How to access the value of an object?

Question

If I define an object like

expr:= A^{\mu \nu} B_{\nu \rho} C^{\rho \lambda};

and perform some manipulation (for instance the following)

rename_dummies(expr);

How do I access the resulting expression to perform further manipulations? For example, I'd like to do something like

\partial_\sigma expr

or

D_{\kappa \mu} expr

or something like that such that the resulting object has the appropriate free indices. Is there a way to do this without explicitly inserting the original value?

Answer 1

Your own answer works, but makes it impossible to use that same expression with a different free index. A more Cadabra-like way of doing this is to declare

{\mu,\nu,\rho,\lambda,\phi,\chi}::Indices.
ex1:= T^{\mu\lambda} = A^{\mu \nu} B_{\nu \rho} C^{\rho \lambda};
ex2 := D_{\kappa \mu\nu} T^{\mu\nu};

and then substitute expr with

 substitute(ex2, ex1);

In Cadabra, equations have names, but those names are not themselves tensors. Those equation names (ex1 and ex2 above) do not carry indices.

Comments

Thanks for your response. But how do I now access the result of the substitution as a tensor?

---

You can do something similar there; e.g. instead of what I wrote for ex2, write

ex2:= S_{\kappa} = D_{\kappa\mu\nu} T^{\mu\nu};

Then you can use this $S_\kappa$ tensor again in another expression and substitute similar to what I did above.

In other words: do not view the Python name for an expression (ex1 and ex2) above as a tensor; rather, write out that tensor explicitly. If you find it confusing, you could e.g. write

 T:= T^{\mu\lambda} = ...

instead of the first line, so use the same name T for the expression and for the tensor that it defines.

This all comes from the fact that in Cadabra there is a difference between the "programming language" (Python) and the "maths language" (LaTeX). Python names do not know about indices.

Answer 2

Couldn't find this in the documentation, but this seems to work:

expr:= A^{\mu \nu} B_{\nu \rho} C^{\rho \lambda};

rename_dummies(expr);

D_{\kappa \mu} @(expr);