fppoulis

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These are replies submitted by fppoulis

@Carl Love 

Thanks, mate! I guess I have already solved...

 

Cheers!

Well... It seems things get solved just by omitting that "(X)" indeed. It will only discard the partial derivatives of b[mu] until you specifi its components. Whenever they are set, it will compute the covariant derivative correctly, as can be checked from the file below.
 

restart

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X = spherical)

{X}

(1)

Parameters(k)

{k}

(2)

Setup(metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = (a(t)*r)^2, (3, 3) = (a(t)*r*sin(theta))^2, (4, 4) = -1})

[metric = {(1, 1) = a(t)^2/(-k*r^2+1), (2, 2) = a(t)^2*r^2, (3, 3) = a(t)^2*r^2*sin(theta)^2, (4, 4) = -1}]

(3)

Define(b[mu])

{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], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(4)

NULL

Define(Db[nu, mu] = D_[nu](b[mu]))NULL

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(5)

Db[definition]

Db[nu, mu] = -Physics:-Christoffel[`~alpha`, mu, nu]*b[alpha]

(6)

Db[nonzero]

Db[mu, nu] = {(1, 1) = (k*r*b[1]+a(t)*(diff(a(t), t))*b[4])/(k*r^2-1), (1, 2) = -b[2]/r, (1, 3) = -b[3]/r, (1, 4) = -(diff(a(t), t))*b[1]/a(t), (2, 1) = -b[2]/r, (2, 2) = -r*(a(t)*r*(diff(a(t), t))*b[4]+b[1]*(k*r^2-1)), (2, 3) = -cos(theta)*b[3]/sin(theta), (2, 4) = -(diff(a(t), t))*b[2]/a(t), (3, 1) = -b[3]/r, (3, 2) = -cos(theta)*b[3]/sin(theta), (3, 3) = -(a(t)*r^2*sin(theta)*(diff(a(t), t))*b[4]+r*b[1]*(k*r^2-1)*sin(theta)-cos(theta)*b[2])*sin(theta), (3, 4) = -(diff(a(t), t))*b[3]/a(t), (4, 1) = -(diff(a(t), t))*b[1]/a(t), (4, 2) = -(diff(a(t), t))*b[2]/a(t), (4, 3) = -(diff(a(t), t))*b[3]/a(t)}

(7)

Define(b[mu] = [0, 0, 0, beta(t)], redo)

{Physics:-D_[mu], Db[nu, mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(8)

Define(Db[nu, mu] = D_[nu](b[mu]), redo)

{Physics:-D_[mu], Db[mu, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(9)

Db[definition]

Db[mu, nu] = Physics:-D_[nu](b[mu], [X])

(10)

expand(Db[definition])

Db[mu, nu] = Physics:-d_[nu](b[mu], [X])-Physics:-Christoffel[`~alpha`, mu, nu]*b[alpha]

(11)

Db[nonzero]

Db[mu, nu] = {(1, 1) = a(t)*(diff(a(t), t))*beta(t)/(k*r^2-1), (2, 2) = -a(t)*r^2*(diff(a(t), t))*beta(t), (3, 3) = -a(t)*r^2*sin(theta)^2*(diff(a(t), t))*beta(t), (4, 4) = diff(beta(t), t)}

(12)

Define(b[mu] = [0, 0, 0, beta], redo)

{Physics:-D_[mu], Db[mu, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(13)

Define(Db[nu, mu] = D_[nu](b[mu]), redo)

{Physics:-D_[mu], Db[mu, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], b[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(14)

Db[definition]

Db[mu, nu] = -Physics:-Christoffel[`~alpha`, mu, nu]*b[alpha]

(15)

expand(Db[definition])

Db[mu, nu] = -Physics:-Christoffel[`~alpha`, mu, nu]*b[alpha]

(16)

Db[nonzero]

Db[mu, nu] = {(1, 1) = a(t)*(diff(a(t), t))*beta/(k*r^2-1), (2, 2) = -a(t)*r^2*(diff(a(t), t))*beta, (3, 3) = -a(t)*r^2*sin(theta)^2*(diff(a(t), t))*beta}

(17)

``


On equation (6) it discards the partial derivative, but as soon as b[mu] components are set, it comes back accordingly, as in (10)-(12).

Download Cov._derivative_of_a_specific_vector_(solved).mw

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