Question: Arrangement of indices after Simplify

April 27 2014 Michael_Watson 40
Maple 18

0

After using Simplify the indices are are arranged in the tensor.  I am using the April 14th update from the Physics R&D page.

 


restart

with(Physics):

Setup(mathematicalnotation = true, coordinatesystems = X)

[coordinatesystems = {X}, mathematicalnotation = true]

(1)

Define(l[mu], eta[mu, nu] = -rhs(g_[Minkowski]))

{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)

declare(l(X))

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

(3)

InitialMetric := g_[mu, nu] = eta[mu, nu]+Physics:-`*`(l[mu](X), l[nu](X)); 1; Define(G[mu, nu] = rhs(InitialMetric))

{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)}

(4)

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)

NULL

We first define the Christoffel symbol in terms of the metric,   `g__μ,ν`.

``

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

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

(6)

SubstituteTensor(g_[mu, nu] = eta[mu, nu]+Physics:-`*`(l[mu](X), l[nu](X)), Physics:-Christoffel[`~rho`, mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~rho`]*(Physics:-d_[nu](Physics:-g_[alpha, mu], [X])+Physics:-d_[mu](Physics:-g_[alpha, nu], [X])-Physics:-d_[alpha](Physics:-g_[mu, nu], [X])), evaluateexpression)

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

(7)

Simplify(SubstituteTensor(Physics:-`*`(l[`~alpha`](X), l[`~rho`](X)) = -Physics:-`*`(l[`~alpha`](X), l[`~rho`](X)), Physics:-Christoffel[`~rho`, mu, nu] = (1/2)*(eta[`~alpha`, `~rho`]+l[`~alpha`](X)*l[`~rho`](X))*(Physics:-d_[nu](l[alpha](X), [X])*l[mu](X)+l[alpha](X)*Physics:-d_[nu](l[mu](X), [X])+Physics:-d_[mu](l[alpha](X), [X])*l[nu](X)+l[alpha](X)*Physics:-d_[mu](l[nu](X), [X])-Physics:-d_[alpha](l[mu](X), [X])*l[nu](X)-l[mu](X)*Physics:-d_[alpha](l[nu](X), [X])), evaluateexpression))

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

(8)

SubstituteTensor(Physics:-`*`(l[alpha](X), eta[`~alpha`, `~rho`]) = l[`~rho`](X), Physics:-Christoffel[`~rho`, mu, nu] = (1/2)*Physics:-d_[nu](l[alpha](X), [X])*l[mu](X)*eta[`~alpha`, `~rho`]-(1/2)*l[mu](X)*Physics:-d_[alpha](l[nu](X), [X])*eta[`~alpha`, `~rho`]+(1/2)*l[alpha](X)*Physics:-d_[nu](l[mu](X), [X])*eta[`~alpha`, `~rho`]+(1/2)*l[alpha](X)*Physics:-d_[mu](l[nu](X), [X])*eta[`~alpha`, `~rho`]+(1/2)*Physics:-d_[mu](l[alpha](X), [X])*l[nu](X)*eta[`~alpha`, `~rho`]-(1/2)*l[nu](X)*Physics:-d_[alpha](l[mu](X), [X])*eta[`~alpha`, `~rho`]-(1/2)*l[`~alpha`](X)*l[`~rho`](X)*Physics:-d_[nu](l[alpha](X), [X])*l[mu](X)+(1/2)*l[`~alpha`](X)*l[`~rho`](X)*l[mu](X)*Physics:-d_[alpha](l[nu](X), [X])-(1/2)*l[`~alpha`](X)*l[`~rho`](X)*l[alpha](X)*Physics:-d_[nu](l[mu](X), [X])-(1/2)*l[`~alpha`](X)*l[`~rho`](X)*l[alpha](X)*Physics:-d_[mu](l[nu](X), [X])-(1/2)*l[`~alpha`](X)*l[`~rho`](X)*Physics:-d_[mu](l[alpha](X), [X])*l[nu](X)+(1/2)*l[`~alpha`](X)*l[`~rho`](X)*l[nu](X)*Physics:-d_[alpha](l[mu](X), [X]))

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

(9)

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

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

(10)

NULL

NULL

Now we can substitute into the null condition for the Ricci tensor, `R__μν`*`#mi("l")`^mu*l^nu = 0.

convert(Physics:-`*`(Physics:-`*`(Ricci[mu, nu], l[`~mu`](X)), l[`~nu`](X)), Christoffel)

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

(11)

NULL

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

(Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~rho`, mu, rho], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~rho`, beta, rho]-Physics:-Christoffel[`~beta`, mu, rho]*Physics:-Christoffel[`~rho`, beta, nu])*l[`~mu`](X)*l[`~nu`](X)

