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

July 07 2014 Michael_Watson 35

2

 

``

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

``

restart

with(Physics):

Setup(mathematicalnotation = true, coordinatesystems = X, usephysicsevaluator = false)

[coordinatesystems = {X}, mathematicalnotation = true, usephysicsevaluator = false]

(1)

Define(l[mu]);

{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)

Define(G[mu, nu] = eta[mu, nu]+Physics:-`*`(l[mu](X), l[nu](X)))

{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)

declare(l(X))

l(x1, x2, x3, x4)*`will now be displayed as`*l

(4)

NULL

NULL

NULL

NULL

Setup(metric = rhs(G[]))

[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_[];

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)

NULL

Christoffel[`~alpha`, mu, nu] = convert(Christoffel[`~alpha`, mu, nu], g_)

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)

Christoffel[beta, mu, nu] = convert(Christoffel[beta, mu, nu], g_)

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)

NULL

SubstituteTensor(g_[mu, nu] = eta[mu, nu]+Physics:-`*`(l[mu](X), l[nu](X)), 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]), evaluateexpression)

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)

NULL

NULL

Christoffel[`~alpha`, mu, nu] = Simplify(Physics:-`*`(eta[`~alpha`, `~beta`]-Physics:-`*`(l[`~alpha`](X), l[`~beta`](X)), rhs(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]))))

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`]

(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`])

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[`~alpha`](X), [X])*l[mu](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[`~alpha`](X), [X])*l[nu](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)

Ricci[mu, nu] = convert(Ricci[mu, nu], Christoffel)

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

(12)

term1 := SubstituteTensor(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[`~alpha`](X), [X])*l[mu](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[`~alpha`](X), [X])*l[nu](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`], Physics:-`*`(Physics:-`*`(d_[alpha](Christoffel[`~alpha`, mu, nu]), l[`~mu`](X)), l[`~nu`](X)))

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

(13)

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

(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)

(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))

(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`]*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_[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)*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)*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)*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)*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_[mu](l[`~alpha`](X), [X])*Physics:-d_[alpha](l[nu](X), [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)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[mu](l[nu](X), [X])+(1/2)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[nu](l[mu](X), [X])

(15)

Simplify((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`]*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_[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)*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)*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)*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)*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_[mu](l[`~alpha`](X), [X])*Physics:-d_[alpha](l[nu](X), [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)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[mu](l[nu](X), [X])+(1/2)*Physics:-d_[alpha](l[`~alpha`](X), [X])*l[`~mu`](X)*l[`~nu`](X)*Physics:-d_[nu](l[mu](X), [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)

(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))"

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?

 

 

NULL

 

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