AI 中文总结
研究维度约化爱因斯坦 - 希尔伯特模型中永恒及蒸发黑洞的孤岛公式与佩奇曲线，通过洛伦兹规范确定广义熵，其与类瓦尔德诺特定理电荷一致，据此得出量子极值面、佩奇时间和佩奇曲线，并探讨向完整四维几何扩展的可能。
AI 中文摘要
我们研究了维度约化爱因斯坦 - 希尔伯特（DREH）模型中渐近平直的永恒和准静态蒸发黑洞解的孤岛公式和佩奇曲线。在此模型中，四维爱因斯坦 - 希尔伯特作用量简化为二维伸缩子引力，量子修正可通过具有大中心荷\(c\)的二维共形场论的波利亚科夫 - 刘维尔（PL）作用量纳入。该引力模型源于四维爱因斯坦 - 希尔伯特引力的\(s\)波近似，提供了一个完全引力的设定来研究孤岛。利用孤岛规则推导了DREH模型中永恒和蒸发黑洞霍金辐射的精细熵。本文通过为广义熵\(S_{\text{gen}}\)提供洛伦兹规范来补充欧几里得方法，广义熵与DREH - PL联合作用的类瓦尔德诺特定理电荷一致。利用此广义熵确定了永恒和蒸发黑洞的量子极值面、佩奇时间和佩奇曲线。最后讨论了纳入真空极化修正扩展到完整四维几何的可能性。
英文摘要
We study the island formula and Page curve for the asymptotically flat eternal and quasi-stationary evaporating black hole solutions within the dimensionally reduced Einstein-Hilbert (DREH) model. In this model, the four-dimensional Einstein-Hilbert action reduces to a two-dimensional dilaton gravity, on top of which the quantum corrections can be incorporated via the Polyakov-Liouville (PL) action for a two-dimensional conformal field theory with a large central charge $c$. This gravity model arises from the s-wave approximation of four-dimensional Einstein-Hilbert gravity and provides a fully gravitational setting in which we study the islands, i.e., we will not require a non-gravitational bath region to collect the radiation, instead it lives in the black hole spacetime itself. The fine-grained entropy of Hawking radiation in the eternal and evaporating black holes within the DREH model was derived using the island rule in \cite{djordjevic2022eternal, djordevic2025evaporating}. The island rule is based on the replica method using the Euclidean gravitational path integral. In this work, we complement the Euclidean approach by providing a Lorentzian prescription for the generalized entropy $S_{\text{gen}}$, in the island formula without invoking the replica trick to compute $S_{\text{gen}}$. The generalized entropy is shown to coincide with Wald-like Noether charge of the combined DREH-PL action. Using this generalized entropy, we determine the quantum extremal surface, the Page time, and the Page curve for the eternal and the evaporating black hole. We conclude with a discussion of the possible extension to the full four-dimensional geometry incorporating vacuum polarization corrections.
Comments23 pages + Appendix, 9 figures
