Mnemonic for Riemann Curvature Tensor

It’s been a year now, I rarely write blog posts. I hope you guys are still following my blog. In this short post, I’m very happy to share the mnemonic that I discovered. Long story short, I don’t like the fact that I have to remember the indices place in Riemann Curvature Tensor. Please comment if you also felt the same. So, I take my time to find a mnemonic for this formula. Voilà: I found it.
Let me first remind you of the formula for Riemann Curvature Tensor,
.
I’m naming partial derivative as P and Christoffel connection
as C. And,
means index to be filled. So, I write right-hand side part of Riemann Curvature Tensor as
.
Our Mnemonic is actually this: you can spell the right-hand side as PC, PC, CC, CC and then insert and
sign consecutively.
So, our initial form looks like this:
.
Now, the only step remaining is to find a way to insert the indices in and
. By looking just the formula, it’s quite easy to remember the place of
and dummy index
which is at
.
So, let’s make a visualization to find a way to insert ,
and
indices. In
, let’s only observe the position of lower indices
as
. For sign, the index goes clockwise direction starting from
as
i.e. and for
sign, the index goes anti-clockwise direction starting from
as
i.e.
. Then replace three square boxes (I mean
on each term of Riemann Curvature Tensor. Surprise! You will see this
.
Feedback?
If you guys have some questions, comments, or insults then, please don’t hesitate to shot me an email or comment below.
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