Cartan-Karlhede algorithm
Encyclopedia
One of the most fundamental problems of Riemannian geometry
is this: given two Riemannian manifold
s of the same dimension, how can one tell if they are locally isometric? This question was addressed by Elwin Christoffel, and completely solved by Élie Cartan
using his exterior calculus
with his method of moving frames.
Cartan's method was adapted and improved for general relativity
by A. Karlhede, who gave the first algorithmic description of what is now called the Cartan–Karlhede algorithm. The algorithm was soon implemented by J. Åman in an early symbolic computation engine, SHEEP (symbolic computation system)
, but the size of the computations proved too challenging for early computer systems to handle.
s fails to distinguish spacetimes as well as they distinguish Riemannian manifold
s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the Lorentz group
SO+(3,R), which is a noncompact Lie group
, while four-dimensional Riemannian manifolds (i.e., with positive definite metric tensor
), have isotropy groups which are subgroups of the compact
Lie group SO(4).
Cartan's method was adapted and improved for general relativity by A. Karlhede, and
implemented by J. Åman in an early symbolic computation engine, SHEEP (symbolic computation system)
.
Cartan showed that at most ten covariant derivatives are needed to compare any two Lorentzian manifolds by his method, but experience shows that far fewer often suffice, and later researchers have lowered his upper bound considerably. It is now known, for example, that
An important unsolved problem is to better predict how many differentiations are really necessary for spacetimes having various properties. For example, somewhere two and five differentiations, at most, are required to compare any two Petrov III vacuum solutions. Overall, it seems to safe to say that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.
Faster implementations of the method running under a modern symbolic computation system available for modern operating system
s in common use, such as Linux
, would also be highly desirable. It has been suggested that the power of this algorithm has not yet been realized, due to insufficient effort to take advantage of recent improvements in differential algebra
. The appearance in the "near future" of a proper on-line database of known solutions has been rumored for decades, but this has not yet come to pass. This is particularly regrettable since it seems very likely that a powerful and convenient database is well within the capability of modern software.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
is this: given two Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s of the same dimension, how can one tell if they are locally isometric? This question was addressed by Elwin Christoffel, and completely solved by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
using his exterior calculus
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
with his method of moving frames.
Cartan's method was adapted and improved for general relativity
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...
by A. Karlhede, who gave the first algorithmic description of what is now called the Cartan–Karlhede algorithm. The algorithm was soon implemented by J. Åman in an early symbolic computation engine, SHEEP (symbolic computation system)
SHEEP (symbolic computation system)
SHEEP is one of the earliest interactive symbolic computation systems. It is specialized for computations with tensors, and was designed for the needs of researchers working with general relativity and other theories involving extensive tensor calculus computations.SHEEP is an open source freeware...
, but the size of the computations proved too challenging for early computer systems to handle.
Physical applications
The Cartan–Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of curvature invariantCurvature invariant
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature...
s fails to distinguish spacetimes as well as they distinguish Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
SO+(3,R), which is a noncompact Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, while four-dimensional Riemannian manifolds (i.e., with positive definite 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...
), have isotropy groups which are subgroups of the compact
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
Lie group SO(4).
Cartan's method was adapted and improved for general relativity by A. Karlhede, and
implemented by J. Åman in an early symbolic computation engine, SHEEP (symbolic computation system)
SHEEP (symbolic computation system)
SHEEP is one of the earliest interactive symbolic computation systems. It is specialized for computations with tensors, and was designed for the needs of researchers working with general relativity and other theories involving extensive tensor calculus computations.SHEEP is an open source freeware...
.
Cartan showed that at most ten covariant derivatives are needed to compare any two Lorentzian manifolds by his method, but experience shows that far fewer often suffice, and later researchers have lowered his upper bound considerably. It is now known, for example, that
- at most two differentiations are required to compare any two Petrov D vacuum solutionVacuum solution (general relativity)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 non-gravitational fields are present.More generally, a...
s, - at most three differentiations are required to compare any two perfect fluid solutionFluid solutionIn 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....
s, - at most one differentiation is required to compare any two null dust solutionNull dust solutionIn mathematical physics, a null dust solution is a Lorentzian manifold in which the Einstein tensor is null...
s.
An important unsolved problem is to better predict how many differentiations are really necessary for spacetimes having various properties. For example, somewhere two and five differentiations, at most, are required to compare any two Petrov III vacuum solutions. Overall, it seems to safe to say that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.
Faster implementations of the method running under a modern symbolic computation system available for modern operating system
Operating system
An operating system is a set of programs that manage computer hardware resources and provide common services for application software. The operating system is the most important type of system software in a computer system...
s in common use, such as Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...
, would also be highly desirable. It has been suggested that the power of this algorithm has not yet been realized, due to insufficient effort to take advantage of recent improvements in differential algebra
Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies the Leibniz product law...
. The appearance in the "near future" of a proper on-line database of known solutions has been rumored for decades, but this has not yet come to pass. This is particularly regrettable since it seems very likely that a powerful and convenient database is well within the capability of modern software.
See also
- Computer algebra systemComputer algebra systemA computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
- Frame fields in general relativityFrame fields in general relativityIn general relativity, a frame field is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime...
- Petrov classificationPetrov classificationIn differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold....
External links
- Interactive Geometric Database includes some data derived from an implementation of the Cartan–Karlhede algorithm.