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Question: Covariant derivative of a specific vector

Question:Covariant derivative of a specific vector

fppoulis 5 Maple 2019
derivative vector covariant physics
December 05 2019
1

Hi there!

Sorry to ask, but I don't know how I can solve it in a smart way. I want to take the covariant derivative of a specific vector, whose components are specifically defined, which can be constant or some functions of the coordinates. But, as soon as I define that specific vector, the covariant derivative fails to compute, saying there are "too many levels of recursion". Let me show what I mean.

I have these two attempts to get this:

 

First attempt

 

restart

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X = spherical)

{X}

(1.1)

Parameters(k)

{k}

(1.2)

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}]

(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)}

(1.4)

NULL

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

(1.5)

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

(1.6)

Db[]

Db[mu, nu] = Matrix(%id = 18446746372997772822)

(1.7)

beta := proc (X) options operator, arrow; b0 end proc``

proc (X) options operator, arrow; b0 end proc

(1.8)

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

(1.9)

Db[]

Error, (in index/PhysicsTensor) too many levels of recursion

 

````

Second attempt

 

restart

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X = spherical)

{X}

(2.1)

Parameters(beta, k)

{beta, k}

(2.2)

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}]

(2.3)

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)}

(2.4)

``

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

{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)}

(2.5)

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

(2.6)

Db[]

Error, (in index/PhysicsTensor) too many levels of recursion

 

````

``

On my first attempt, expression (1.7) is almost fine, but β is supposed to be constant (and a function of "t" in the future), so I set it as a constant function. But then it causes that problem. Therefore, on my second attempt, I tried to set β as a parameter since the beginning, but it was no good either.


If I were to take the covariant derivative as "D_[nu](b[mu])" (without the "(X)"), it would not have that specific problem, but would not be a good solution, since the covariant derivative would discard the partial derivative of b[mu] and, if it would have some dependence on the coordinates, it would give a wrong result.


The only way it seems to work is by defining the tensor "Db[mu,nu]" explicitly as "d_[mu](b[nu](X)) - Christoffel[~alpha, mu, nu]*b[alpha]". (It seems that the problem comes from that "(X)" next to "b[alpha]" in the connection term.) But that would be an awful way to circumvent the problem. Isn't there any better way to get this?


Can someone help me with this, please?


Thanks for any help!

 

Download Cov._derivative_of_a_specific_vector.mw

 
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