I am taking a course in General Relativity at university and am struggling with getting to grips with tensors which are new to me. I am struggling with the idea of taking the covariant derivative of a covariant derivative.
The object I am dealing with is given in my lecture notes as follows:
$$∇_\beta{}∇_\gamma{}\omega{}_\delta{}=\partial_\beta(\partial_\gamma\omega_\delta-\Gamma_{\gamma\delta}^\sigma\omega_\sigma)-\ \Gamma_{\beta\gamma}^\rho(\partial_\rho\omega_\delta-\Gamma_{\rho\delta}^\sigma\omega_\sigma)-\ \Gamma_{\beta\delta}^\rho(\partial_\gamma\omega_\rho-\Gamma_{\gamma\rho}^\sigma\omega_\sigma)\tag1$$
However, I am struggling to arrive at the same terms as the lecture notes. I understand that the first covariant derivative $∇_\gamma\omega_\delta$ is given as:
$$∇_\gamma\omega_\delta=\partial_\gamma\omega_\delta-\Gamma_{\gamma\delta}^\sigma\omega_\sigma$$
Where I become stuck is taking the covariant derivative of $∇_\gamma\omega_\delta$ as my efforts result in the following:
$$∇_\beta∇_\gamma\omega_\delta=\partial_\beta(\partial_\gamma\omega_\delta-\Gamma_{\gamma\delta}^\sigma\omega_\sigma) -X$$
Where X is some term with a Christoffel symbol in front but I cannot write it out as I am unsure of how to correctly assign the indices and also how you would end up with two terms.
I would really appreciate it if someone could clearly break down how I would arrive at the expression in (1) keeping in mind that tensors are a new concept to me.
I have seemed to work it out.
$$∇_\gamma{}\omega_\delta=\partial_\gamma\omega_\delta-\Gamma_{\gamma\delta}^\sigma\omega_\sigma=A_{\gamma\delta}\tag2$$
Now taking the covariant derivative of the result of (2) with respect to $x^\beta$.
$$∇_\beta{}A_{\gamma\delta}=\partial_\beta{}A_{\gamma\delta}-\Gamma_{\beta\gamma}^\rho{}A_{\rho\delta}-\Gamma_{\delta\beta}^\rho{}A_{\gamma{}\rho}\tag3$$
Now substituting in the tensor relationship that is $A_{\gamma\delta}$ into (3) and making sure we change the indices appropriately gives:
$$∇_\beta{}A_{\gamma\delta}=\partial_\beta(\partial_\gamma\omega_\delta-\Gamma_{\gamma\delta}^\sigma\omega_\sigma)-\Gamma_{\beta\gamma}^\rho(\partial_\rho\omega_\delta-\Gamma_{\rho\delta}^\sigma\omega_\sigma)\ -\Gamma_{\delta\beta}^\rho(\partial_\gamma\omega_\rho-\Gamma_{\gamma\rho}^\sigma\omega_\sigma)\tag4$$
This matches what was given in the original question and was arrived at using a method of working out the covariant derivative for object's with multiple indices which was given in previous lecture notes.