Gluing different gravitational models: $f(R)$ case

TL;DR

This paper tackles how to join two distinct $f(R)$ gravity theories across a non-null hypersurface. Using a variational framework, it introduces an auxiliary field $\Phi = \partial f/\partial R$ (with Legendre transform $U(\Phi)$) to derive generalized junction conditions and analyzes both Jordan and Einstein frames to ensure boundary consistency. The main results include a tensor junction condition involving jumps in $K_{\mu\nu}$ weighted by $\Phi$ and jumps in $\nabla_\alpha \Phi$, the possibility of a Ricci scalar discontinuity across the interface, and the demonstration that the two frames are dynamically equivalent under conformal maps when the fields are properly related. A key physical takeaway is that continuity of $\partial f/\partial R$ (rather than $R$ itself) is required for consistent matching, enabling the construction of composite spacetimes with adjacent gravitational theories.

Abstract

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 $\partial f(R)/\partial R$, the extrinsic curvature $K_{μν}$, 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.