Treating Maxwell soft hair as a transition function that relates U(1) gauge fields living in the asymptotic region and near-horizon region, the U(1) gauge parameter $\varphi (x^a)$ naturally becomes a good label of those Maxwell soft hair degrees of freedom. This interpretation also builds the connection between Maxwell soft hair and U(1) edge modes living in the intermediate region, which admits a well-defined effective action description. We study the statistical properties by Euclidean path integral, which concludes that the soft hair density of state increases with black hole temperature. Hawking radiations increase black hole entropy by creating entanglements, while the measurements of soft modes project the black hole onto lower entropy states. The competition between phase spaces of Hawking radiations and soft hair measurements gives rise to one version of the Page curve consistent with the unitary evolution of the black hole.

中文翻译：

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软黑洞信息悖论：来自麦克斯韦黑洞软毛的页面曲线

将麦克斯韦软毛作为一个过渡函数，将位于渐近区和近视距区的 U(1) 规范场相关联，U(1) 规范参数 $\phi (X^{\mathrm{一种}})$自然而然地成为了那些麦克斯韦柔软的头发自由度的好标签。这种解释还建立了 Maxwell 软毛和位于中间区域的 U(1) 边缘模式之间的联系，这允许定义明确的有效动作描述。我们通过欧几里得路径积分研究了统计特性，得出的结论是软毛态密度随着黑洞温度的增加而增加。霍金辐射通过产生纠缠来增加黑洞熵，而软模式的测量将黑洞投射到低熵态。霍金辐射的相空间和软毛测量之间的竞争产生了一种与黑洞单一演化一致的佩奇曲线。