arXiv 2207.092712022-07gr-qchep-th

Barrow熵如何修正引力--兼评Tsallis熵

How Barrow Entropy Modifies Gravity: With Comments on Tsallis Entropy

Sofia Di Gennaro, Hao Xu, Yen Chin Ong

备注：: 6 pages: added comparisons with the Tsallis entropy case and discussions on the Bekenstein bound; references updated; version accepted by EPJC
DOI：: 10.1140/epjc/s10052-022-11040-2

摘要：

巴罗提出视界熵的面积定律可能会因为量子引力效应而接受“分形修正”$S\propto A^{1+Δ/2}$，用$0\leqslant Δ\leqslant 1$来度量与标准面积定律的偏差。虽然这种修正在文献中已经被广泛研究，但其相应的引力理论尚未被讨论。我们遵循Jacobson的方法导出了修正的引力理论（被解释为有效理论），发现在静止情况下，所得到的理论与广义相对论的区别仅在于一个重新标度的宇宙学常数。因此，在渐近平坦的静止时空中，这个理论与广义相对论是相同的。当没有视界时，该理论不适用;多视界的情况是复杂的。我们强调的重要性，确定正确的热力学质量的理论与修改热力学，以避免不一致。对有效理论之外的Hawking蒸发率也作了评述。此外，我们表明，贝肯斯坦界是满意的，如果热力学质量被用作能量，直到一个常数的前因子。我们也简要地评论一下Tsallis熵的情况。有趣的是，对于后者，贝肯斯坦边界成立的要求对非广延参数$δ> 1/2$施加了一个下限，不幸的是，这排除了先前提出的宇宙膨胀可以单独使用正常物质场加速的可能性。

英文摘要：

Barrow proposed that the area law of the horizon entropy might receive a "fractal correction" $S\propto A^{1+Δ/2}$ due to quantum gravitational effects, with $0\leqslant Δ\leqslant 1$ measures the deviation from the standard area law. While such a modification has been widely studied in the literature, its corresponding theory of gravity has not been discussed. We follow Jacobson's approach to derive the modified gravity theory (interpreted as an effective theory), and find that in the stationary case the resulting theory only differs from general relativity by a re-scaled cosmological constant. Consequently in asymptotically flat stationary spacetimes the theory is identical to general relativity. The theory is not applicable when there is no horizon; the multi-horizon case is complicated. We emphasize on the importance of identifying the correct thermodynamic mass in a theory with modified thermodynamics to avoid inconsistencies. We also comment on the Hawking evaporation rate beyond the effective theory. In addition, we show that the Bekenstein bound is satisfied if the thermodynamic mass is used as the energy, up to a constant prefactor. We briefly comment on the Tsallis entropy case as well. Interestingly, for the latter, the requirement that Bekenstein bound holds imposes a lower bound on the non-extensive parameter: $δ> 1/2$, which unfortunately rules out the previously suggested possibility that the expansion of the universe can accelerate with normal matter field alone.

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