Worldsheet diffeomorphism

A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which $X^{\mu }$ is a scalar and $g_{\alpha \beta }$ is a rank-two tensor.

On a future-directed patch, choose target time as worldsheet time, $X^0=t$. The momentum equation then gives $P^0=1/e$ and $X^{\prime 0}=0$. Freedom to vary target time at an open-string endpoint sets its shift to zero, simplifying the endpoint momentum-flux condition.

Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as $X^0=\tau$ for a string.

For a string, this local static gauge identifies two embedding coordinates with the worldsheet coordinates, $X^0=t$ and $X^1=\sigma$. The remaining coordinates describe a graph over that coordinate plane. Folded embeddings may require more than one patch.

For $X=(t,\sigma ,k(\sigma +t))$, the vertical coordinate velocity is $k$, but its normal projection has magnitude $|k|/\sqrt{1+k^2}<1$. Tangential motion merely changes string labels. The induced worldsheet metric has determinant $-1$ even when constant-label curves become spacelike for $|k|>1$.

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.