Geometric Selection Rules for Singularity Formation in Modified Gravity

TL;DR

This paper explores how polynomial degeneracies of curvature invariants serve as geometric rules that influence the formation of singularities in modified gravity theories, showing that singularities are not always inevitable.

Contribution

It introduces a novel perspective on singularity formation by linking algebraic degeneracies of curvature invariants to geometric selection rules in modified gravity.

Findings

Degeneracies impose constraints on energy-momentum tensors.

Singularities are generally avoided by curvature or scalar anisotropies.

Singularity formation occurs only along specific algebraically admissible branches.

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.