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

TL;DR

This work shows that the π Lagrangian in the DGP decoupling limit, though yielding second-order equations of motion, requires Gibbons-Hawking-York–type boundary terms for a well-posed variational principle in the presence of boundaries. The authors derive the necessary boundary term both from the brane-localized π action and from the bulk Einstein-Hilbert action, including corner contributions, and demonstrate that the same term naturally reproduces the π action. They then use these boundary terms to obtain Israel-like junction conditions for π across a sheet and extend the construction to general galileons, where the boundary term at order n is the (n−1)-th order galileon, with corresponding junction conditions. The results clarify boundary dynamics in brane-world and cascading DGP setups, and suggest potential connections to semiclassical gravity quantities such as entropy in the decoupling limit. The methods provide a consistent framework for handling higher-derivative but second-order field theories with boundaries in higher dimensions.

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.

Figures (1)

• Figure 1: Setup