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

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 $c\mathrm{\ne }\tilde{c}$ the generalized entropy is not invariant under local Lorentz boosts.) It is shown that a certain second null derivative of the entropy, $S_{\text{gen}}\mathrm{"}+(6\mathrm{/}c)(S_{\text{out}}^{\mathrm{\prime }}{)}^2$, 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.