Generalized Israel Junction Conditions for a Fourth-Order Brane World

TL;DR

This paper develops a consistent framework for junction conditions in a fourth-order gravity theory on a brane, avoiding mathematical singularities and establishing equivalence with a reduced second-order theory.

Contribution

It introduces a novel regularization approach for junction conditions in higher-order gravity and proves their equivalence to a second-order formulation with an extra tensor field.

Findings

Regularization method for higher derivatives at the brane

Equivalent second-order theory with an extra tensor field

Unified junction conditions for different continuity assumptions

Abstract

We discuss a general fourth-order theory of gravity on the brane. In general, the formulation of the junction conditions (except for Euler characteristics such as Gauss-Bonnet term) leads to the higher powers of the delta function and requires regularization. We suggest the way to avoid such a problem by imposing the metric and its first derivative to be regular at the brane, while the second derivative to have a kink, the third derivative of the metric to have a step function discontinuity, and no sooner as the fourth derivative of the metric to give the delta function contribution to the field equations. Alternatively, we discuss the reduction of the fourth-order gravity to the second-order theory by introducing an extra tensor field. We formulate the appropriate junction conditions on the brane. We prove the equivalence of both theories. In particular, we prove the equivalence of the junction conditions with different assumptions related to the continuity of the metric along the brane.