The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

s. Given two

s of the same dimension, it is not always obvious whether they are

locally isometric In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Pr ...

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...

, using his

exterior calculus In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

with his method of moving frames, showed that it is always possible to compare the manifolds.

Carl Brans Carl Henry Brans (; born December 13, 1935) is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory. Biography ...

developed the method further, and the first practical implementation was presented by in 1980. The main strategy of the algorithm is to take

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...

s of the

Riemann 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. ...

. Cartan showed that in ''n'' dimensions at most ''n''(''n''+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the

Petrov classification In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is ...

. The potentially large number of derivatives can be computationally prohibitive. The algorithm was implemented in an early symbolic computation engine,

SHEEP Sheep or domestic sheep (''Ovis aries'') are domesticated, ruminant mammals typically kept as livestock. Although the term ''sheep'' can apply to other species in the genus ''Ovis'', in everyday usage it almost always refers to domesticated sh ...

, but the size of the computations proved too challenging for early computer systems to handle. For most problems considered, far fewer derivatives than the maximum are actually required, and the algorithm is more manageable on modern computers. On the other hand, no publicly available version exists in more modern software.

Physical applications

The Cartan–Karlhede algorithm has important applications in

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. ...

. One reason for this is that the simpler notion of

curvature invariant In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors form ...

s fails to distinguish spacetimes as well as they distinguish

s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the

Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch phy ...

SO⁺(1,3), which is a ''noncompact''

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

, while four-dimensional Riemannian manifolds (i.e., with

positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite ...

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 allo ...

), have isotropy groups which are subgroups of the

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Lie group SO(4). In 4 dimensions, Karlhede's improvement to Cartan's program reduces the maximal number of covariant derivatives of the Riemann tensor needed to compare metrics to 7. In the worst case, this requires 3156 independent tensor components. There are known models of spacetime requiring all 7 covariant derivatives. For certain special families of spacetime models, however, often far fewer often suffice. It is now known, for example, that *at most one differentiation is required to compare any two null dust solutions, *at most two differentiations are required to compare any two Petrov D

vacuum solution In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or n ...

s, *at most three differentiations are required to compare any two perfect

fluid solution In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often ...

External links

Interactive Geometric Database
includes some data derived from an implementation of the Cartan–Karlhede algorithm.

References

{{DEFAULTSORT:Cartan-Karlhede algorithm Riemannian geometry Mathematical methods in general relativity

Physical applications

See also

External links

References