I am working my way through Carroll's text on GR and am having trouble understanding what it means when an index is inside/outside parentheses. For example, in his discussion of geodesic deviation, he defines the relative velocity of geodesics as:
$$V^\mu=(\nabla_TS)^\mu$$
Can someone help me understand this notation?
The idea is that $\nabla_T S$ is a tensor field (in this case, a vector field), and $\big(\nabla_T S\big)^\mu$ is its $\mu^{th}$ component. The parentheses are meant to distinguish this from $\nabla_TS^\mu \overset{?}=\nabla_T \big(S^\mu\big)$, which in the present context does not actually make sense, but which could nevertheless be confusing to students learning about tensors for the first time.
There is (at least) two equivalent ways of performing such calculations on tensors.
One would have the covariant derivative acting on the whole tensor field, for example on a (1,1)-valence one:
$$T = T^{\mu}_{\;\;\;\nu}\partial_{\mu} \otimes dx^{\nu} $$
$$\nabla_{\alpha}T = \nabla_{\alpha}(T^{\mu}_{\;\;\;\nu}\partial_{\mu} \otimes dx^{\nu}) = \nabla_{\alpha}(T^{\mu}_{\;\;\;\nu}) \partial_{\mu} \otimes dx^{\nu} + T^{\mu}_{\;\;\;\nu} \nabla_{\alpha} \partial_{\mu} \otimes dx^{\nu} + T^{\mu}_{\;\;\;\nu} \partial_{\mu} \otimes \nabla_{\alpha} dx^{\nu},$$
where $\nabla_{\alpha}(T^{\mu}_{\;\;\;\nu}) = \partial_{\alpha}T^{\mu}_{\;\;\;\nu} $ because tensor field components are functions, and the covariant derivatives of coordinate vector fields and 1-forms are:
$$ \nabla_{\alpha}\partial_{\mu} = \Gamma^{\lambda}_{\alpha\mu}\partial_{\lambda}$$
$$ \nabla_{\alpha} dx^{\nu} = - \Gamma^{\nu}_{\alpha\lambda} dx^{\lambda}$$
If you change the dummy (summation) indices appropriately, you will get:
$$ \nabla_{\alpha }T = \big{[} \partial_{\alpha}T^{\mu}_{\;\;\;\nu} + T^{\lambda}_{\;\;\;\nu}\Gamma^{\mu}_{\alpha\lambda} - T^{\mu}_{\;\;\;\lambda} \Gamma^{\lambda}_{\alpha\nu}\big{]} \partial_{\mu}\otimes dx^{\nu} \equiv (\nabla_{\alpha}T)^{\mu}_{\;\;\;\nu}\partial_{\mu}\otimes dx^{\nu}.$$
In the above, the final expression for the tensor components of the derived field is present: $(\nabla_{\alpha}T)^{\mu}_{\;\;\;\nu}$. In the community one would typically write it as $\nabla_{\alpha}T^{\mu}_{\;\;\;\nu}$ as it is clear from the context that we do mean first the derivative, then reading off its components. You can also use the semicolon notation $(\nabla_{\alpha}T)^{\mu}_{\;\;\;\nu}=T^{\mu}_{\;\;\;\nu;\alpha}$, where it is again understood that first the covariant derivative along $\partial_{\alpha}$ is taken, and then the tensor components are read off.
