#3 Nambu-Goto action

Given a d-dimensional manifold \mathcal{M}_{d} with a metric G_{\mu\nu}(X), a string is a one-dimensional object moving inside \mathcal{M}_{d}. The Nambu-Goto action describes the dynamics of these objects. It’s strongly inspired in the relativistic dynamics of a point particle.

A point particle describes a curve when it’s moving through space and time. Such curve (its trajectory) depends on a single parameter, usually the proper time. Likewise, a string describes a two-dimensional surface called world sheet, and depends on two parameters X^{\mu}(\tau,\sigma). Even though the role of both parameters is arbitrary, one can think of \tau as the proper time, and, for any given \tau, \sigma would label each one of the points that the string is made of.

The parametrization of the world sheet X^{\mu}(\tau,\sigma)

X^{\mu}(\tau,\sigma): \ \mathbb{R}\times[0,l]\xrightarrow{\quad\quad} \mathcal{M}_{d}

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\tau,\sigma)\quad\quad\quad X^{\mu}(\tau,\sigma)

maps each pair (\tau,\sigma), which labels a point of the world sheet, to a point of \mathcal{M}_{d}.

Nambu-Goto action is the generalization of the point particle action to higher dimensions. The action of a point particle is defined as proportional to the length of the trajectory of the particle through spacetime. Such geometric quantity is the same for every observer and independent of the parameter that describes the curve:

S_{pp}=-m\int \sqrt{G_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}\ {\rm d}\tau=-m\int\sqrt{G_{\mu\nu}\frac{{\rm d}{x}^{\mu}}{{\rm d}\tau}\frac{{\rm d}{x}^{\nu}}{{\rm d}\tau}}\ {\rm d}\tau

Therefore, the action for a string would be proportional to the surface of the world sheet. The volume 2-form is defined as \omega=\sqrt{-h}\ {\rm d}\tau\wedge{\rm d}\sigma, where h is the determinant of the induced metric over the world sheet. This metric is thus the pull-back through X^{\mu} of G_{\mu\nu}, i.e

h_{ab}u^{a}v^{b}=h(u,v)\equiv(X^{*}G)(u,v)=G(X_{*}u,X_{*}v)=G_{\mu\nu}u^{a}\partial_{a}X^{\mu}v^{b}\partial_{b}X^{\nu}

Then

h_{ab}=G_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu},

so

-\det{h}=(\partial_{\tau}X^{\mu}\partial_{\sigma}X^{\nu}G_{\mu\nu})^2-(\partial_{\tau}X^{\mu})^2(\partial_{\sigma} X^{\mu})^2 = (\partial_{\tau}X^{\mu}\partial_{\sigma}X_{\mu})^2-(\partial_{\tau}X^{\mu})^2(\partial_{\sigma} X^{\mu})^2

Finally we have

S_{NG}=-T\int {\rm d}\tau{\rm d}\sigma\sqrt{(\partial_{\tau}X^{\mu}\partial_{\sigma}X_{\mu})^2-(\partial_{\tau}X^{\mu})^2(\partial_{\sigma} X^{\mu})^2}.

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