# Different way to demonstrate the equivalence of Nambu-Goto to Polyakov action?

The Nambu-Goto Action and Polyakov action are \begin{eqnarray} S_{NG} &=& -\frac{T}{c}\int d^2 \xi \sqrt{-h} \\ S_{poly} &=& -\frac{1}{2}\frac{T}{c} \int d^2\xi \sqrt{-h} h^{ab} g_{\mu\nu}\partial_a x^\mu \partial_b x^\nu \end{eqnarray} where $h_{ab}$ is the string metric and $h_{ab}=g_{\mu\nu}\partial_ax^\mu\partial_bx^\nu$ where $g_{\mu\nu}$ is the metric in the target space.