$1/2$.\
[]{data-label="Square"}](general.eps){width="17cm"}

Energy, Momentum and Angular Momentum
=====================================

By varying the action by $g_{\mu\nu}$ we see that the string energy-momentum tensor $T^{\mu \nu}$ is given by $$\begin{aligned}
    \sqrt{-g} T^{\mu \nu}(x^\lambda)  
        &= -\mu \int d^2\zeta \Theta^{\mu \nu}(\zeta^c) 
        \delta^{(4)}\left(x^\lambda-x^\lambda(\zeta^c) \right), 
\label{EMTofCS0} \\
    &\Theta^{\mu \nu}=\sqrt{-\gamma}\gamma^{ab}\partial_a x^\mu\partial_bx^\nu,\end{aligned}$$ where $x^\lambda(\zeta^c)$ is the solution of $\Sigma$.

In the inertial reference system , we can rewrite the energy-momentum tenser for the stationary rotating strings as $$\begin{aligned}
    \sqrt{-g} T^{\bar\mu \bar\nu} (\bar t, \bar \rho, \bar \varphi, \bar z)  
    = -\mu \int d\tau d\sigma\ \Theta^{\bar\mu\bar\nu}(\sigma) \
    \delta\left(\bar t-\tau\right)\delta\left(\bar\rho-\bar\rho(\sigma)\right) 
        \delta\left(\bar\varphi-\bar\varphi(\tau, \sigma)\right)
            \delta\left(\bar z-\bar z(\sigma)\right). 
\label{EMTofCS} \end{aligned}$$ From , and , we obtain $\Theta^{\bar\mu\bar\nu}$ explicitly. (See Appendix B.)

Now, we define the string energy, $E$, the angular momentum, $J$, and the momentum along the rotating axis, $P$. We consider infinitely long strings with periodic structure, i.e., $\ell$ is assumed to be a rational number, then we define $E, J$ and $P$ for one period, $\bar z \sim \bar z+Z_p$ as $$\begin{aligned}
    E&:= \int_{\rho_{min}}^{\rho_{max}}d\bar\rho\int_0^{2 \pi}d\bar\varphi 
        \int_0^{Z_p} d\bar z 
            \sqrt{-g}~T^{\bar t}_{\ \bar \nu}(-\partial_{\bar t})^{\bar\nu}
        = \mu \int_0^{n_l\sigma_p}d\sigma\ 
            \Theta^{\bar t}_{~{\bar t}}(\sigma), 
\\
    J &:= \int_{\rho_{min}}^{\rho_{max}} d\bar\rho \int_0^{2 \pi} d\bar\varphi \int_0^{Z_p} d\bar z 
                \sqrt{-g}~ T^{\bar t}_{\ \bar \nu}(\partial_{\bar\varphi})^{\bar\nu} 
        = -\mu \int_0^{n_l\sigma_p} d\sigma \ 
            \Theta^{\bar t}_{~{\bar\varphi}}(\sigma), 
