Chandrasekaran, Penington and Witten (CPW) used the crossed product construction in modular theory to associate an entropy with the algebra of observables in the Schwarzschild black hole exterior. This entropy was shown to equal the generalized entropy modulo a constant. They also proved a version of the generalized second law (GSL) in Einstein gravity. We summarize these developments and our generalization of these results to static black holes in an arbitrary diffeomorphism invariant theory of gravity [1] — this article is a summary of that work and is based on a talk at the QGatRRI conference in Sep.2023. The algebra entropy again equals the generalized entropy modulo a constant where the generalized entropy now contains the Wald entropy. A version of the GSL follows by employing the arguments of CPW to this case.

Chandrasekaran、Penington 和 Witten (CPW) 使用模理论中的交叉积构造将熵与史瓦西黑洞外部可观测量的代数联系起来。该熵被证明等于广义熵模常数。他们还证明了爱因斯坦引力中广义第二定律（GSL）的一个版本。我们总结了这些进展，并将这些结果推广到任意微分同胚不变引力理论中的静态黑洞 [1]——本文是该工作的总结，基于 2023 年 9 月 QGatRRI 会议上的一次演讲。代数熵再次等于广义熵模常数，其中广义熵现在包含 Wald 熵。 GSL 的一个版本采用了 CPW 对此案例的论证。