The Nambu-Goto Action

Let us generalise the world-line formulation of a classical relativistic particle to two-dimensional world-sheets.

Orthonormal coordinates: $X^{\mu }=\{X^0,X^1,\dots ,X^{D-1}\}$ on flat D-dimensionalMinkowski spacetime ${\mathbb{R}}^{1,D-1}$.

Coordinates: ${\sigma }^a=({\sigma }^0,{\sigma }^1)$ on the string world-sheet $\Sigma$.

Trajectory of a string: $X:\Sigma \to {\mathbb{R}}^{1,D-1}$ with coordinate functions $X^{\mu }(\sigma )$.

In other words,

$X:\Sigma \to {\mathbb{R}}^{1,D-1}$

$X:\sigma \mapsto X^{\mu }(\sigma )$

The natural generalisation on the relativistic action principle is

$S=-T{\int }_{\Sigma }d(\text{area})=-T{\int }_{\Sigma }{d}^2\sigma \sqrt{-\text{det}(\gamma )}$ where ${\gamma }_{ab}={\partial }_a{X}^{\mu }{\partial }_b{X}^{\nu }{\eta }_{\mu \nu }$ and ${\partial }_a{X}^{\mu }=\frac{\partial X^{\mu }}{\partial {\sigma }^a}$.

The symmetric tensor ${\gamma }_{ab}$ is metric on $\Sigma$ induced by $X:\Sigma \to {\mathbb{R}}^{1,D-1}$ and the real parameter T>0 is known as the string tension.

The action (known as Nambu-Goto Action) is proportional to the area of the world-sheet and therefore, classical solutions $X^{\mu }(\sigma )$ minimise the world-sheet area.