Physics[Tetrads][gamma_] - represent and compute the Ricci rotation coefficients
Physics[Tetrads][lambda_] - represent a linear combination of the Ricci rotation coefficients - see reference
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Calling Sequence
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gamma_[a, b, c]
gamma_[a, b, c, matrix]
gamma_[keyword]
lambda_[a, b, c]
lambda_[a, b, c, matrix]
lambda_[keyword]
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Parameters
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_mu, nu_
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the spacetime indices related to a global system of references, these are names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves
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_a, b, c_
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the tetrad indices related to a local system of references, as names representing integer numbers the same way as the spacetime indices
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keyword
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optional, it can be definition, matrix, nonzero
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Description
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The components of the gamma_[a, b, c] tensor are the Ricci rotation coefficients and the components of the lambda_[a, b, c] tensor are linear combinations of them, according to the definitions in the Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2 (definitions (98.9) and (98.10)). Thus,
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where , represented in Maple by D_[nu](e_[a,mu]), is the covariant derivative of the tetrad, and
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From these definitions it follows that is antisymmetric in its first pair of indices, and the covariant derivatives entering the definition of can be replaced by non-covariant ones using the d_ operator. You can retrieve these definitions directly in the worksheet entering gamma_[definition] and lambda_[definition].
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Both gamma_ and lambda_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account.
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Examples
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Set the spacetime to be curved, for instance set the Schwarzschild metric in spherical coordinates (see g_):
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The definition of the Ricci rotation coefficients and related
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Isolating the covariant derivative that appears on the right-hand side, of the definitions of , we have
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The and tensors are related by formulas (98.10) and (98.11) of Landau's book referenced at the end, this relationship can be obtained as follows
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Simplifying the repeated indices,
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Substituting the free indices in this expression and adding as follows, then isolating , we obtain the inverse of this relationship
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The nonzero Ricci coefficients and related components for the Schwarzschild metric
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The value of
The components of
The Riemann tensor in the tetrad system of coordinates can be expressed in terms of the Ricci rotation coefficients using formula (98.13) of Landau's book
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You can verify identities like this one by taking left-hand side minus right-hand side and computing an Array of components of this tensorial expression using TensorArray, then get its elements (components) using ArrayElems; recalling, ArrayElems returns a set with the elements of an Array, omitting all those elements equal to zero, so we expect an empty set here:
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Let's verify in the same way the expression of the covariant derivative in terms of the Ricci rotation coefficients
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See Also
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Array, ArrayElems, d_, D_, e_, eta_, g_, IsTetrad, l_, m_, mb_, n_, NullTetrad, OrthonormalTetrad, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, TensorArray, Tetrads, TransformTetrad
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References
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Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
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Compatibility
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The Physics[Tetrads][gamma_] and Physics[Tetrads][lambda_] commands were introduced in Maple 2015.
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