Gluing different gravitational models: f(R) case

This paper presents a comprehensive analysis of junction conditions for gluing different f ( R ) gravitational theories across a non-null hypersurface. Using the variational approach, we systematically derive the junction conditions for both general f ( R ) theories and the special case of Einstein gravity, for comparison. We demonstrate that when joining two distinct f ( R ) theories, the junction conditions require continuity of ∂ f(R)/∂ R ∂ f ( R ) / ∂ R , the extrinsic curvature Kμ ν K μ ν , while allowing for discontinuities in the Ricci Scalar R . Furthermore, we establish the equivalence between Jordan and Einstein frame formulations through careful treatment of conformal transformations; Our results reveal that different f ( R ) theories can be consistently matched provided specific relations between their functional forms and geometric quantities are satisfied at the interface.