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

PhD researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT and Machine Learning related apps and softwares from time to time. Can code in most of the popular languages. Like to share my knowledge in Physics and applications using this Blog and a YouTube channel.