论文标题
规范合奏中五维施瓦茨柴尔德黑洞的热力学
Thermodynamics of five-dimensional Schwarzschild black holes in the canonical ensemble
论文作者
论文摘要
我们使用York的形式主义研究了规范合奏中五维Schwarzschild黑洞的热力学。在固定尺寸$ r $和固定温度$ t $的空腔内，在$πrt = 1 $的阈值中，黑洞可以处于热平衡状态。对于两个特定的黑洞，一个小小的黑色半径$ r _ {+1} $以及一个半径的大黑洞$ r _ {+2} $，可以实现此热平衡。在五个维度中，Radii $ r _ {+1} $和$ r _ {+2} $具有精确的表达式。通过路径整体形式主义和分区功能，人们获得了动作和自由能。这导致了系统的热能和熵，后者结果由Bekenstein-Hawking区域定律$ s $ s = \ frac {a _ {+}} {4} $，其中$ a _+$是黑洞的表面积。当放置热浴以等于或小于光子轨道的半径$ r $放置时，热容量为正，这意味着热力学稳定性。这意味着小黑洞不稳定，大洞是稳定的。通用的自由能用于证明经典的热平面通过$ r _ {+1} $转移以在稳定的$ r _ {+2} $上定居。值得注意的是，当腔半径等于buchdahl半径时，较大$ r _ {+2} $黑洞的自由能为零。提到了由于instanton解决方案中的路径积分中的扰动而引起的不稳定性的关系。量子热扁平空间具有负自由能，我们找到了大黑洞，量子热平面或两者都是基态的条件。显示相应的相图。使用给定能量$ e $的状态$ν$的密度，我们还发现大黑洞的熵$ r _ {+2} $。此外，我们在五维热力学与约克的四维结果之间建立了联系。
We study the thermodynamics of a five-dimensional Schwarzschild black hole in the canonical ensemble using York's formalism. Inside a cavity of fixed size $r$ and fixed temperature $T$, there is a threshold at $πr T = 1$ above which a black hole can be in thermal equilibrium. This thermal equilibrium can be achieved for two specific black holes, a small black hole of horizon radius $r_{+1}$, and a large black hole of radius $r_{+2}$. In five dimensions, the radii $r_{+1}$ and $r_{+2}$ have an exact expression. Through the path integral formalism and the partition function, one obtains the action and the free energy. This leads to the thermal energy and entropy of the system, the latter turning out to be given by the Bekenstein-Hawking area law $S = \frac{A_{+}}{4}$, where $A_+$ is the black hole's surface area. The heat capacity is positive when the heat bath is placed at a radius $r$ that is equal or less than the photonic orbit, implying thermodynamic stability. This means that the small black hole is unstable and the large one is stable. A generalized free energy is used to show that it is feasible that classical hot flat space transits through $r_{+1}$ to settle at the stable $r_{+2}$. Remarkably, the free energy of the larger $r_{+2}$ black hole is zero when the cavity radius is equal to the Buchdahl radius. The relation to the instabilities that arise due to perturbations in the path integral in the instanton solution is mentioned. Quantum hot flat space has negative free energy and we find the conditions for which the large black hole, quantum hot flat space, or both are the ground state. The corresponding phase diagram is displayed. Using the density of states $ν$ at a given energy $E$ we also find that the entropy of the large black hole $r_{+2}$. In addition, we make the connection between the five-dimensional thermodynamics and York's four-dimensional results.
