Regge calculus
Encyclopedia
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 the early 1960s.
The starting point for Regge's work is the fact that every 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
.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, Regge calculus is a formalism for producing simplicial approximations
Simplicial manifold
In mathematics, the term simplicial manifold commonly refers to either of two different types of objects, which combine attributes of a simplex with those of a manifold. Briefly; a simplex is a generalization of the concept of a triangle into forms with more, or fewer, than two dimensions. ...
of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge
Tullio Regge
Tullio Regge is an Italian theoretical physicist. He obtained a degree in physics from the University of Turin in 1952, and a Ph.D. in physics from the University of Rochester in 1957 under the direction of Robert Marshak. From 1958 to 1959 Regge held a post at the Max Planck Institute for...
in the early 1960s.
The starting point for Regge's work is the fact that every Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
can be expressed in terms of deficit angles
Defect (geometry)
In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...
associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
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
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
.
The deficit angles can be computed directly from the various edge
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor
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 standard way to express curvature of Riemannian manifolds...
can be computed from the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
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
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
equation), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in numerical relativity
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 and many other phenomena governed by Einstein's Theory...
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
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
.
External links
- Regge calculus on ScienceWorldScienceWorldScienceWorld, also known as Eric Weisstein's World of Science, is a Web site that opened to the general public in January 2002. As of November 2007, ScienceWorld includes more than 4,000 entries in fields of science including astronomy, chemistry, physics, as well as biographies of many scientists...