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:
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
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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