Mnemonic for Riemann Curvature Tensor

15. March 2020 Differential Geometry, General Relativity Comments
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,

R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\nu \sigma} - \partial_{\nu} \Gamma^{\rho}_{\mu \sigma} + \Gamma^{\rho}_{\mu \lambda}\Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda}\Gamma^{\lambda}_{\mu \sigma}.

I’m naming partial derivative \partial_{\square} as P and Christoffel connection \Gamma^{\square}_{\square\square} as C. And, \square means index to be filled. So, I write right-hand side part of Riemann Curvature Tensor as

R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu} = + \text{PC} - \text{PC} + \text{CC} - \text{CC}.

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:

R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu} = + \hspace{0.2cm} \partial_{\square} \Gamma^{\square}_{\square\square} \hspace{0.2cm} - \hspace{0.2cm} \partial_{\square} \Gamma^{\square}_{\square\square} \hspace{0.2cm} + \hspace{0.2cm} \Gamma^{\square}_{\square\square} \Gamma^{\square}_{\square\square} \hspace{0.2cm} - \hspace{0.2cm} \Gamma^{\square}_{\square\square} \Gamma^{\square}_{\square\square}.

Now, the only step remaining is to find a way to insert the indices in \partial_{\square} and \Gamma^{\square}_{\square\square}. By looking just the formula, it’s quite easy to remember the place of \rho and dummy index \lambda which is at

R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu} = + \hspace{0.2cm} \partial_{\square} \Gamma^{\rho}_{\square\square} \hspace{0.2cm} - \hspace{0.2cm} \partial_{\square} \Gamma^{\rho}_{\square\square} \hspace{0.2cm} + \hspace{0.2cm} \Gamma^{\rho}_{\square\lambda} \Gamma^{\lambda}_{\square\square} \hspace{0.2cm} - \hspace{0.2cm} \Gamma^{\rho}_{\square\lambda} \Gamma^{\lambda}_{\square\square}.

So, let’s make a visualization to find a way to insert \sigma, \mu and \nu indices. In R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu}, let’s only observe the position of lower indices \sigma \mu \nu as

. For + sign, the index goes clockwise direction starting from \mu as

i.e. \mu\nu\sigma and for - sign, the index goes anti-clockwise direction starting from \nu as

i.e. \nu\mu\sigma. Then replace three square boxes (I mean \square) on each term of Riemann Curvature Tensor. Surprise! You will see this

R^{\rho}_{\hspace{0.15cm}\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\nu \sigma} - \partial_{\nu} \Gamma^{\rho}_{\mu \sigma} + \Gamma^{\rho}_{\mu \lambda}\Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda}\Gamma^{\lambda}_{\mu \sigma}.

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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