On Thermodynamics and Phase Space of Near Horizon Extremal Geometries

Abstract

Near Horizon Extremal Geometries (NHEG), are geometries which may appear in the near horizon region of the extremal black holes. These geometries have $SL(2,\mathbb{R})\times U(1{)}^n$ 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.

Ph.D thesis, a pedagogical review of NHEG dynamics and phase space, link to the Persian version is included, effectively 103 pages, 17 figures

References in corpus (7)

• Bekenstein-Hawking Entropy and Strange Metals
• Near-horizon symmetries of extremal black holes
• CFT Duals for Extreme Black Holes
• No Dynamics in the Extremal Kerr Throat
• Nonperturbative black hole entropy and Kloosterman sums
• Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes
• Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra

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• First law of black hole thermodynamics and Smarr formula with a cosmological constant
• Source and Response Soft Charges for Maxwell Theory on $Ad{S}_d$
• Topics in Asymptotic Symmetries and Infrared Effects
• Conserved charge of a gravity theory with $p$-form gauge fields and its property under Kaluza-Klein reduction
• Near-Horizon Extremal Geometries: Coadjoint Orbits and Quantization
• Generalized Off-Shell ADT Conserved Charges in the Presence of Matter Chern-Simons Term