Cartan-Karlhede algorithm
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One of the most fundamental problems of Riemannian geometry is this: given two Riemannian manifolds 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.
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[edit] Physical Applications
The Cartan-Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of curvature invariants fails to distinguish spacetimes as well as they distinguish Riemannian manifolds. 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
- at most two differentiations are required to compare any two Petrov D vacuum solutions,
- at most three differentiations are required to compare any two perfect fluid solutions,
- at most one differentiation is required to compare any two null dust solutions.
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 systems 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.
[edit] See also
[edit] External links
- Interactive Geometric Database includes some data derived from an implementation of the Cartan-Karlhede algorithm.
[edit] References
- Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard (2003). Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7. Chapter 9 offers an excellent overview of the basic idea of the Cartan method and contains a useful table of upper bounds, more extensive than the one above.
- Pollney, D.; Skea, J. F.; and d'Inverno, Ray (2000). "Classifying geometries in general relativity (three parts)". Class. Quant. Grav. 17: 643–663, 2267–2280, 2885–2902. doi:. A research paper describing the authors' database holding classifications of exact solutions up to local isometry.
- Olver, Peter J. (1995). Equivalents, Invariants, and Symmetry. Cambridge: Cambridge University Press. ISBN 0-521-47811-1. An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.
