SubstitutionTensor command not seeing eta[~alpha,~beta]

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Question:SubstitutionTensor command not seeing eta[~alpha,~beta]

Posted: Michael_Watson 125

July 07 2014

The TensorSubstittuition is not finding all of the eta[~mu,~mu]*l[nu](X)=l[~mu](X) substitutions.

(1)

${Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], l[mu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(2)

${Physics:-Dgamma[mu], G[mu, nu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], l[mu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(3)

(4)

$[metric = {(1, 1) = 1+l[1](X)^2, (1, 2) = l[1](X)*l[2](X), (1, 3) = l[1](X)*l[3](X), (1, 4) = l[1](X)*l[4](X), (2, 2) = 1+l[2](X)^2, (2, 3) = l[2](X)*l[3](X), (2, 4) = l[2](X)*l[4](X), (3, 3) = 1+l[3](X)^2, (3, 4) = l[3](X)*l[4](X), (4, 4) = -1+l[4](X)^2}]$

(5)

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = 1+l[1](X)^2, (1, 2) = l[1](X)*l[2](X), (1, 3) = l[1](X)*l[3](X), (1, 4) = l[1](X)*l[4](X), (2, 1) = l[1](X)*l[2](X), (2, 2) = 1+l[2](X)^2, (2, 3) = l[2](X)*l[3](X), (2, 4) = l[2](X)*l[4](X), (3, 1) = l[1](X)*l[3](X), (3, 2) = l[2](X)*l[3](X), (3, 3) = 1+l[3](X)^2, (3, 4) = l[3](X)*l[4](X), (4, 1) = l[1](X)*l[4](X), (4, 2) = l[2](X)*l[4](X), (4, 3) = l[3](X)*l[4](X), (4, 4) = -1+l[4](X)^2}))$

(6)

Physics:-Christoffel[`~alpha`, mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~beta`]*(Physics:-d_[nu](Physics:-g_[beta, mu], [X])+Physics:-d_[mu](Physics:-g_[beta, nu], [X])-Physics:-d_[beta](Physics:-g_[mu, nu], [X]))

(7)

Physics:-Christoffel[beta, mu, nu] = (1/2)*Physics:-d_[nu](Physics:-g_[beta, mu], [X])+(1/2)*Physics:-d_[mu](Physics:-g_[beta, nu], [X])-(1/2)*Physics:-d_[beta](Physics:-g_[mu, nu], [X])

(8)

Physics:-Christoffel[beta, mu, nu] = (1/2)*Physics:-d_[nu](l[beta](X), [X])*l[mu](X)+(1/2)*l[beta](X)*Physics:-d_[nu](l[mu](X), [X])+(1/2)*Physics:-d_[mu](l[beta](X), [X])*l[nu](X)+(1/2)*l[beta](X)*Physics:-d_[mu](l[nu](X), [X])-(1/2)*Physics:-d_[beta](l[mu](X), [X])*l[nu](X)-(1/2)*l[mu](X)*Physics:-d_[beta](l[nu](X), [X])

(9)

(10)

$SubstituteTensor({Physics:-`*`(eta[`~mu`, `~nu`], d_[gamma](l[nu](X))) = d_[gamma](l[`~mu`](X)), Physics:-`*`(l[nu](X), eta[`~mu`, `~nu`]) = l[`~mu`](X), Physics:-`*`(l[`~mu`](X), l[mu](X)) = 0}, Physics:-Christoffel[`~alpha`, mu, nu] = -(1/2)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[nu](l[beta](X), [X])*l[mu](X)+(1/2)*Physics:-d_[nu](l[beta](X), [X])*l[mu](X)*eta[`~alpha`, `~beta`]-(1/2)*l[`~alpha`](X)*l[`~beta`](X)*l[beta](X)*Physics:-d_[nu](l[mu](X), [X])+(1/2)*l[beta](X)*Physics:-d_[nu](l[mu](X), [X])*eta[`~alpha`, `~beta`]-(1/2)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[mu](l[beta](X), [X])*l[nu](X)+(1/2)*Physics:-d_[mu](l[beta](X), [X])*l[nu](X)*eta[`~alpha`, `~beta`]-(1/2)*l[`~alpha`](X)*l[`~beta`](X)*l[beta](X)*Physics:-d_[mu](l[nu](X), [X])+(1/2)*l[beta](X)*Physics:-d_[mu](l[nu](X), [X])*eta[`~alpha`, `~beta`]+(1/2)*l[`~alpha`](X)*l[`~beta`](X)*l[nu](X)*Physics:-d_[beta](l[mu](X), [X])-(1/2)*l[nu](X)*Physics:-d_[beta](l[mu](X), [X])*eta[`~alpha`, `~beta`]+(1/2)*l[`~alpha`](X)*l[`~beta`](X)*l[mu](X)*Physics:-d_[beta](l[nu](X), [X])-(1/2)*l[mu](X)*Physics:-d_[beta](l[nu](X), [X])*eta[`~alpha`, `~beta`])$

(11)

(12)

(13)

(14)

