| Revision as of 10:31, 21 May 2005 editRussBlau (talk | contribs)7,732 editsm sort stub← Previous edit | Revision as of 21:07, 29 May 2005 edit undoHillman (talk | contribs)11,881 edits Improved intuitive explanation, added references and linksNext edit → | ||
| Line 1: | Line 1: | ||
| In ], '''Regge calculus''' is a |
In ], '''Regge calculus''' is a formalism for producing ] of spacetimes which are solutions to the ]. The calculus was introduced by the Italian theoretician ] in the early 1960s. | ||
| The starting point for Regge's work is the fact that every ] admits a ] into ]. Furthermore, the ] ] can be expressed in terms of ] associated with ''2-faces'' where arrangements of ''4-simplices'' meet. These 2-faces play the same role as the ] 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'' ], whereas a vertex with a negative angular deficit angle represents a concentration of ''negative'' ]. | |||
| The deficit angles can be computed directly from the various ] lengths in the triangulation, which is equivalent to saying that the ] can be computed from the ] of a Lorentzian manifold. Regge showed that the ] can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial ] according to the vacuum field equation. | |||
| The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain ]), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in ] such as simulating the collision of two ]. | |||
| The elegant idea behind Regge calculus virtually cries out for generalization. In particular, Regge calculus been adapted to study quantum gravity. | |||
| ==Reference== | |||
| * {{Journal reference | Author=Renate Loll | Title=Discrete approaches to quantum gravity in four dimensions | Journal=Living Rev. Relativity | Year=1998 | Volume=1 | Pages=13}} Available at . See ''section 3''. | |||
| * {{Book reference | Author=Misner, Charles W.; Thorne, Kip S. & Wheeler, John Archibald | Title=Gravitation | Publisher=San Francisco: W. H. Freeman | Year =1973 | ID=ISBN 0-7167-0344-0}} See ''chapter 42''. | |||
| * {{Journal reference | Author=T. Regge | Title=General relativity without coordinates | Journal=Phys. Rev. | Year=1961 | Volume=108 | Pages=558-571}} | |||
| {{physics-stub}} | |||
| ==External links== | ==External links== | ||
| * on ] | * on ] | ||
Revision as of 21:07, 29 May 2005
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 angle 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 virtually cries out for generalization. In particular, Regge calculus been adapted to study quantum gravity.
Reference
- Template:Journal reference Available at "Living Reviews of Relativity". See section 3.
- . ISBN 0-7167-0344-0.
{{cite book}}: Missing or empty|title=(help); Unknown parameter|Author=ignored (|author=suggested) (help); Unknown parameter|Publisher=ignored (|publisher=suggested) (help); Unknown parameter|Title=ignored (|title=suggested) (help); Unknown parameter|Year=ignored (|year=suggested) (help) See chapter 42.