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| ==Overview== | ==Overview== | ||
| 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 represents a concentration of ''negative'' Gaussian curvature. | The starting point for Regge's work is the fact that every four dimensional time orientable ] 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 represents a concentration of ''negative'' Gaussian curvature. | ||
| 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 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. | ||
Revision as of 07:02, 29 September 2020
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 1961.
Overview
The starting point for Regge's work is the fact that every four dimensional time orientable 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
- Numerical relativity
- Quantum gravity
- 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
Notes
- Tullio E. Regge (1961). "General relativity without coordinates". Nuovo Cimento. 19 (3): 558–571. Bibcode:1961NCim...19..558R. doi:10.1007/BF02733251. S2CID 120696638. Available (subscribers only) at Il Nuovo Cimento
References
- 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.
{{cite journal}}: Cite journal requires|journal=(help) - Misner, Charles W. Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.
{{cite book}}: CS1 maint: multiple names: authors list (link) See chapter 42. - Herbert W. Hamber (2009). Hamber, Herbert W (ed.). 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.
- James B. Hartle (1985). "Simplicial MiniSuperSpace I. General Discussion". Journal of Mathematical Physics. 26 (4): 804–812. Bibcode:1985JMP....26..804H. doi:10.1063/1.526571.
- Ruth M. Williams; Philip A. Tuckey (1992). "Regge calculus: a brief review and bibliography". Class. Quantum Grav. 9 (5): 1409–1422. Bibcode:1992CQGra...9.1409W. doi:10.1088/0264-9381/9/5/021.
{{cite journal}}: Unknown parameter|lastauthoramp=ignored (|name-list-style=suggested) (help) Available (subscribers only) at "Classical and Quantum Gravity". - Tullio E. Regge and Ruth M. Williams (2000). "Discrete Structures in Gravity". Journal of Mathematical Physics. 41 (6): 3964–3984. arXiv:gr-qc/0012035. Bibcode:2000JMP....41.3964R. doi:10.1063/1.533333. S2CID 118957627. Available at .
- 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: 375–439.
{{cite journal}}: Cite journal requires|journal=(help) - Adrian P. Gentle (2002). "Regge calculus: a unique tool for numerical relativity". Gen. Rel. Grav. 34 (10): 1701–1718. doi:10.1023/A:1020128425143. S2CID 119090423. eprint
- Renate Loll (1998). "Discrete approaches to quantum gravity in four dimensions". Living Rev. Relativ. 1 (1): 13. arXiv:gr-qc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13. PMC 5253799. PMID 28191826.
{{cite journal}}: CS1 maint: unflagged free DOI (link) Available at "Living Reviews of Relativity". See section 3. - J. W. Barrett (1987). "The geometry of classical Regge calculus". Class. Quantum 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".