Regge calculus

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In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes which are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in the early 1960s.

The starting point for Regge's work is the fact that every Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetime curvature can be expressed in terms of deficit angles associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices where arrangements of triangles meet in a triangulation of a 2-manifold, which is easier to visualize. Here a vertex with a positive angular deficit represents a concentration of positive Gaussian curvature, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature.

The deficit angles can be computed directly from the various edge lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor can be computed from the metric tensor of a Lorentzian manifold. Regge showed that the vacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial spacelike hyperslice according to the vacuum field equation.

The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain constraint equation), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in numerical relativity such as simulating the collision of two black holes.

The elegant idea behind Regge motivates to construct a generalization. In particular, Regge calculus has been adapted to study quantum gravity.

[edit] References

Adrian P. Gentle (2002). "Regge calculus: a unique tool for numerical relativity". Gen. Rel. Grav. 34: 1701-1718. eprint

Renate Loll (1998). "Discrete approaches to quantum gravity in four dimensions". Living Rev. Relativity 1: 13. Available at "Living Reviews of Relativity". See section 3.

Ruth M. Williams, Philip A. Tuckey (1992). "Regge calculus: a brief review and bibliography". Class. Quant. Grav. 9: 1409-1422. Available at "Classical and Quantum Gravity".

J. W. Barrett (1987). "The geometry of classical Regge calculus". Class. Quant. Grav. 4: 1565-1576. Available at "Classical and Quantum Gravity".

Misner, Charles W. Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0. See chapter 42.