Virasoro constraint under the static gauge

Question

I am reading an article on introduction to string theory.

Consider an open string of length $L$, rotating around its center of mass with angular velocity $\omega$. Here we fix the gauge by the static gauge $\eta_\mu X^\mu = \tau$, where $\eta_\mu= (1,0,\cdots, 0)$. It claims that,

"under the static gauge, the time-dependent and space-dependent part of the Virasoro constraint decouple, so we can write the constraint as an equation only dependent of the spatial part: $$ (\dot{X}\pm X')^2 = 1." $$

I thought the above equation is the Virasoro constraint, but why it changed from $$ (\dot{X}\pm X')^2 = 0 $$ to the above equation by applying the static gauge? Or if I misunderstood the text.

Answer 1

The equations of motion are $$ T_{ab} = \partial_a X^\mu \partial_b X_\mu - \frac{1}{2} \gamma_{ab} \gamma^{cd} \partial_c X^\mu \partial_d X_\mu = 0 . $$ Working this out in $(\tau,\sigma)$ coordinates, we get $$ T_{\tau\tau} = T_{\sigma\sigma} = \frac{1}{2} [ {\dot X}^\mu {\dot X}_\mu + X'^\mu X'_\mu ] = 0 , \qquad T_{\tau\sigma} = {\dot X}^\mu X'_\mu . $$ These equations together imply $$ ( {\dot X}^\mu \pm X'^\mu ) ( {\dot X}_\mu \pm X'_\mu ) = 0 . $$ Then using the gauge $X^0 = \tau$, we find $$ | \dot{\vec{X}} \pm \vec{X}'|^2 = ( {\dot X}^0 \pm X'^0 )^2 = 1 . $$