Ellipsix Informatics: WTF is an einbein anyway?

2010

Aug

WTF is an einbein anyway?

Posted by David on August 15, 2010 — Comments

I just realized that in my last post I sort of neglected to address the main question. So what is an einbein? Turns out the answer is on the next page of the Green, Schwartz, and Witten textbook: the einbein is the induced metric, normally written $h_{\alpha\beta}$ , where $\alpha$ and $\beta$ range over the coordinates in the parametrization of the worldline/worldsheet/whatever. A one-dimensional worldline is parametrized by only one coordinate, $\tau$ , so the induced metric has only one component, $h_{\tau\tau} \equiv -e^2$ .

Naturally, all this emerges from the general string/brane action,

$S = -\frac{T}{2}\int\udc^{n+1}\sigma\sqrt{\abs{h}}h^{\alpha\beta}g_{\mu\nu}\partial_\alpha X^\mu \partial_\beta X^\nu$

The set of brane coordinates $\sigma$ is just $\tau$ , the induced metric determinant is $h = -e^2$ , and the inverse induced metric has only the one component, $h^{\tau\tau} = -e^{-2}$ . Also, $\dot{x}^2$ is equal to