Induced Curvature Singularities in Modified Theories of Gravity

Modified theories of gravity, while offering potential solutions to long-standing cosmic puzzles such as dark energy and the hierarchy problem, can introduce highly complex spacetime structures. This work investigates the emergence of “induced curvature singularities” at finite radii in static, spherically symmetric geometries within these extended theories. Unlike the event horizons of regular black holes, which are merely coordinate singularities, these induced singularities represent genuine curvature divergences where the Kretschmann scalar K tends toward infinity (K → ∞).

These divergences typically stem from non-General Relativity (non-GR) terms that are present in the field equations. In this study, we explore the precise mathematical conditions required for their formation, identifying their physical mechanisms, such as poles in effective mass functions and the appearance of branch points in the metric solutions. Our analysis further explores the profound physical implications of these structures, specifically focusing on geodesic incompleteness for both timelike and null observers, as well as the presence of unbounded tidal forces that would lead to spaghettification.

By distinguishing these true singularities from standard coordinate horizons, we emphasize the crucial role of metric function regularity and the violation of effective energy conditions. This investigation aims to provide a diagnostic tool for testing the viability of modified gravity models in the strong-field regime. Our results suggest that certain modified gravity frameworks may be physically untenable due to the unavoidable formation of these singular surfaces at finite distances from the source.