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Physics[SubstituteTensorIndices] - perform substitution of covariant/contravariant tensor indices in tensorial expressions
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Calling Sequence
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SubstituteTensorIndices(mu = .., ~nu = .., ... expression)
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Parameters
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expression
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any algebraic tensorial expression typically having some free and some repeated indices
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mu = ... , ~nu = .., ...
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the substitution equations, can be a sequence, or a set or list of them
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evaluatetensor = ...
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optional - can be true (default) or false, to indicate whether to evaluate the tensors after substituting on them
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evaluateexpression = ...
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optional - can be true or false (default), to indicate whether to evaluate expression after substituting the indices in its tensors
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covariantandcontravariant = ...
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optional - can be true (default) or false, to indicate whether to substitute both covariant and contravariant indices when only one of them is given as substitution equation
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Description
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The SubstituteTensorIndices substitutes indices in tensors. Nowhere else are the indices substituted, and the substitution can be performed in a covariant index, the corresponding contravariant one, or in both.
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The tensors where indices are substituted are re-evaluated after substitution; this re-evaluation can optionally be suppressed giving the argument evaluatetensor = false. The expression where these re-evaluated tensors are introduce is by default not re-evaluated; you can change that passing the optional argument evaluateexpression = true.
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To check and determine the free and repeated indices of an expression use Check.
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Compatibility
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The Physics[SubstituteTensorIndices] command was introduced in Maple 16.
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Examples
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![[mathematicalnotation = true]](/support/helpjp/helpview.aspx?si=4949/file02569/math118.png)
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Define a couple of arbitrary spacetime tensors for exploration purposes
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![{A, B, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}](/support/helpjp/helpview.aspx?si=4949/file02569/math130.png)
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Enter, for example, this tensorial expression
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![g_[alpha, mu]*A[mu]*g_[alpha, nu]*B[nu, sigma, sigma]](/support/helpjp/helpview.aspx?si=4949/file02569/math136.png)
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![B[nu, sigma, `~sigma`]*Physics:-g_[alpha, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`]](/support/helpjp/helpview.aspx?si=4949/file02569/math139.png)
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To check the repeated and free indices in an expression use Check
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![Check(B[nu, sigma, `~sigma`]*Physics:-g_[alpha, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`], all)](/support/helpjp/helpview.aspx?si=4949/file02569/math148.png)
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![[{alpha, mu, nu, sigma}], {}](/support/helpjp/helpview.aspx?si=4949/file02569/math154.png)
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So (3) has no free indices. Substitute now  : the standard subs command will only substitute the covariant 
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![subs(alpha = rho, B[nu, sigma, `~sigma`]*Physics:-g_[alpha, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`])](/support/helpjp/helpview.aspx?si=4949/file02569/math168.png)
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![B[nu, sigma, `~sigma`]*Physics:-g_[rho, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`]](/support/helpjp/helpview.aspx?si=4949/file02569/math171.png)
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Consequently, the resulting expression is not equivalent to (3): it now has free indices
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![Check(B[nu, sigma, `~sigma`]*Physics:-g_[rho, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`], all)](/support/helpjp/helpview.aspx?si=4949/file02569/math179.png)
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![[{mu, nu, sigma}], {rho, `~alpha`}](/support/helpjp/helpview.aspx?si=4949/file02569/math185.png)
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To substitute both covariant and contravariant repeated indices obtaining an expression equivalent to original one use
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![SubstituteTensorIndices(alpha = rho, B[nu, sigma, `~sigma`]*Physics:-g_[alpha, mu]*Physics:-g_[`~alpha`, `~nu`]*A[`~mu`])](/support/helpjp/helpview.aspx?si=4949/file02569/math191.png)
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![B[nu, sigma, `~sigma`]*Physics:-g_[mu, rho]*Physics:-g_[`~nu`, `~rho`]*A[`~mu`]](/support/helpjp/helpview.aspx?si=4949/file02569/math194.png)
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![Check(B[nu, sigma, `~sigma`]*Physics:-g_[mu, rho]*Physics:-g_[`~nu`, `~rho`]*A[`~mu`], all)](/support/helpjp/helpview.aspx?si=4949/file02569/math198.png)
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![[{mu, nu, rho, sigma}], {}](/support/helpjp/helpview.aspx?si=4949/file02569/math204.png)
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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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