Regge calculus: Difference between revisions
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In [[general relativity]], '''Regge calculus''' is a formalism for producing [[Simplicial manifold|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. |
In [[general relativity]], '''Regge calculus''' is a formalism for producing [[Simplicial manifold|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. |
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The starting point for Regge's work is the fact that every [[Lorentzian manifold]] admits a [[Triangulation (advanced geometry)|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 [[vertex (geometry)|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 starting point for Regge's work is the fact that every [[Lorentzian manifold]] admits a [[Triangulation (advanced geometry)|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 [[vertex (geometry)|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]]. |
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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 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. |
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Revision as of 00:58, 2 March 2008
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 Calculus has motivated the construction of further generalizations of this idea. In particular, Regge calculus has been adapted to study quantum gravity.
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 (subscribers only) at "Classical and Quantum Gravity".
- J. W. Barrett (1987). "The geometry of classical Regge calculus". Class. Quant. Grav. 4: 1565–1576. Available (subscribers only) 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.
{{cite book}}: CS1 maint: multiple names: authors list (link) See chapter 42.
- T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19: 558–571. Available (subscribers only) at Il Nuovo Cimento