Geometric Selection Rules for Singularity Formation in Modified Gravity

TL;DR

This work shows that polynomial degeneracies among curvature invariants, encoded as syzygies of the Ricci and Weyl tensors, impose algebraic constraints that restrict singularity formation in modified gravity. By translating these geometric identities into constraints on the effective stress-energy tensor for metric $f(R)$ gravity and scalar-tensor theories, the authors demonstrate that curvature- and scalar-induced anisotropies generally obstruct the vanishing conditions required for singularity development, unless the matter content or couplings are finely tuned to restore isotropy. In Brans-Dicke and generalized Brans-Dicke theories, a running coupling or chameleon-like interactions can screen anisotropy and allow singular end-states only within a restricted class of configurations, with GR recovered in the appropriate limit. The analysis, tied to the Raychaudhuri equation, provides a geometric pre-selection principle for singularities, suggesting that in many realistic modified gravity models, singularities are not generic outcomes of collapse but are constrained to algebraically admissible branches dictated by underlying curvature invariants.

Abstract

We argue that the polynomial degeneracies of curvature invariants can act as geometric selection rules for spacetime singularities in modified theories of gravity. The degeneracies arise purely from the algebraic structure of Riemannian geometry and impose non-trivial constraints on the effective energy-momentum tensor. We derive these constraints for metric $f(R)$ gravity and a wide class of scalar-tensor theories to show that a singularity formation is generally occluded by curvature and/or scalar-induced anisotropies. Therefore, formation of a singularity in modified theories of gravity is not always a generic outcome but can occur only along algebraically admissible branches selected by Riemannian curvature invariants.