Thermodynamics of Apparent Horizon in Brane World Scenario

TL;DR

This paper explores the thermodynamics of apparent horizons in brane world scenarios, deriving entropy expressions that unify different dimensional limits and confirming the consistency of horizon entropy calculations from brane and bulk perspectives.

Contribution

It provides a unified entropy formula for apparent horizons in brane worlds and demonstrates the equivalence of brane and bulk entropy calculations, linking Friedmann equations to thermodynamics.

Findings

Entropy reduces to n-dimensional area formula at large horizon radius.

Entropy obeys (n+1)-dimensional area formula at small horizon radius.

Friedmann equation can be expressed as a thermodynamic first law.

Abstract

In this paper we discuss thermodynamics of apparent horizon of an $n$-dimensional Friedmann-Robertson-Walker (FRW) universe embedded in an $(n+1)$-dimensional AdS spacetime. By using the method of unified first law, we give the explicit entropy expression of the apparent horizon of the FRW universe. In the large horizon radius limit, this entropy reduces to the $n$-dimensional area formula, while in the small horizon radius limit, it obeys the $(n+1)$-dimensional area formula. We also discuss the corresponding bulk geometry and study the apparent horizon extended into the bulk. We calculate the entropy of this apparent horizon by using the area formula of the $(n+1)$-dimensional bulk. It turns out that both methods give the same result for the apparent horizon entropy. In addition, we show that the Friedmann equation on the brane can be rewritten to a form of the first law, $dE=TdS +WdV$, at the apparent horizon.