The contracted Bianchi Identity in GTR can be written as:

where is the Einstein Tensor such that,

and is the Ricci Tensor and is the Ricci Scalar.

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 :

with ,

with ,

and with

[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,

Therefore,

Since, and are dummy indices, therefore, we can write

Hence Proved.

### References:

http://mathworld.wolfram.com/ContractedBianchiIdentities.html

https://math.stackexchange.com/questions/756031/proving-the-contracted-bianchi-identity

https://en.wikipedia.org/wiki/Proofs_involving_covariant_derivatives