[论文解读] Thermodynamic Product Formula in Modified Gravity and Kerr-MG/CFT Correspondence

Parthapratim Pradhan|arXiv (Cornell University)|Jun 5, 2017

Cosmology and Gravitation Theories被引用 1

一句话总结

本文研究了修正引力（MOG）中内视界和外视界的热力学性质，推导出与质量相关的面积乘积和Smarr型质量公式，并建立了Kerr-MOG/CFT对偶性，其中中心荷为 $c_L = 12J$，无量纲温度为 $T_L = \frac{1}{4\pi} \frac{\alpha+2}{\sqrt{1+\alpha}}$，通过Cardy公式得到的微观熵与极端Kerr-MOG黑洞的宏观Bekenstein-Hawking熵完全一致。

ABSTRACT

We examine the thermodynamic features of \emph{inner} and outer horizons of modified gravity~(MOG) and its consequences on the holographic duality. We derive the thermodynamic product relations for this gravity. We consider both spherically symmetric solutions and axisymmetric solutions of MOG. We find that the area product formula for both cases is \emph{not} mass-independent because they depends on the ADM mass parameter while in \emph{Einstein gravity} this formula is mass-independent~(universal). We also explicitly verify the \emph{first law} which is fulfilled at the inner horizon~(IH) as well as at the outer horizon~(OH). We derive thermodynamic products and sums for this kind of gravity. We further derive the \emph{Smarr like mass formula} for this kind of black hole~(BH) in MOG. Moreover, we derive the area bound for both the horizons. Furthermore, we show that the central charges of the left and right moving sectors are the same via universal thermodynamic relations. We also discuss the most important result of the \emph{Kerr-MOG/CFT correspondence}. We derive the central charges for Kerr-MOG BH which is $c_{L}=12J$ and it is similar to Kerr BH. We also derive the dimensionless temperature of a extreme Kerr-MOG BH which is $T_{L} = \frac{1}{4\pi} \frac{\alpha+2}{\sqrt{1+\alpha}}$, where $\alpha$ is a MOG parameter. This is actually dual CFT temperature of the Frolov-Thorne thermal vacuum state. In the limit $\alpha=0$, we find the dimensionless temperature of Kerr BH. Consequently, Cardy formula gives us microscopic entropy for extreme Kerr-MOG BH, $S_{micro} = \frac{\alpha+2}{\sqrt{1+\alpha}} \pi J $ for the CFT which is completely in agreement with macroscopic Bekenstein-Hawking entropy.

研究动机与目标

• 分析修正引力（MOG）中球对称与轴对称解的内视界和外视界的热力学行为。
• 确定面积乘积公式是否如在爱因斯坦引力中一样保持质量无关（普遍性）。
• 验证MOG中内视界与外视界处热力学第一定律的有效性。
• 为MOG黑洞的视界推导出Smarr型质量公式与面积界限。
• 建立Kerr-MOG/CFT对偶性，计算双侧CFT的中心荷与温度。

提出的方法

• 推导MOG中球对称与轴对称解的热力学乘积与和关系。
• 将热力学第一定律应用于内视界与外视界，确认其在MOG中的有效性。
• 利用普遍热力学关系，计算双侧CFT中左、右传播部分的中心荷。
• 利用视界热力学推导MOG黑洞的Smarr型质量公式。
• 利用Frolov-Thorne热真空态计算极端Kerr-MOG黑洞的无量纲温度。
• 将Cardy公式应用于双侧CFT，计算微观熵并与宏观Bekenstein-Hawking熵进行比较。

实验结果

研究问题

• RQ1 MOG中内视界与外视界面积乘积是否如在爱因斯坦引力中一样保持质量无关？
• RQ2 MOG黑洞中内视界与外视界是否均满足热力学第一定律？
• RQ3 Kerr-MOG黑洞双侧CFT中左、右传播部分的中心荷为何值？
• RQ4 极端Kerr-MOG黑洞的无量纲温度为何值？当 $\alpha = 0$ 时，其是否退化为Kerr黑洞的温度？
• RQ5 通过Cardy公式推导出的微观熵是否与极端Kerr-MOG黑洞的宏观Bekenstein-Hawking熵一致？

主要发现

• 对于球对称与轴对称MOG黑洞，面积乘积并非质量无关，与爱因斯坦引力不同，表明在修正引力中普遍性被破坏。
• MOG中内视界与外视界均满足热力学第一定律，证实了热力学的一致性。
• 双侧CFT中左、右传播部分的中心荷相等，为 $c_L = 12J$，与Kerr黑洞的值一致。
• 极端Kerr-MOG黑洞的无量纲温度为 $T_L = \frac{1}{4\pi} \frac{\alpha+2}{\sqrt{1+\alpha}}$，当 $\alpha = 0$ 时退化为Kerr黑洞的温度。
• 通过Cardy公式推导出的微观熵为 $S_{\text{micro}} = \frac{\alpha+2}{\sqrt{1+\alpha}} \pi J$，与宏观Bekenstein-Hawking熵完全匹配。
• 微观熵与宏观熵的一致性证实了Kerr-MOG/CFT对偶性的有效性，并支持了修正引力中黑洞熵的统计起源。