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# Regge calculus

In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that 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 calculus has motivated the construction of further generalizations of this idea. In particular, Regge calculus has been adapted to study quantum gravity.

See also

Euclidean quantum gravity

Piecewise linear manifold

Euclidean simplex

Path integral formulation

Lattice gauge theory

Wheeler-DeWitt equation

Mathematics of general relativity

Causal dynamical triangulation

Ricci calculus

References

Tullio E. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571. doi:10.1007/BF02733251. Available (subscribers only) at Il Nuovo Cimento

John Archibald Wheeler (1965). "Geometrodynamics and the Issue of the Final State, in "Relativity Groups and Topology"". Les Houches Lecture Notes 1963, Gordon and Breach.

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

Herbert W. Hamber (2009). Quantum Gravitation - The Feynman Path Integral Approach. Springer Publishing. doi:10.1007/978-3-540-85293-3. ISBN 978-3-540-85292-6. Chapters 4 and 6. [1] [2]

James B. Hartle (1985). "Simplicial MiniSuperSpace I. General Discussion". Jour. Math. Physics 26: 804–812. Bibcode:1985JMP....26..804H. doi:10.1063/1.526571.

Ruth M. Williams and Philip A. Tuckey (1992). "Regge calculus: a brief review and bibliography". Class. Quant. Grav. 9 (5): 1409–1422. Bibcode:1992CQGra...9.1409W. doi:10.1088/0264-9381/9/5/021. Available (subscribers only) at "Classical and Quantum Gravity".

Tullio E. Regge and Ruth M. Williams (2000). "Discrete Structures in Gravity". Jour. Math. Phys. 41: 3964–3984. arXiv:gr-qc/0012035. Bibcode:2000JMP....41.3964R. doi:10.1063/1.533333. Available at [3].

Herbert W. Hamber (1984). "Simplicial Quantum Gravity, in the Les Houches Summer School on Critical Phenomena, Random Systems and Gauge Theories, Session XLIII". North Holland Elsevier. pp. 375–439. [4]

Adrian P. Gentle (2002). "Regge calculus: a unique tool for numerical relativity". Gen. Rel. Grav. 34 (10): 1701–1718. doi:10.1023/A:1020128425143. eprint

Renate Loll (1998). "Discrete approaches to quantum gravity in four dimensions". Living Rev. Relativity 1: 13. arXiv:gr-qc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13. Available at "Living Reviews of Relativity". See section 3.

J. W. Barrett (1987). "The geometry of classical Regge calculus". Class. Quant. Grav. 4 (6): 1565–1576. Bibcode:1987CQGra...4.1565B. doi:10.1088/0264-9381/4/6/015. Available (subscribers only) at "Classical and Quantum Gravity".

External links

Regge calculus on ScienceWorld

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