从广义熵重构$f(R)$引力：精确拉格朗日量

中文摘要 AI 辅助

广义视界熵被广泛用作贝肯斯坦-霍金面积定律的理论修正，且通过沃尔德构造，它们也可编码对基础引力动力学的修正。我们从给定的熵-面积关系重构度规$f(R)$引力，表明该过程本质上依赖于分支，具体取决于所需的面积-曲率映射。在最大对称分支上，满足精确关系$A=48\pi/R$，重构简化为一次单积分，且可非微扰地完成。我们得到了若干广义熵对应的闭式拉格朗日量，表明熵项$a~S_{\rm BH}^{q}$会产生与$R^{2-q}$成正比的曲率项。具体而言，卡尼亚达基斯（Kaniadakis）熵会产生$1/R$修正，而对数熵则生成$R^2\ln R$项。我们进一步推导了一个与分支无关的判据$\partial_{R}^{2} f=(ds/dR)d(S/s)/ds$，该判据将多尔戈夫-川崎（Dolgov-Kawasaki）稳定性直接与熵泛函关联，同时在最大对称分支上给出$m_{\rm sc}^2=S'(s)/(3f_{RR})$。与固定质量的史瓦西-德西特分支的比较显示，重构得到的拉格朗日量不同，且稳定性性质相反。最后，弱孤立视界的 boost 荷重现了原始广义熵。这些结果建立了广义视界热力学与修正引力动力学之间直接的非微扰联系。

英文摘要

Generalized horizon entropies are widely used as theoretical modifications of the Bekenstein-Hawking area law, but through the Wald construction they may also encode modifications of the underlying gravitational dynamics. We reconstruct metric $f(R)$ gravity from prescribed entropy-area relations and show that the procedure is intrinsically branch dependent through the required area-curvature map. On the maximally symmetric branch, where $A=48π/R$ exactly, the reconstruction reduces to a single quadrature and can be performed non-perturbatively. We obtain closed-form Lagrangians for several generalized entropies and show that an entropy term $a~S_{BH}^{q}$ generates a curvature term proportional to $R^{2-q}$. In particular, Kaniadakis entropy produces a $1/R$ correction, while logarithmic entropy generates an $R^2\ln R$ term. We further derive a branch-independent criterion, $\partial_{R}^{2} f=(ds/dR)d(S/s)/ds$, relating Dolgov-Kawasaki stability directly to the entropy functional, together with $m_{\rm sc}^2=S'(s)/(3f_{RR})$ on the maximally symmetric branch. Comparison with the fixed-mass Schwarzschild-de Sitter branch reveals different reconstructed Lagrangians and reversed stability properties. Finally, the weak-isolated-horizon boost charge reproduces the original generalized entropy. These results establish a direct non-perturbative link between generalized horizon thermodynamics and modified gravitational dynamics.

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15 pages