In this post, I will be demonstrating a simple way to prove the contracted Bianchi Identity.
We start with the usual Bianchi Identity in GTR:
where is the Riemann Tensor,
and is the covariant derivative.
We see that there are 5 indices in the above tensors. So we go ahead and contract alternate indices twice, to get a single index tensor.
Remember the following properties of the Riemann Tensor:
Now, swap (interchange) the following in the first term of :
[Remember Riemann Tensor is pair symmetric]
Perform a similar swapping of indices, as in the last step, on the second and third term too, to get:
Operate on the above equation,
[Note that is covariantly constant and can be taken inside the covariant derivative]
Now, operate on the above,
Since, and are dummy indices, therefore, we can write