I am doing some manipulations with tensors in curved background. There are, however, coordinates, where Lorentz symmetry is manifest. So, on one hand, I am dealing with particular coordinates and, on the other hand, I don't want to deal with tensors component by component, because there is Lorentz symmetry around that simplifies things.
For example, we have a vector field of the form $$A_\mu = a_{\mu}F(x^\nu x_\nu,x^\lambda b_\lambda)$$ where $a_\mu$ and $b_\lambda$ are constant covectors and $x^\nu$ is a coordinate and $F$ - some unknown function of scalar arguments. I would like to be able to evaluate efficiently expressions like $D_\mu A_\nu$ where the Christoffel symbols are also expressed in terms of $x^\mu$ and some constant parameters/Lorentz vectors (it would be nice to implement evaluation of the Christoffel symbols automatically, but anyway I already computed them). In computations, I would like Mathematica to do things like $$\partial_\mu F(x^2,bx)=\frac{\partial F}{\partial x^2}2x_\mu + \frac{\partial F}{\partial bx}b_\mu, \qquad \partial_\mu x^\nu =\delta^\nu_{\mu}$$ and so on and then to be able to contract indices. What is the best way to achieve that? I assume, I can use xCoba and introduce (co)vector fields for $a$, $x$ and $b$. But then Mathematica will not take into account, for example, that $\partial_\mu a_\nu =0$ unless I assign to $a_\mu$'s components some constant values $a_1$, $a_2$, $a_3$ and $a_4$. On the other hand, if I do assign some concrete values to the components, I am afraid, I will get $a^\mu b_\mu$ in the form $a^1b_1+a^2b_2+a^3b_3+a^4b_4$, which is a mess. Or, may be it is better to deal with Mathematica's built in tensor manipulations?
I looked through suggested similar questions and it seems that Cartesian tensor gradient will do the job with differentiation. Essentially, to the best of my understanding, I am supposed to teach Mathematica how derivatives act on each expression that one encounters. But then I will also need to contract indices, e.g. $$x^\mu\eta_{\mu\nu}\to x_\nu, \qquad x^\mu x^\mu \eta_{\mu\nu} \to x^2.$$ Should I also then teach Mathematica all contraction rules? How does it work in practice?