(12)

  Do the first term

 

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

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

(13)

NULL

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

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

(14)

 

 

Do same thing with the first term but use the Simplify command

 

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

Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])*l[`~mu`](X)*l[`~nu`](X) = -Physics:-d_[alpha2](l[`~alpha3`](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha5`](X)*l[`~rho`](X)*l[rho](X)+Physics:-d_[alpha4](l[`~alpha5`](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha3`](X)*l[`~alpha4`](X)-Physics:-d_[alpha](l[alpha3](X), [X])*eta[`~alpha`, `~alpha1`]*Physics:-d_[alpha1](l[alpha2](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+eta[`~alpha`, `~alpha5`]*Physics:-d_[alpha2](l[alpha](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)-eta[`~alpha5`, `~alpha3`]*Physics:-d_[alpha3](Physics:-d_[alpha5](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha6`](X)+eta[`~alpha6`, `~alpha5`]*Physics:-d_[alpha3](Physics:-d_[alpha5](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](l[alpha3](X), [X])*Physics:-d_[rho](l[`~rho`](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](Physics:-d_[alpha3](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)*l[`~alpha6`](X)

(15)

SubstituteTensor(Physics:-`*`(l[`~nu`](X), l[nu](X)) = 0, Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])*l[`~mu`](X)*l[`~nu`](X) = -Physics:-d_[alpha2](l[`~alpha3`](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha5`](X)*l[`~rho`](X)*l[rho](X)+Physics:-d_[alpha4](l[`~alpha5`](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha3`](X)*l[`~alpha4`](X)-Physics:-d_[alpha](l[alpha3](X), [X])*eta[`~alpha`, `~alpha1`]*Physics:-d_[alpha1](l[alpha2](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+eta[`~alpha`, `~alpha5`]*Physics:-d_[alpha2](l[alpha](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)-eta[`~alpha5`, `~alpha3`]*Physics:-d_[alpha3](Physics:-d_[alpha5](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha6`](X)+eta[`~alpha6`, `~alpha5`]*Physics:-d_[alpha3](Physics:-d_[alpha5](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[alpha2](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](l[alpha3](X), [X])*Physics:-d_[rho](l[`~rho`](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](Physics:-d_[alpha3](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)*l[`~alpha6`](X))

Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])*l[`~mu`](X)*l[`~nu`](X) = -Physics:-d_[alpha](l[alpha3](X), [X])*eta[`~alpha`, `~alpha1`]*Physics:-d_[alpha1](l[alpha2](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+eta[`~alpha`, `~alpha5`]*Physics:-d_[alpha2](l[alpha](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](l[alpha3](X), [X])*Physics:-d_[rho](l[`~rho`](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](Physics:-d_[alpha3](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)*l[`~alpha6`](X)

(16)

 

Simplify command does make the algebra easier, but the indices are not the same. Now, equation 16 should correspond to equation 14, but there is no combination of alphas that is consistent.  The variables alpha1 and alpha5 must be rho. One term is always wrong when I try to change the other indices.

alpha2 and alpha3 must be either mu or nu based on the first term.  But alpha6 should also be either mu or nu based on the last term, however that will make alpha2 and alpha3 either (nu and rho) or (mu and rho).  Neither combination makes all of the terms consistent with (14).  Very frustrating.

 

SubstituteTensorIndices({alpha1 = rho, alpha5 = rho}, Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])*l[`~mu`](X)*l[`~nu`](X) = -Physics:-d_[alpha](l[alpha3](X), [X])*eta[`~alpha`, `~alpha1`]*Physics:-d_[alpha1](l[alpha2](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+eta[`~alpha`, `~alpha5`]*Physics:-d_[alpha2](l[alpha](X), [X])*Physics:-d_[alpha5](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](l[alpha3](X), [X])*Physics:-d_[rho](l[`~rho`](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](Physics:-d_[alpha3](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)*l[`~alpha6`](X))

Physics:-d_[rho](Physics:-Christoffel[`~rho`, mu, nu], [X])*l[`~mu`](X)*l[`~nu`](X) = -Physics:-d_[alpha](l[alpha3](X), [X])*eta[`~alpha`, `~rho`]*Physics:-d_[rho](l[alpha2](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+eta[`~alpha`, `~rho`]*Physics:-d_[alpha2](l[alpha](X), [X])*Physics:-d_[rho](l[alpha3](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](l[alpha3](X), [X])*Physics:-d_[rho](l[`~rho`](X), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)+Physics:-d_[alpha2](Physics:-d_[alpha3](l[alpha6](X), [X]), [X])*l[`~alpha2`](X)*l[`~alpha3`](X)*l[`~alpha6`](X)

(17)

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