[论文解读] On Thermodynamics and Phase Space of Near Horizon Extremal Geometries

Kamal Hajian|arXiv (Cornell University)|Aug 14, 2015

Black Holes and Theoretical Physics参考文献 57被引用 14

一句话总结

该论文建立了近视界极端几何（NHEGs）的热力学定律与经典相空间结构，NHEGs是具有SL(2,R)×U(1)^n等距对称性的极端黑洞的近视界极限。通过协变相空间方法，推导出辛结构与守恒荷，其构成一个无限维代数——'NHEG代数'，其中心荷等于黑洞熵，从而为极端黑洞熵提供了微观统计解释。

ABSTRACT

Near Horizon Extremal Geometries (NHEG), are geometries which may appear in the near horizon region of the extremal black holes. These geometries have $SL(2,\mathbb{R})\! imes\!U(1)^n$ isometry, and constitute a family of solutions to the theory under consideration. In the first part of this report, their thermodynamic properties are reviewed, and their three universal laws are derived. In addition, at the end of the first part, the role of these laws in black hole thermodynamics is presented. In the second part of this thesis, we review building their classical phase space in the Einstein-Hilbert theory. The elements in the NHEG phase space manifold are built by appropriately chosen coordinate transformations of the original metric. These coordinate transformations are generated by some vector fields, dubbed "symplectic symmetry generators." To fully specify the phase space, we also need to identify the symplectic structure. In order to fix the symplectic structure, we use the formulation of Covariant Phase Space method. The symplectic structure has two parts, the Lee-Wald term and a boundary contribution. The latter is fixed requiring on-shell vanishing of the symplectic current, which guarantees the conservation and integrability of the symplectic structure, and leads to the new concept of "symplectic symmetry." Given the symplectic structure, we construct the corresponding conserved charges, the "symplectic symmetry generators." We also specify the explicit expression of the charges as a functional over the phase space. These symmetry generators constitute the "NHEG algebra," which is an infinite dimensional algebra (may be viewed as a generalized Virasoro), and admits a central extension which is equal to the black hole entropy.

研究动机与目标

• 推导近视界极端几何（NHEGs）的三条普适热力学定律，将黑洞热力学推广至极端视界。
• 在爱因斯坦-希尔伯特引力中，利用由辛对称性生成元生成的微分同胚，构建NHEGs的经典相空间。
• 通过协变相空间方法确定辛结构，通过在壳条件下辛流为零，确保辛结构的守恒性与可积性。
• 将守恒荷识别为哈密顿生成元，并证明其构成NHEG代数，即一个具有中心扩张的无限维代数。
• 证明NHEG代数的中心荷精确等于相应极端黑洞的贝肯斯坦-霍金熵。

提出的方法

• 采用协变相空间方法，推导NHEG相空间的辛结构，将其分解为体项Lee-Wald项与边界贡献项。
• 通过要求辛流在壳条件下为零，固定边界贡献，从而确保辛结构的守恒性与可积性。
• 将辛对称性生成元定义为生成保持辛结构的微分同胚的向量场，从而引入'辛对称性'概念。
• 将守恒荷构造为相空间上的泛函，对应于这些生成元，并证明其在泊松括号下封闭形成NHEG代数。
• 将NHEG代数识别为具有中心扩张的广义维拉索罗代数，并显式计算其结果与黑洞熵一致。
• 通过度量的参数变分，将哈密顿生成元与温度、化学势等热力学变量联系起来。

实验结果

研究问题

• RQ1 黑洞热力学的三条普适定律如何推广至近视界极端几何？
• RQ2 在爱因斯坦-希尔伯特引力中，NHEGs的经典相空间结构是怎样的？
• RQ3 NHEG相空间的辛结构如何确定？在壳条件下辛流为零起到什么作用？
• RQ4 守恒荷（辛对称性生成元）的代数结构是什么？其与黑洞熵有何关联？
• RQ5 NHEG代数的中心荷能否被识别为极端黑洞的贝肯斯坦-霍金熵？

主要发现

• 为NHEGs推导出三条普适热力学定律，其中第三定律反映了极端性条件。
• NHEG相空间的辛结构由体项Lee-Wald项与边界贡献项组成，后者通过要求在壳条件下辛流为零而被固定。
• 与辛对称性生成元相关的守恒荷构成NHEG代数，这是一个与广义维拉索罗代数同构的无限维代数。
• NHEG代数的中心荷被显式计算，结果等于相应极端黑洞的贝肯斯坦-霍金熵。
• 极端黑洞的熵在相空间中被实现为哈密顿生成元，证实了其热力学意义。
• 该构造通过NHEG代数为极端黑洞熵提供了微观统计解释，支持了Kerr/CFT对应框架。