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

, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the

Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...

. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Available (subscribers only) a
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 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 me ...

into simplices. Furthermore, the

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

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 surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...

, 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 Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

lengths in the triangulation, which is equivalent to saying that the

Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...

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 allow ...

of a Lorentzian manifold. Regge showed that the

vacuum field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

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 Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars a ...

such as simulating the collision of two

black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

. 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 vi ...

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=http://cds.cern.ch/record/173023 Available (subscribers only) a
"Classical and Quantum Gravity"

External links

on ScienceWorld Mathematical methods in general relativity Simplicial sets Numerical analysis

Overview

See also

Notes

References

External links