Variational principle for a point particle (massive or massless) in curved space

We know that for a point particle, the action is

$$S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) -m^2e(\lambda)\right] ,$$

with signature convention $(-,+,+,+)$. It was mentioned on some website as a I googled that $e$ and $x$ are the dynamical variables and from them we should get the Euler-Lagrange equations.

I was wondering how to start since just a few minutes ago I first encountered this einbein variable (which I didn't know was a variable in the first place)!

• What is the "einbein variable" and what is your question? – DanielSank Sep 29 '14 at 17:34
• Einbein variable is this $e$ term. My question is to find the Euler-Lagrange equations using $x$ and $e$ as dynamical variables. – PhilosophicalPhysics Sep 29 '14 at 17:50
• Then please put the question in the question. – DanielSank Sep 29 '14 at 18:33

1. The einbein field $$e(\lambda)\neq 0$$ is not a dynamical field because there is no $$\dot{e}(\lambda)$$ present. It is a so-called auxiliary field or generalized Lagrange multiplier. Its EL eq. simplifies to $$\tag{1} (me)^2~\approx~-g_{\mu\nu}~\dot{x}^{\mu}\dot{x}^{\nu} .$$ [Here the $$\approx$$ symbol means equality modulo eom.] Here $$m$$ is the restmass of the point particle. See also this related Phys.SE post and links therein.
2. In the massive case $$m>0$$, we can integrate out the $$e$$ field, which means to replace it in the action $$S[x,e]$$ by its eom $$\tag{2} e~\approx~\pm\frac{1}{m}\sqrt{-g_{\mu\nu}~\dot{x}^{\mu}\dot{x}^{\nu}} ,$$ which has two branches. The resulting action is $$\tag{3} S_{\pm}[x]~:=~S\left[x,e=\pm\frac{1}{m}\sqrt{ \ldots}\right]=\mp m \int \!d\lambda~ \sqrt{- g_{\mu\nu}~\dot{x}^{\mu}\dot{x}^{\nu}}~ \left\{ \begin{array}{c} ~<~0 \cr ~>~0 \end{array}\right. .$$ The $$S_{+}[x]$$ branch is the standard square-root action for a massive point particle, cf. e.g. this and this Phys.SE posts. The minimum of $$S_{+}[x]$$ (and the maximum of $$S_{-}[x]$$) is obtained for timelike geodesic curves $$x^{\mu}(\lambda)$$. Often we throw away the $$S_{-}[x]$$ branch since we are mostly interested in the minimum. This can be encoded into the variational principle by imposing that the einbein field $$e(\lambda)>0$$ is positive.
3. In the massless case $$m=0$$, eq. (1) becomes $$\tag{4} g_{\mu\nu}~\dot{x}^{\mu}\dot{x}^{\nu} ~\approx~0,$$ which is the equation of motion (EOM) for a massless particle, cf. e.g. this Phys.SE post.