in the form $$\begin{aligned}
  ds_{red}^2 
    &= (1-q^2-\Omega^2\rho^2)
    \left(d\rho^2 +\frac{\rho^2}{1-\Omega^2\rho^2}d\tilde\varphi^2\right)\\
    &= F(x^A) \left(d r^2 +r^2 d\varphi^2\right),
\label{conformal_metric}
\end{aligned}$$ where $$\begin{aligned}
  &F= \frac{1-q^2-\Omega^2\rho^2}{1-\Omega^2\rho^2}
    (1+\sqrt{1-\Omega^2\rho^2})^2 e^{-2\sqrt{1-\Omega^2\rho^2}}, 
\label{conformal}\\
    &r=\frac{\rho}{1+\sqrt{1-\Omega^2\rho^2}} e^{\sqrt{1-\Omega^2\rho^2}}. 
\end{aligned}$$

Using the argument from to inversely, we see that geodesics in the metric is equivalent to the orbits of the particle whose dynamics is governed by the action $$\begin{aligned}
    S = \int \left[p_A {x}'^A 
            - N_{(2)}{\cal H} \right] d\sigma 
\end{aligned}$$ with $$\begin{aligned}
    &{\cal H} = \frac12\delta^{AB}p_A p_B + U(r), 
\\
    &U(r)=-\frac12 F(\rho(r)), 
\end{aligned}$$ where the particle should satisfy the Hamiltonian constraint ${\cal H} =0$. The shape of potential $U(r)$ is shown in Fig.\[potential\].

![Potentials for $q=1/5, 2/5$ and $1/2$ in two-dimensional flat space. []{data-label="potential"}](potential.eps){width="12cm"}

The components of $\Theta^{\mu\nu}$
===================================

The components of $\Theta^{\mu\nu}$ are explicitly expressed in the following: $$\begin{aligned}
\Theta^{tt}  &= -(1-\ell^2), \\
\Theta^{t\rho}  &= \frac{\ell\, \Omega }{2 \rho }
(\rho_\mathrm{max}^2 -\rho_\mathrm{min}^2) \sin \left(2|\Omega|\sigma \right),  \\
\rho  \, \Theta^{t\varphi}  
    &= -\frac{\Omega^2 \rho^2  - \ell^2}{\Omega \rho }, \\
\Theta^{tz}  &= q\ell \mathrm{sign}(\Omega), \\
\Theta^{\rho\rho}  
    &= \frac{\Omega^2 }{4\rho^2 } 
 (\rho_\mathrm{max}^2 -\rho_\mathrm{min}^2)^2\sin^2 \left(2|\Omega|\sigma\right), \\
\rho  \, \Theta^{\rho\varphi}  
    &= \frac{\ell \, }{2\rho^2}
 (\rho_\mathrm{max}^2 -\rho_\mathrm{min}^2)\sin \left(2|\Omega|\sigma\right),  \\
\Theta^{\rho z}  
    &= \frac{q|\Omega|}{2\rho}(\rho_\mathrm{max}^2 -\rho_\mathrm{min}^2)
        \sin \left(2|\Omega|\sigma\right),  \\
\rho^2  \,\Theta^{\varphi \varphi}  
    &= -\Omega^2 \rho^2
\left( 1- \frac{\ell^2}{\Omega^4 \rho^4 } \right), 