Editing Regge calculus

{{general relativity}}
In [[general relativity]], '''Regge calculus''' is a formalism for producing [[Simplicial manifold|simplicial approximations]] of spacetimes that are solutions to the [[Einstein field equation]].  The calculus was introduced by the Italian theoretician [[Tullio Regge]] in 1961.<ref>{{cite journal | author=Tullio E. Regge | title=General relativity without coordinates | journal=Nuovo Cimento  | year=1961 | volume=19 | issue=3 | pages=558–571 | doi=10.1007/BF02733251| bibcode=1961NCim...19..558R | s2cid=120696638 | author-link=Tullio E. Regge }} Available (subscribers only) at [https://doi.org/10.1007%2FBF02733251 Il Nuovo Cimento]</ref>

==Overview==
The starting point for Regge's work is the fact that every four dimensional time orientable [[Lorentzian manifold]] admits a [[Triangulation (geometry)|triangulation]] into [[simplices]].  Furthermore, the [[spacetime]] [[curvature]] can be expressed in terms of [[Defect (geometry)|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 deficit angles can be computed directly from the various [[edge (geometry)|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 (mathematics)|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==
{{Div col|colwidth=20em}}
*[[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]]
*[[Twisted geometries]]
{{Div col end}}

==Notes==
{{reflist}}

==References==
* {{cite journal | author= John Archibald Wheeler | title= Geometrodynamics and the Issue of the Final State, in "Relativity Groups and Topology" | publisher= Les Houches Lecture Notes 1963, Gordon and Breach | year=1965 | author-link= John Archibald Wheeler }} 
* {{cite book | author=Misner, Charles W. Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | publisher=San Francisco: W. H. Freeman | year =1973 | isbn=978-0-7167-0344-0}}  See ''chapter 42''.
* {{cite book | author= Herbert W. Hamber | editor1-first= Herbert W | editor1-last= Hamber | title= Quantum Gravitation - The Feynman Path Integral Approach | publisher = Springer Publishing | year=2009 |  doi=10.1007/978-3-540-85293-3 | isbn=978-3-540-85292-6| url= https://cds.cern.ch/record/1233211 }} Chapters 4 and 6.  [https://link.springer.com/book/10.1007%2F978-3-540-85293-3] [https://arxiv.org/abs/0704.2895]
* {{cite journal | author= James B. Hartle | title= Simplicial MiniSuperSpace I. General Discussion | journal=  Journal of Mathematical Physics| year=1985 | volume=26 | issue= 4 | pages=804–812 | doi=10.1063/1.526571|bibcode = 1985JMP....26..804H }} 
* {{cite journal |author1=Ruth M. Williams  |author2=Philip A. Tuckey  |name-list-style=amp | title=Regge calculus: a brief review and bibliography | journal=Class. Quantum Grav. | year=1992 | volume=9 | issue= 5 | pages=1409–1422 | doi=10.1088/0264-9381/9/5/021|bibcode = 1992CQGra...9.1409W |s2cid=250776873 |url=https://cds.cern.ch/record/227081  }}  Available (subscribers only) at [http://www.iop.org/EJ/abstract/-search=10468506.3/0264-9381/9/5/021 "Classical and Quantum Gravity"].
* {{cite journal | author= [[Tullio E. Regge]] and Ruth M. Williams | title= Discrete Structures in Gravity | journal= Journal of Mathematical Physics | year=2000 | volume=41 | issue= 6 | pages=3964–3984 | doi=10.1063/1.533333 |arxiv = gr-qc/0012035 |bibcode = 2000JMP....41.3964R | s2cid= 118957627 }}  Available at [https://arxiv.org/abs/gr-qc/0012035].  
* {{ cite journal | author = Herbert W. Hamber | title = Simplicial Quantum Gravity, in the Les Houches Summer School on Critical Phenomena, Random Systems and Gauge Theories, Session XLIII | year = 1984 | publisher = North Holland Elsevier | pages =375–439 }} [http://aeneas.ps.uci.edu/lesh_scan.pdf]
* {{cite journal | author=Adrian P. Gentle | title=Regge calculus: a unique tool for numerical relativity | journal=Gen. Rel. Grav. | year=2002 | volume=34 | issue=10 | pages=1701–1718 | doi=10.1023/A:1020128425143| s2cid=119090423 | url=https://cds.cern.ch/record/784592 }} [http://www.arxiv.org/abs/gr-qc/0408006 eprint]
* {{cite journal | author=Renate Loll | title=Discrete approaches to quantum gravity in four dimensions | journal=Living Rev. Relativ. | year=1998 | volume=1 | issue=1 | pages=13|arxiv = gr-qc/9805049 |bibcode = 1998LRR.....1...13L |doi = 10.12942/lrr-1998-13 | doi-access=free | pmid=28191826 | pmc=5253799 }}  Available at [https://web.archive.org/web/20050429050753/http://relativity.livingreviews.org/Articles/lrr-1998-13/index.html "Living Reviews of Relativity"]. See ''section 3''.
* {{cite journal | author= J. W. Barrett | title=The geometry of classical Regge calculus | journal=Class. Quantum Grav. | year=1987 | volume=4 | issue= 6 | pages=1565–1576 | doi=10.1088/0264-9381/4/6/015|bibcode = 1987CQGra...4.1565B | s2cid=250783980 | url=https://cds.cern.ch/record/173023 }}  Available (subscribers only) at [http://www.iop.org/EJ/abstract/-search=10468854.14/0264-9381/4/6/015 "Classical and Quantum Gravity"].

==External links==
*[http://scienceworld.wolfram.com/physics/ReggeCalculus.html Regge calculus] on [[ScienceWorld]]

[[Category:Mathematical methods in general relativity]]
[[Category:Simplicial sets]]
[[Category:Numerical analysis]]