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In the derivation of the geodesic, one starts with the integral of the line element (arclength):

$$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$

The integrand is then substituted into the Euler-Lagrange equation, which simplifies to the geodesic equation.

My question is would it be possible to start with the area element instead of the line element, and thereby arrive at a surface area geodesic equation, which I guess would be an equation for the surface area of a given patch of manifold?

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5  
This is the Nambu Goto action, and there is a deep theory of minimal surfaces attached to it, because there is a harmonic parametrization. –  Ron Maimon Dec 24 '11 at 0:23
    
Instead of using the area element, would it be possible to use Green's theorem? –  ben Dec 24 '11 at 4:13
    
More on Nambu-Goto action: physics.stackexchange.com/search?q=Nambu+Goto –  Qmechanic Apr 20 at 18:11

2 Answers 2

A particle traces out a wordline, hence the action is proportional to the arc length,

$$S = \int \mathrm{d}s \, \sqrt{g_{\mu\nu}\dot{x}^\mu \dot{x}^\nu}$$

Varying the action, as you stated, gives rise to the Euler-Lagrange equations which are equivalent to the geodesic equations. If we considered an action proportional to area, we would be describing a string as it traces a wordsheet rather than a line. If we denote $\sigma^\alpha$ the collective intrinsic coordinates on the sheet, then the induced metric is simply given by,

$$\gamma_{\alpha \beta} = \frac{\partial X^\mu}{\partial \sigma^\alpha}\frac{\partial X^\nu}{\partial \sigma^\beta} g_{\mu\nu}$$

where $X^\mu$ denote the embedding functions, and $g_{\mu \nu}$ is the metric of the target space. Naively, we would then write the action as,

$$S = -T\int \mathrm{d}^2 \sigma \, \sqrt{-\det\gamma} = -T \int \mathrm{d}^2 \sigma \, \sqrt{(\dot{X} \cdot X')^2 -(\dot{X})^2 (X')^2}$$

with implicit contraction with the metric, primes denote differentiation with respect to $\sigma$, and dots represented with respect to $\tau$, as we chose $\sigma^\alpha = (\tau, \sigma)$. The constant $T$ has dimensions of tension; whilst unrelated to the question, in the Nambu-Goto action it is given by,

$$T=\frac{1}{2\pi \alpha'}$$

where $\alpha'$ is the universal Regge slope. The $m^2$ against $J$ plot for hadrons gives rise to a linear regression line, which has a gradient $\alpha'$, hence the name. Historically, string theory was first explored as a means to explain strong interactions, rather than quantum gravity.


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Source: David Tong's lecture notes on string theory, at http://www.damtp.cam.ac.uk/user/tong/string.

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This is somwhat outside my area, but Elie Cartan developed a form of differential geometry based on the area element rather than the line element. The main reference in French is E. Cartan. Les espaces métriques fondés sur la notion d’aire. There is a relatively recent arxiv article in English The concept of orthogonality in Cartan's geometry based on the concept of area that extends these ideas. The abstract says

In 1931 Elie Cartan constructed a geometry which was rarely considered. Cartan proposed a way to define an infinitesimal metric ds starting from a variational problem on hypersurfaces in an n-dimensional manifold $\mathcal{M}$. This distance depends not only of the point $\text{m}\in\mathcal{M}$ but on the orientation of a hyperplane in the tangent space $T_{\text{m}}\mathcal{M}$. His first step is a natural definition of the orthogonal direction to such tangent hyperplane. In this paper we extend it, starting form considerations from the calculus of variations.

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