Boundary Terms and Junction Conditions for the DGP Pi-Lagrangian and Galileon

TL;DR

This paper derives boundary terms and junction conditions for the DGP Pi-Lagrangian and galileons, ensuring a well-posed variational principle and extending the understanding of boundary effects in higher-dimensional brane models.

Contribution

It introduces the necessary boundary terms for the DGP Pi-Lagrangian and galileons, generalizing the Gibbons-Hawking-York boundary terms for these higher derivative theories.

Findings

Derived boundary terms from the Pi-Lagrangian for well-posedness.

Calculated boundary terms from the bulk Einstein-Hilbert action.

Established junction conditions for Pi and galileons across sources.

Abstract

In the decoupling limit of DGP, Pi describes the brane-bending degree of freedom. It obeys second order equations of motion, yet it is governed by a higher derivative Lagrangian. We show that, analogously to the Einstein-Hilbert action for GR, the Pi-Lagrangian requires Gibbons-Hawking-York type boundary terms to render the variational principle well-posed. These terms are important if there are other boundaries present besides the DGP brane, such as in higher dimensional cascading DGP models. We derive the necessary boundary terms in two ways. First, we derive them directly from the brane-localized Pi-Lagrangian by demanding well-posedness of the action. Second, we calculate them directly from the bulk, taking into account the Gibbons-Hawking-York terms in the bulk Einstein-Hilbert action. As an application, we use the new boundary terms to derive Israel junction conditions for Pi across a sheet-like source. In addition, we calculate boundary terms and junction conditions for the galileons which generalize the DGP Pi-lagrangian, showing that the boundary term for the n-th order galileon is the (n-1)-th order galileon.