Contraction with delta[-a,-b]

[Sergei Savin]

Hi all! 

I am working in Euclidean metric, where people do not care about raising and lowering indices. Very simple example of an expression is T_{a,b,c,d} = d_{ab} d_{cd} + d_{ac} d_{bd} where d_{ab} stands for Kronecker delta. I need further to contract T with another tensor, say, A^{a,b,c,d}. 

Thus, I must implement euclidean metric on my manifold. 

What I am doing is (very simple 2-dim example)

Needs["xAct`xCoba`"]
DefManifold[M, 2, {a, b, c, d, e, f, g, h}]
DefChart[Cart, M, {1, 2}, {x[], y[]}]
g = CTensor[{{1, 0}, {0, 1}}, {-Cart, -Cart}];
SetCMetric[g, Cart]
MetricCompute[g, Cart, All, CVSimplify -> Simplify]

What I want to get as an output is basically 

T^{ab} d_{bc} =T^a_c

I am doing the following

DefScalarFunction[{T11, T12, T21, T22}]
T = CTensor[{{T11, T12}, {T21, T22}}, {-Cart, -Cart}];
T[b, c] delta[-a, -b]

and get 

Any help is appreciated.

Thank you in advance,

Sergei.

[Alfonso Jacinto García Parrado Gómez Lobo]

Hi,

If you use DefScalarFunction to introduce the components of your tensor T, then you should use InChart in order to get the components as functions of the coordinates:

T = CTensor[{{InChart[T11, Cart] @@ ScalarsOfChart@Cart,
     InChart[T12, Cart] @@ ScalarsOfChart@Cart}, {InChart[T21, Cart] @@
       ScalarsOfChart@Cart,
     InChart[T22, Cart] @@ ScalarsOfChart@Cart}}, {-Cart, -Cart}];

Is this perhaps the problem which you asked assistance for ?

Alfonso.
 

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Sent: Friday, September 30, 2016 4:38 PM
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Subject: [xAct: 1877] Contraction with delta[-a,-b]

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[Sergei Savin]

Hi Alfonso,

thank you for the answer. 

Not exactly. 

What I need help with is to do contractions with delta Kronecker  in Euclidean metric such that I do not care about lowering and raising indices, e.g. T_{ab} delta_{bc} =T_{ac}.

I do not need dependence on coordinates. I could do this with xTensor but for this I must write indices of deltas in special order which is impossible in my case (example above). I used DefScalarFunction simply because I have no idea how to do this, I've read documentation and also tried to find the answer here.

Thanks in advance,

Sergei.

воскресенье, 2 октября 2016 г., 16:05:19 UTC+2 пользователь Alfonso Jacinto García Parrado Gómez Lobo написал:

[Thomas Bäckdahl]

Hi!

xTensor is always using the metric (g in our case) to move indices up and down.
(By the way you should not use g ans an index and as a metric so remove it from the list of indices in DefManifold.)

Now, if you have a Cartesian metric, then g[-a,-b] is the same as delta[-a,-b]. However, to make xAct understand what you mean you should write g[-a,-b] instead of delta[-a,-b], because delta[-a,-b] has no meaning in xAct. That should hopefully solve your problem. As xAct cares about if indices are up or down, so you will have to type them in like that. Hence, contractions are always between an up index and a down index. If they don't appear like that in your original expression, you can raise and lower them with g[a,b] or g[-a,-b], which in your case will not change the meaning because you have a Cartesian metric. Without a Cartesian metric, such expressions would not have a covariant meaning.

Regards

Thomas

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