general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, 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. Available (subscribers only) a
Il Nuovo Cimento
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Overview

The starting point for Regge's work is the fact that every four dimensional time orientable

Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...

admits a

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

into

simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

. Furthermore, the

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

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 In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

, 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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

can be computed from the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

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 A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

. 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 Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

Notes

References

* * See ''chapter 42''. * Chapters 4 and 6.

* * Available (subscribers only) a
"Classical and Quantum Gravity"
* Available a

*

*
eprint
* Available a

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) a
"Classical and Quantum Gravity"

External links

ScienceWorld Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include ...

Mathematical methods in general relativity Simplicial sets Numerical analysis

Overview

See also

Notes

References

External links