- Regge calculus
In
general relativity , Regge calculus is a formalism for producing simplicial approximations of spacetimes which are solutions to theEinstein field equation . The calculus was introduced by the Italian theoreticianTullio Regge in the early 1960s.The starting point for Regge's work is the fact that every
Lorentzian manifold admits a triangulation intosimplices . Furthermore, thespacetime curvature can be expressed in terms ofdeficit 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 theRiemann curvature tensor can be computed from themetric tensor of a Lorentzian manifold. Regge showed that thevacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initialspacelike 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 innumerical relativity such as simulating the collision of twoblack 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
* [http://www.arxiv.org/abs/gr-qc/0408006 eprint]
* Available at [http://relativity.livingreviews.org/Articles/lrr-1998-13/index.html "Living Reviews of Relativity"] . See "section 3".
* Available (subscribers only) at [http://www.iop.org/EJ/abstract/-search=10468506.3/0264-9381/9/5/021 "Classical and Quantum Gravity"] .
* Available (subscribers only) at [http://www.iop.org/EJ/abstract/-search=10468854.14/0264-9381/4/6/015 "Classical and Quantum Gravity"] .
* See "chapter 42".
* Available (subscribers only) at [http://www.springerlink.com/content/5382k74272mpu8n7/ Il Nuovo Cimento]
ee also
*
Mathematics of general relativity External links
* [http://scienceworld.wolfram.com/physics/ReggeCalculus.html Regge calculus] on
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