- Cartan-Karlhede algorithm
One of the most fundamental problems of
Riemannian geometry is this: given twoRiemannian manifold s of the same dimension, how can one tell if they are locally isometric? This question was addressed byElwin Christoffel , and completely solved byÉlie Cartan using his exterior calculus with his method ofmoving 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.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 distinguishRiemannian manifold s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of theLorentz group SO+(3,R), which is a "noncompact"Lie group , while four-dimensional Riemannian manifolds (i.e., withpositive definite metric tensor ), have isotropy groups which are subgroups of thecompact Lie group SO(4).Cartan's method was adapted and improved for general relativity by A. Karlhede, andimplemented 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 perfectfluid solution s,
*at most one differentiation is required to compare any twonull dust solution 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 s in common use, such asLinux , 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 indifferential 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.ee also
*
Computer algebra system
*Frame fields in general relativity
*Petrov classification External links
* [http://130.15.26.66/servlet/GRDB2.GRDBServlet Interactive Geometric Database] includes some data derived from an implementation of the Cartan-Karlhede algorithm.
References
*cite book | author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard| title=Exact Solutions to Einstein's Field Equations (2nd ed.) | location=Cambridge | publisher=Cambridge University Press | year=2003 | id=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.
*cite journal | author=Pollney, D.; Skea, J. F.; and d'Inverno, Ray | title=Classifying geometries in general relativity (three parts) | journal=Class. Quant. Grav. | year=2000 | volume=17 | pages=643–663, 2267–2280, 2885–2902 | doi=10.1088/0264-9381/17/3/306 A research paper describing the authors' database holding classifications of exact solutions up to local isometry.
*cite book | author=Olver, Peter J. | title=Equivalents, Invariants, and Symmetry | location=Cambridge | publisher=Cambridge University Press | year=1995 | id=ISBN 0-521-47811-1 An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.
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