Black hole enthalpy and an entropy inequality for the thermodynamic volume
Black hole enthalpy and an entropy inequality for the thermodynamic volume
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DOI：
10.1103/physrevd.84.024037
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发表时间：
2010-12
影响因子：
5
通讯作者：
M. Cvetič;G. Gibbons;D. Kubizňák;C. Pope
中科院分区：
文献类型：
--
作者：
M. Cvetič;G. Gibbons;D. Kubizňák;C. Pope
在宇宙学常数或规范耦合常数作为真空期望值的理论中，其变化应包含在黑洞热力学第一定律中。这变成了$de=Tds+\Omega_i Dj_i+\Phi_\Alpha d Q_\Alpha+\Theta d\Lambda$，其中$E$现在是时空的热焓，而$\Lambda$的热力学共轭$\Theta$与事件视界内的有效体积$V=-\FRAC{16\pi\Theta}{D-2}$‘成正比。在这里，我们使用第一定律或Smarr关系，计算了各种不同的$D$维带电旋转渐近Add黑洞时空的$\Theta$和$V$。我们将我们的表达式与Kastor，Ray和Traschen的一个建议进行了比较，该建议涉及Komar积分和Killing势，这些建议是由共形KillingYano张量构造的。我们猜想体积$V$和视界面积$A$满足等式$R\EQUV((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\，({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$，其中${\cal A}_{D-2}$是单位为$(D-2)$-球体的体积，我们证明了这对许多黑洞都成立，并且对Schwarzschild-ads是饱和的。有趣的是，这个不等式是欧几里得$(D-1)$空间中体积$V$的等周不等式的‘逆’，欧几里得$(D-1)$空间由面积为$A$的曲面所界定，对于面积为$R\le 1的曲面，$R\le1$。我们的猜想可以解释为：对于Schwarzschild-ADS，给定‘体积’$V$的视界内的熵是最大化的。热力学定义$V$需要一个宇宙常数(或规范耦合常数)。然而，除7维外，在$\Lambda$或$g$为零的地方存在光滑极限，即使对于渐近平坦的黑洞，也提供了$V$的定义。
In a theory where the cosmological constant $\Lambda$ or the gauge coupling constant $g$ arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically AdS black hole spacetimes, using the first law or the Smarr relation. We compare our expressions with those obtained by implementing a suggestion of Kastor, Ray and Traschen, involving Komar integrals and Killing potentials, which we construct from conformal Killing-Yano tensors. We conjecture that the volume $V$ and the horizon area $A$ satisfy the inequality $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ is the volume of the unit $(D-2)$-sphere, and we show that this is obeyed for a wide variety of black holes, and saturated for Schwarzschild-AdS. Intriguingly, this inequality is the 'inverse' of the isoperimetric inequality for a volume $V$ in Euclidean $(D-1)$ space bounded by a surface of area $A$, for which $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.