Thermodynamical Properties of Apparent Horizon in Warped DGP Braneworld

TL;DR

This paper derives thermodynamic properties of the apparent horizon in warped DGP braneworld models, showing how entropy expressions relate to bulk and brane geometries and the effects of intrinsic curvature.

Contribution

It provides a unified thermodynamic framework for the Friedmann equations in warped DGP models, including entropy expressions with and without intrinsic curvature terms.

Findings

Entropy on the brane includes two parts when intrinsic curvature is present.

Entropy expressions differ between bulk and brane in warped DGP models.

Friedmann equations can be rewritten as the first law of thermodynamics at the apparent horizon.

Abstract

In this paper we first obtain Friedmann equations for the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk, with intrinsic curvature term of the brane included in the action (DGP model). Then, we show that one can always rewrite the Friedmann equations in the form of the first law of thermodynamics, $dE=TdS+WdV$, at apparent horizon on the brane, regardless of whether there is the intrinsic curvature term on the brane or a cosmological constant in the bulk. Using the first law, we extract the entropy expression of the apparent horizon on the brane. We also show that in the case without the intrinsic curvature term, the entropy expressions are the same by using the apparent horizon on the brane and by using the bulk geometry. When the intrinsic curvature appears, the entropy of apparent horizon on the brane has two parts, one part follows the $n$-dimensional area formula on the brane, and the other part is the same as the entropy in the case without the intrinsic curvature term. As an interesting result, in the warped DGP model, the entropy expression in the bulk and on the brane are not the same. This is reasonable, since in this model gravity on the brane has two parts, one induced from the $(n+1)$-dimensional bulk gravity and the other due to the intrinsic curvature term on the brane.