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

Aron C. Wall

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.

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

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 . (When the generalized entropy is not invariant under local Lorentz boosts.) It is shown that a certain second null derivative of the entropy, , 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 . This indicates that the generalized second law should be defined using the causal horizon.

Paper Structure

This paper contains 7 sections, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gravitational Aspects
  3. The Entropy Anomaly in Flat Spacetime
  4. Conformal Transformations on the Horizon
  5. The Second Law on the Causal Horizon
  6. Other Kinds of Horizons
  7. Prospects

Figures (2)

  • Figure 1: A spacetime interval $(x,\,y)$ of proper length $r$, lying inside of a parallelogram representing its causal domain $D$. There is an infinite entanglement entropy in $D$, or equivalently on $(x,\,y)$. $r_1$ and $r_2$ are the length-scales associated with UV cutoffs. If one wishes to consider the effects of boosting the cutoff length, one can also view the cutoff as a vector with null components $u_1$ and $v_1$.
  • Figure 2: The spatial slices $ACDB$ and $ADB$ contain exactly the same information as each other. So naively one would expect that they also contain the same (renormalized) entanglement entropy $S$. But consider e.g. a left-moving chiral field, for which $c > \tilde{c} = 0$. All information travels to the left at the speed of light. After cutting off the entanglement entropy at a fixed proper distance $r_1$ from point $A$, one finds $S(ACDB) < S(ADB)$ because some of the entropy has propagated leftward past the cutoff. Since the entropy depends on the boost angle at which it is measured, it fails to be invariant under local Lorentz symmetry. The opposite sign entropy change would occur for right-moving fields. When $c = \tilde{c}$, the entropy is the same in all reference frames.