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

TL;DR

This paper derives junction conditions for connecting different $f(R)$ gravitational theories across a hypersurface, highlighting the continuity requirements and frame equivalence, thus enabling consistent matching of diverse $f(R)$ models.

Contribution

It systematically establishes junction conditions for arbitrary $f(R)$ theories and demonstrates their equivalence in Jordan and Einstein frames through conformal transformations.

Findings

Continuity of $rac{ ext{d}f(R)}{ ext{d}R}$ and extrinsic curvature is required.

Discontinuities in Ricci Scalar $R$ are permitted at the junction.

Frame transformations preserve the matching conditions under specific relations.

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 $K_{\mu \nu}$, 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.