# Mnemonic for Riemann Curvature Tensor

Its been a year now, I rarely write blog posts. I was busy fixing my personal matter. I hope you guys are still following my blog. In this short post, I’m very happy to share mnemonic that I discovered while I was taking a course on General Relativity as the first semester (2019/2020) Master student in Theoretical Physics at Jagiellonian University. 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 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:

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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

.

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