress tensor that mixes physical matter with additional gravitational degrees of freedom, obscuring which sector actually controls null focusing, horizon multiplicity, and the disappearance of trapped regions. We develop a geometry-first alternative in spherical symmetry. For a normalized pair of future-directed radial null vectors, two longitudinal Ricci projections and one independent transverse Einstein projection obey exact area-weighted transport laws. These imply a trapping-persistence theorem: after a nondegenerate future-outer marginal sphere, a regular ingoing generator cannot exit the trapped region while the ingoing longitudinal and transverse convergence channels remain nonnegative. In metric $f(R)$ gravity, with $F\equiv df/dR>0$, we then construct an entropy-resolved hierarchy based on the local Wald-density variable $F{r}^2$. In the longitudinal entropy-focusing law, the scalaron second derivative cancels exactly, leaving a nonpositive scalar-gradient square together with the physical matter projection; in the independent transverse entropy-transport law, all scalaron derivative terms cancel. Entropy trapping is consequently governed by the physical matter condition $T_{nn}\ge 0$ and an algebraic mixed threshold rather than by a fictitious effective fluid. The resulting hierarchy cleanly separates matter positivity, geometric focusing, marginal-surface recovery, and gravitational-entropy evolution, and identifies the precise local reversals required for regular inner horizons, throats, and trapped-to-antitrapped transitions.

Close

PrevNext

View figure in article

• Alternative gravity theories
• General relativity
• Gravitation