Information AboutCartan-karlhede Algorithm |
| CATEGORIES ABOUT CARTAN-KARLHEDE ALGORITHM | |
| riemannian geometry | |
| mathematical methods in general relativity | |
|
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. PHYSICAL APPLICATIONS The Cartan-Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of Curvature Invariant 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. SEE ALSO EXTERNAL LINKS
REFERENCES
|
|
|