Helically twisted spacetime: study of geometric and wave optics, and physical analysis

TL;DR

This work presents an exact, four-dimensional spacetime with a built-in helical twist encoded by $ds^2 = -dt^2 + dr^2 + r^2 d\phi^2 + (dz + \omega r d\phi)^2$, showing a negative energy density near the axis and a violation of the weak energy condition. It analyzes both geodesic motion and wave optics in this background, obtaining stable photon orbits, helically modulated geodesics, and a torsion-controlling impact on deflection angles and mode confinement. The study develops a Hamiltonian framework and Poincaré analysis to map phase-space structure as the torsion parameter $\omega$ is varied, revealing a transition from regular to mixed dynamics. In the wave regime, the effective potential and refractive index $n_r(r)$ depend on $\omega$, $\ell$, and $k_z$, producing bound and scattering optical modes that could be realized in twisted metamaterials or liquid-crystal analogues, thus linking curvature–torsion in gravity to condensed-matter systems.

Abstract

We analyse a stationary, cylindrically symmetric spacetime endowed with an intrinsic helical twist, $ds^{2} = -dt^{2} + dr^{2} + r^{2} dφ^{2} + (dz + ω\, r\,dφ)^{2}$. Solving the Einstein equations exactly yields an anisotropic energy-momentum tensor whose density is negative and decays as $r^{-2}$, thus violating the weak energy condition near the axis. Three notable features emerge: (i) axis-centred negative energy; (ii) unequal transverse stresses; (iii) a torsional momentum flux $T_{φz}ω^{3}/r$. We identify stable photon orbits and deflection angle, fully helical geodesics, and torsion-controlled wave optics modes, suggesting laboratory analogues in twisted liquid-crystal and photonic systems. The coupling between the torsion parameter $ω$ and other physical parameters leads to significant effects, altering the motion along the positive or negative $z$-axis. These results make the twisted helical metric a useful test bed for studying the interplay of curvature, torsion, and matter in both gravitational and condensed-matter contexts.

Figures (11)

• Figure 1: Geodesic trajectories in a helically twisted spacetime with torsion. (a) For $\omega = 0.7$, the evolution parameter spans $\tau \in [0, 200]$, and the initial conditions are $r(0) = 0.1$, $\phi(0) = 0.1$, $z(0) = 0$, $\dot{r}(0) = 0.1$, $\dot{\phi}(0) = 1.0$, and $\dot{z}(0) = 1.0$. (b) For $\omega = 0.16$, with the same evolution interval $\tau \in [0, 200]$ and the same initial configuration.
• Figure 2: Geodesic motion in helicoidal spacetime with torsion for two values of the angular parameter $\omega$. (a) For $\omega = 3$, the evolution parameter spans the interval $\tau \in [0, 300]$, and the initial conditions are $r(0) = 0.1$, $\phi(0) = 0.5$, $z(0) = 0$, $\dot{r}(0) = 0.01$, $\dot{\phi}(0) = 1.5$, and $\dot{z}(0) = 0.2$. (b) For $\omega = 0.15$, with evolution parameter in $\tau \in [0, 800]$, and initial conditions $r(0) = 0.2$, $\phi(0) = 2.7$, $z(0) = 0$, $\dot{r}(0) = 0.01$, $\dot{\phi}(0) = 2.7$, and $\dot{z}(0) = 3.5$.
• Figure 3: Geodesic trajectories in the helicoidal spacetime for fixed angular parameter $\omega = 0.49$. In panel (a), the initial conditions are $r(0) = 0.5$, $\phi(0) = 0.5$, $z(0) = 0$, $\dot{r}(0) = 0.05$, $\dot{\phi}(0) = 1.9$, and $\dot{z}(0) = 2.7$. In panel (b), the trajectory evolves from a region much closer to the axis, with $r(0) = 0.01$, and other initial conditions as $\phi(0) = 0.1$, $z(0) = 0$, $\dot{r}(0) = 0.05$, $\dot{\phi}(0) = 1.9$, and $\dot{z}(0) = 2.7$. In both cases, the evolution parameter runs in the interval $\tau \in [0, 100]$. In panel (c), the same initial radial and angular values of panel (b) are considered, but with a negative vertical velocity, $\dot{z}(0) = -2.7$ (i.e., $p_z<0$).
• Figure 4: Poincaré maps on the plane $z=z_0=2$. (a) Outbound crossings ($p_r>0$) collapse into a narrow vertical strip centred at $r\simeq3.0$, indicating that the orbit meets the reference plane near the outer turning point of each revolution. (b) Inbound crossings ($p_r<0$) populate a much broader band, $1\lesssim r\lesssim3$; the pattern evolves from thin, nearly integrable curves at small $\omega$ to wider, quasi-periodic layers as $\omega$ increases.
• Figure 5: Poincaré sections on the plane $z=z_{0}=2$ for orbits with negative axial momentum $p_{z}<0$. (a) Outbound crossings ($\dot z<0$, $p_{r}>0$) cluster in a narrow vertical strip around $r\simeq3$, mirroring the behaviour seen for $p_{z}>0$ but shifted in phase because the trajectory meets the reference plane while descending. (b) Inbound crossings ($\dot z<0$, $p_{r}<0$) populate a very thin band just below $p_{r}=0$, indicating that the repulsive coupling term $-2\omega L p_{z}/r$ prevents the particle from penetrating deeply into the potential well. Colours encode the torsion strength $\omega=0.3\text{-}0.8$. Together with Fig. \ref{['fig:poincare']}, these panels show that changing the sign of $p_{z}$ rearranges the phase-space islands without destroying the overall layer structure, confirming that the product $\omega L p_{z}$ is the key control parameter.