Causal Entropy Bound for a Spacelike Region

The identification of a causal-connection scale motivates us to propose a new covariant bound on entropy within a generic spacelike region. This “causal entropy bound,” scaling as $\sqrt{\mathrm{EV}}$, and thus lying around the geometric mean of Bekenstein's $S/\mathrm{ER}$ and holographic $S/A$ bounds, is checked in various “critical” situations. In the case of limited gravity, Bekenstein's bound is the strongest while naive holography is the weakest. In the case of strong gravity, our bound and Bousso's holographic bound are stronger than Bekenstein's, while naive holography is too tight, and hence typically wrong.