1. 2010
    Aug
    15

    WTF is an einbein anyway?

    Posted by David Zaslavsky on

    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

    $\dot{x}^2 = g_{\mu\nu}\partial_\tau X^\mu \partial_\tau X^\nu

    Substituting all this in,

    $$S = \frac{T}{2}\int\udc\tau e^{-1}\dot{x}^2$$

    Now I just have to arbitrarily …

  2. 2010
    Aug
    14

    The origin of the einbein

    Posted by David Zaslavsky on — Edited

    You know that feeling you get when you’re reading a textbook and the authors start pulling equations out of thin air on the second page of the second chapter? No? Well then, my congratulations on having a life.

    But seriously, I came across a gem of this sort in Superstring Theory volume 1 by Green, Schwartz, and Witten. Starting with the action for the relativistic point particle,

    $$S = -m\int\udc s = -m\int\udc\tau\sqrt{-\dot{x}^2}$$

    they introduce an “einbein” \(e(\tau)\) to eliminate the problems that (1) the formula has a square root and (2) it doesn’t apply to massless particles. Fine, but WTF is an einbein anyway? And where do you come up with the equation in the very next sentence,

    $$S = \frac{1}{2}\int\bigl(e^{-1}\dot{x}^2 - em^2\bigr)\udc\tau$$

    I’ve been wondering about this for far too long. Here’s an explanation: we know experimentally that massless particles travel at the speed of light, along null geodesics. But the spacetime interval \(\udc s\) along a null geodesic is zero. So when you try to compute the classical point-particle action \(-m\int \udc s …