# Black Hole Entropy, Constraints, and Symmetry in Quantum Gravity

Open Access

- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 22, 2006
- Committee Members:
- Abhay Vasant Ashtekar, Committee Chair
- Martin Bojowald, Committee Member
- Benjamin Owen, Committee Member
- Luen Chau Li, Committee Member

- Keywords:
- loop quantum gravity
- black hole entropy
- symmetry reduction

- Abstract:
- This thesis addresses two major problems in the area of quantum gravity. The first regards an extension of the statistical mechanical derivation of the Bekenstein-Hawking entropy from loop quantum gravity in [12]. Let us review what was accomplished in [12]. In [12], equilibrium black holes were modeled by isolated horizons. We recall the following terminology: When the intrinsic geometry of an isolated horizon is spherically symmetric, we call it ``type I'. When the intrinsic geometry of an isolated horizon is only axisymmetric, we call it ``type II'. The quantum geometry and entropy of extit{type I} isolated horizons were investigated in [12]. The first part of this thesis generalizes the investigations in [12] to the type II case, thereby encompassing distortions and rotations of the horizon. In particular, the leading term in the entropy of large horizons is again given by one fourth of the horizon area, using the extit{same} value of the Barbero-Immirzi parameter as was used in the type I case. The second problem addressed in this thesis regards how to define `symmetric state' in quantum gravity (as well as quantum field theory more generally). The question is approached via a discussion on the relation between symmetry reduction before and after quantization of a field theory. A toy model field theory is used: the axisymmetric Klein-Gordon field. We consider three possible notions of symmetry at the quantum level: invariance under the group action, and two notions derived from imposing symmetry as a system of constraints a la Dirac. One of the latter two turns out to be the most appropriate notion of symmetry in the sense that it satisfies a number of physical criteria, including the commutativity of quantization and symmetry reduction. Somewhat surprisingly, the requirement of invariance under the symmetry group action is <i>not</i> appropriate for this purpose. A generalization of the physically selected notion of symmetry to loop quantum gravity is presented and briefly discussed.