Wednesday, January 04, 2012

General Relativity flash-cards using Anki

Anki is a neat spaced repetition system that allows you to maximize memorization efficiency, or something like that. Interestingly, I came across it on the Clojure group, and there are already decks available for learning Clojure. It accepts LaTeX, so I've decided to make a flash-deck of some handy formulas that pop up in General Relativity, because there's enough to learn without forgetting!

(Also, it's handy to have LaTeX snippets someplace semi-permanent.)

Bianchi identity:

$\nabla_{[\lambda}R_{\rho\sigma]\mu\nu}=0$

Christoffel symbol:

$\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)$

Covariant derivative of a 1-form:

$\nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}-\Gamma_{\mu\nu}^{\lambda}\omega_{\lambda}$

Covariant derivative of a vector:

$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma_{\mu\lambda}^{\nu}V^{\lambda}$

Covariant form of Maxwell's equations:


$\partial_{\mu}F^{\nu\mu}=J^{\nu}$

$\partial_{[\mu}F_{\nu\lambda]}=0$

for

$J^{\nu}=\left(\rho,J^{x},J^{y},J^{z}\right)$

and

$F_{\mu\nu}=\left(\begin{array}{cccc}
0 & -E_{1} & -E_{2} & -E_{3}\\
E_{1} & 0 & B_{3} & -B_{2}\\
E_{2} & -B_{3} & 0 & B_{1}\\
E_{3} & B_{2} & -B_{1} & 0
\end{array}\right)$

Riemann tensor:

$R_{\sigma\mu\nu}^{\rho}=\partial_{\mu}\Gamma_{\nu\sigma}^{\rho}-\partial_{\nu}\Gamma_{\mu\sigma}^{\rho}+\Gamma_{\mu\lambda}^{\rho}\Gamma_{\nu\sigma}^{\lambda}-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\mu\sigma}^{\lambda}$

Properties of the Riemann tensor:


$R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}$

$R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\nu\mu}$

$R_{\rho\sigma\mu\nu}=R_{\mu\nu\rho\sigma}$

$R_{\rho[\sigma\mu\nu]}=0$


Ricci tensor:

$R_{\mu\nu}=R_{\mu\lambda\nu}^{\lambda}$

Ricci scalar:

$R=R_{\mu}^{\mu}=g^{\mu\nu}R_{\mu\nu}$

Einstein tensor:

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$

Formulae from Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity

Anki synchronizes with DropBox, but it's a bit involved. When I get my deck synchronized and uploaded, I will post a link to it.

Update: https://ankiweb.net/shared/info/1777635479

1 comment:

joshua@milehighcode.com said...

Thanks for the link-back Adam!!