Hyperbolic geometry in 't Hooft's approach to (2 + 1)-dimensional gravity
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In 't Hooft's polygon model of (2 + 1)-dimensional gravity coupled to point particles, the initial data are constrained by the vertex equations and the particle equations. We show that the constraint equations correspond to a hyperbolic geometry. In particular, we derive that the hyperbolic group of motions is the discrete analogue of the diffeomorphisms in the continuum theory. The hyperbolic model can be extended to point particles, but they spoil the gauge invariance. Using this insight we calculate consistent sets of initial data for the model.
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The first discrete formulation designed explicitly for 2+1 dimensions was developed by 't Hooft et al.
“The first discrete formulation designed explicitly for 2 + 1 dimensions was developed by’ t Hooft et al. [250, 252, 253, 124, 274, 156].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [156]
The Introduction says ’t Hooft introduced a polygon approach to (2+1)-dimensional gravity, and Section 2 describes its tessellation into polygonal tiles. This supports the attribution of a discrete formulation to ’t Hooft, but the paper does not establish that it was the first such formulation. (LLM, conf=0.85)
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Hollmann, H. R., & Williams, R. M.
- Affiliations
- University of Cambridge (GB)
- ORCID
- Williams, Ruth M
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1999-01-01
- Volume
- 16
- Issue
- 5
- Pages
- 1503-1518
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1988364498
- Cited by
- 5
- References
- 0
- FWCI
- 0.26
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- three-dimensional gravity, hyperbolic geometry, constraint equations, point particles, gauge invariance, initial data
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