How to define a tensor in xAct for Mathematica?

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.

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.

• 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 $$u^\mu \longrightarrow(0,1,cos(\phi),\sin(\phi))$$ Commented May 1, 2017 at 21:29

To define a specific tensor by the assignation of every single component

u = CTensor[{0, 1,cos[ϕ[]],sin[ϕ[]]}, {ch}]

• Showing a specific example of the usage of your code would be helpful. Commented Aug 16, 2023 at 12:22