$SubstituteTensor({Physics:-`*`(eta[`~mu`, `~nu`], d_[gamma](l[nu](X))) = d_[gamma](l[`~mu`](X)), Physics:-`*`(eta[`~mu`, `~nu`], l[nu](X)) = l[`~mu`](X), Physics:-`*`(l[`~mu`](X), l[mu](X)) = 0}, (1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*Physics:-d_[alpha](l[beta](X), [X])*Physics:-d_[nu](l[mu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*l[beta](X)*Physics:-d_[alpha](Physics:-d_[nu](l[mu](X), [X]), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*Physics:-d_[alpha](l[beta](X), [X])*Physics:-d_[mu](l[nu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*l[beta](X)*Physics:-d_[alpha](Physics:-d_[mu](l[nu](X), [X]), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*Physics:-d_[alpha](Physics:-d_[beta](l[mu](X), [X]), [X])*l[nu](X)-(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*Physics:-d_[beta](l[mu](X), [X])*Physics:-d_[alpha](l[nu](X), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*Physics:-d_[alpha](l[mu](X), [X])*Physics:-d_[beta](l[nu](X), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*eta[`~alpha`, `~beta`]*l[mu](X)*Physics:-d_[alpha](Physics:-d_[beta](l[nu](X), [X]), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](Physics:-d_[mu](l[`~alpha`](X), [X]), [X])*l[nu](X)+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[mu](l[`~alpha`](X), [X])*Physics:-d_[alpha](l[nu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](Physics:-d_[nu](l[`~alpha`](X), [X]), [X])*l[mu](X)+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[nu](l[`~alpha`](X), [X])*Physics:-d_[alpha](l[mu](X), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[alpha](Physics:-d_[nu](l[beta](X), [X]), [X])*l[mu](X)-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[nu](l[beta](X), [X])*Physics:-d_[alpha](l[mu](X), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~beta`](X)*Physics:-d_[nu](l[beta](X), [X])*l[mu](X)+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~beta`](X)*l[mu](X)*Physics:-d_[beta](l[nu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*Physics:-d_[alpha](l[`~beta`](X), [X])*l[mu](X)*Physics:-d_[beta](l[nu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[alpha](l[mu](X), [X])*Physics:-d_[beta](l[nu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*l[mu](X)*Physics:-d_[alpha](Physics:-d_[beta](l[nu](X), [X]), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~beta`](X)*l[nu](X)*Physics:-d_[beta](l[mu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*Physics:-d_[alpha](l[`~beta`](X), [X])*l[nu](X)*Physics:-d_[beta](l[mu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[alpha](l[nu](X), [X])*Physics:-d_[beta](l[mu](X), [X])+(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*l[nu](X)*Physics:-d_[alpha](Physics:-d_[beta](l[mu](X), [X]), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~beta`](X)*Physics:-d_[mu](l[beta](X), [X])*l[nu](X)-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*Physics:-d_[alpha](l[`~beta`](X), [X])*Physics:-d_[mu](l[beta](X), [X])*l[nu](X)-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[alpha](Physics:-d_[mu](l[beta](X), [X]), [X])*l[nu](X)-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*l[`~beta`](X)*Physics:-d_[mu](l[beta](X), [X])*Physics:-d_[alpha](l[nu](X), [X])-(1/2)*l[`~mu`](X)*l[`~nu`](X)*l[`~alpha`](X)*Physics:-d_[alpha](l[`~beta`](X), [X])*Physics:-d_[nu](l[beta](X), [X])*l[mu](X))$

(15)

(16)

$"SubstituteTensor({l[~mu](X)*l[mu](X)=0,eta[~mu,~nu]*l[nu](X)=l[~mu](X),eta[~mu,~nu]*d_[gamma](l[nu](X))=d_[gamma](l[~mu](X))},l[~lambda](X)*Physics:-d_[beta](Physics:-d_[lambda](l[mu](X),[X]),[X])*l[alpha](X)*l[~mu](X)*eta[~alpha,~beta]-eta[~beta,~kappa]*Physics:-d_[beta](l[lambda](X),[X])*Physics:-d_[kappa](l[nu](X),[X])*l[~lambda](X)*l[~nu](X)+Physics:-d_[alpha](l[~alpha](X),[X])*Physics:-d_[kappa](l[nu](X),[X])*l[~kappa](X)*l[~nu](X)+Physics:-d_[kappa](l[nu](X),[X])*Physics:-d_[mu](l[~kappa](X),[X])*l[~mu](X)*l[~nu](X))"$

(17)

The first term should change eta[~alpha,~beta]*l[alpha] -> l[~beta]. I made eta symmetric so the order should not matter. I notice that the command seems to work when the substitution is at the beginning of the term. Do I need to use the new RepositionRepeatedIndicesAsIn command?

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Question: SubstitutionTensor command not seeing eta[~alpha,~beta]

Question:SubstitutionTensor command not seeing eta[~alpha,~beta]

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Question: SubstitutionTensor command not seeing eta[~alpha,~beta]

Question:SubstitutionTensor command not seeing eta[~alpha,~beta]

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