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I'm working on a cosmological model for general relativity, and I need to define a tensor and assign values to this. For example, a tensor $A_{\mu\nu}$ that is a function of other tensors:

$$A_{\mu\nu}=R_{\mu\alpha}g^{\mu \alpha}+G^{\mu\beta}T_{\beta\nu}$$

And I need to use this expression for $A_{\mu\nu}$ to calculate the covariant derivative of this. I have the components of every tensor of this expression ($T_{00}=\cos(\theta), T_{11}=\sin(\theta)$, etc..), but I need express $A_{\mu\nu}$ in this way.

How can I do this in xAct? because in the tutorials of xCoba I didn't find something like this.

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2 Answers 2

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You can do the following (there might be more alternatives):

1 Define a tensor to represent A_{\mu\nu} (I assume that M4 is your manifold):

In[]:= DefTensor[A[-mu,-nu],M4]

2 Type your expression with the correct index configuration (there are indices misplaced in your original expression):

In[]:= expr = A[-mu,-nu]==R[-mu,-alpha]g[-nu,alpha]+G[-mu,beta]T[-beta,-nu]

3 Use xCoba commands to compute the components of A[-mu,-nu] (I assume that B is your basis):

In[]:= ToBasis[B]/@expr;
       TraceBasisDummy/@%;
       ComponentArray@%;
       ToValues@%;

4 The last output should be a list giving all the components you are looking for in the base B if all the components of the tensors in the R.H.S have been already computed.

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    $\begingroup$ Thank you!, It works!! @Alfonso But I have another question. How do you define a specific tensor by assignation of every single component?? For example, I need to define $$u^\mu \longrightarrow(0,1,cos(\phi),\sin(\phi))$$ $\endgroup$ Commented May 1, 2017 at 21:29
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To define a specific tensor by the assignation of every single component

u = CTensor[{0, 1,cos[ϕ[]],sin[ϕ[]]}, {ch}]
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    $\begingroup$ Showing a specific example of the usage of your code would be helpful. $\endgroup$
    – bbgodfrey
    Commented Aug 16, 2023 at 12:22

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