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

We analyse a stationary, cylindrically symmetric spacetime endowed with an intrinsic helical twist, $d{s}^2=-d{t}^2+d{r}^2+r^{2}d{\phi }^2+(dz+\omega rd\phi {)}^2$. 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 $T_{\phi z}{\omega }^3\mathrm{/}r$. 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 $\omega$ 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.