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Hyperbolic geometry in ’t Hooft’s approach to \((2+1)\)-dimensional gravity. (English) Zbl 0942.83049

’t Hooft is the 1999 Nobel price winner in physics of \((2+1)\)-dimensional gravity.
Author’s summary: In ’t Hooft’s polygon model [G. ’t Hooft, Causality in \((2+1)\)-dimensional gravity, Classical. Quantum Gravity 9, 1335-1348 (1992)] 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.

MSC:

83C80 Analogues of general relativity in lower dimensions
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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