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Metric Equivalence Problem


1. Find a complete system of invariants, or

2. Decide when two metrics differ only by a coordinate transformation.

The most common statement of the problem is, "Given metrics g and g^', does there exist a coordinate transformation from one to the other?" Christoffel (1869) and Lipschitz (1870) showed how to decide this question for two Riemannian metrics.

The solution by É. Cartan requires computation of the 10th order covariant derivatives. The demonstration was simplified by A. Karlhede using the tetrad formalism so that only seventh order covariant derivatives need be computed. however, in many common cases, the first or second-order derivatives are sufficient to answer the question.


See also

Equivalent Metrics

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References

Christoffel, E. B. "Über die Transformation der homogenen Differentialausdrücke zweiten Grades." J. für Math. 70, 46-70, 1869.Karlhede, A. and Lindström, U. "Finding Space-Time Geometries without Using a Metric." Gen. Relativity Gravitation 15, 597-610, 1983.Lipschitz, R. "Untersuchungen in betreff der ganzen homogenen Functionen von n Differentialen." J. für Math 70, 71-102, 1870.

Referenced on Wolfram|Alpha

Metric Equivalence Problem

Cite this as:

Weisstein, Eric W. "Metric Equivalence Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MetricEquivalenceProblem.html

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