Express in values, the action of Covariant derivative on a scalar field.

[Jaswin Kasi]

Let Trial[] be a scalar field, I want to look at what happens when I act upon by CD[-a] where I can give a to be some no. 

But I am not getting the result in values. 

I have tried the following line 

CD[{-0, -Global}] Trial[] // ToValues // Simplify

Still I get the following 

E^(-I Rotation t[]) CD[{0, -Global}] Cos[

r[]]^(2 b) Hypergeometric2F1[AA, BB, CC, Sin[

r[]]^2] Sin[

r[]]^(2 a) SphericalHarmonicY[l, m, \[Theta][], \[Phi][]]

What do I do ? 

[Alfonso Jacinto García Parrado Gómez Lobo]

Could you please send a (not too long) notebook with your explicit computation ? Thanks.

Alfonso.

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Sent: Tuesday, May 03, 2016 12:01 PM
To: xAct Tensor Computer Algebra
Subject: [xAct: 1766] Express in values, the action of Covariant derivative on a scalar field.

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[Jaswin Kasi]

Okay since I found that using the usual tensors according to xCoba is very messy, I thought of using CTensors. 

So to be particular about what I am doing, I'll mention the whole problem and attach the semi-working code. 

I have a AdS4 space, with global coordinates. I have a massive free scalar field coupled to the curved space. Now I can solve the equations of motion and find the solutions to the free scalar field, which I have already done. 

Now I need to find the Energy-Momentum tensor for these scalar field solutions. 

The scalar fields solutions turn out to be e^{-iwt}SphericalHarmonics[l,m,theta,phi](...)

where (...) is some Hypergeometric function. 

Just to see if the code works fine, I am trying act act $\Nabla^2_{S^2}$ on SphericalHarmonicsY[l,m,theta,phi] which should give me the l(l+1)SphericalHarmonicsY[l,m,theta,phi] but I am not getting it. There are terms with basis with different colours, so what do I do ? 

Having said that I am not even able to do a simple computation like, finding the covariant derivative on just one of the coordinate say r[]. That too is giving be some basis. What do I do ?  

[Alfonso Jacinto García Parrado Gómez Lobo]

Hi,

>Just to see if the code works fine, I am trying act act $\Nabla^2_{S^2}$ on SphericalHarmonicsY[l,m,theta,phi] >which should give me the l(l+1)SphericalHarmonicsY[l,m,theta,phi] but I am not getting it. There are terms with >basis with different colours, so what do I do ? 

You can get rid of the terms with basis by using TraceBasisDummy. However, I'm not sure that you are using the right expression for the Laplacian on the sphere. I attach your notebook with some additions to do the computation you want. For that end I use the package xTerior which allows you to compute the Hodge Laplacian for any metric and in particular the metric of the 2-sphere.

>Having said that I am not even able to do a simple computation like, finding the covariant derivative on >just one of the coordinate say r[]. That too is giving be some basis. What do I do ?  

The covariant derivative of r[] is a co-vector which can be expanded in the co-base defined by the coordinate chart. That's why you get the basis elements.

Alfonso.

[Jaswin Kasi]

Dear Alfonso, 

Thanks for the help. I realised my mistake, also I saw that simplification of SphericalHarmonicY in Mathematica is not so good, there are reports of not satisfying the eigenvalue equation properly. 

On Tuesday, May 3, 2016 at 7:56:35 PM UTC+5:30, Jaswin Kasi wrote: