Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon

TL;DR

This paper demonstrates that in 1+1 dimensional conformal vacua, the generalized entropy increases along causal horizons, supporting the generalized second law when using the causal horizon as the defining boundary.

Contribution

It shows that a specific second null derivative of the generalized entropy remains sign-invariant under conformal transformations, establishing the causal horizon as the proper boundary for the second law.

Findings

Generalized entropy increases on causal horizons in conformal vacua.

Alternative horizon definitions can have decreasing entropy, but causal horizons do not.

The second null derivative of entropy is conformally invariant and sign-preserving.

Abstract

The anomalous conformal transformation law of the generalized entropy is found for dilaton gravity coupled to a 1+1 conformal matter sector with central charges $c = \tilde{c}$. (When $c \ne \tilde{c}$ the generalized entropy is not invariant under local Lorentz boosts.) It is shown that a certain second null derivative of the entropy, $S_\text{gen}" + (6/c)(S_\text{out}')^2$, is primary, and therefore retains its sign under a general conformal transformation. Consequently all conformal vacua have increasing entropy on causal horizons. Alternative definitions of the horizon, including apparent or dynamical horizons, can have decreasing entropy in any dimension $D \ge 2$. This indicates that the generalized second law should be defined using the causal horizon.