Rotating Barrow entropy-inspired wormholes versus black holes: null trajectories, shadow morphology, and exotic matter
We construct a rotating traversable wormhole supported by a regular, entropy-inspired matter distribution and investigate its geometry, null geodesics, shadow morphology, and exotic matter content. We derive the geometry from the Barrow deformation of the Bekenstein--Hawking area law, characterized by an exponent $0<n\le1$, which replaces point-like divergences with a regular matter profile. The resulting energy density is negative everywhere and scales as $ρ_B\propto-r^{-n-3}$, while the shape function smoothly satisfies the flare-out condition at the throat. We introduce slow rotation via $N(r,θ)=\exp[αJ^2\cos^2θ/r^2]$ and $ω(r)=2J/r^3$; both remain finite at the throat, ensuring a horizon-free, traversable geometry. Equatorial null geodesics yield a photon-sphere condition distinct from the standard black-hole relation. In the static limit, the photon sphere sits at the throat, $r_{\rm ph}=r_{\rm th}$; for slow rotation, the prograde branch moves to $r_{\rm ph}^2\simeq 6J$, while the retrograde branch is pushed toward the throat. The critical impact parameters are $\mathscr{B}_{\pm}\approx\pm r_{\rm ph}+2J/r_{\rm ph}$. For $M=1$, $r_{\rm th}=1$, $J/M^2=0.5$, this gives $r_{\rm ph}\approx1.732$, $\mathscr{B}_{+}\approx1.90$, $\mathscr{B}_{-}\approx-1.55$, and a fractional asymmetry $\mathcal{A}\approx0.10$. The weak and null energy conditions are violated, with the strongest violation near the throat, while the Barrow exponent controls the localization of exotic matter. The volume-integral quantifier $I_V$ remains finite when $r_{\rm th}^{n+1}=4M/[(n+2)π^{n/2}]$. Comparisons with Kerr and regular black holes indicate that shadow diameter alone may be degenerate, whereas photon-ring structure and rotational asymmetry provide complementary signatures capable of distinguishing this entropy-inspired wormhole from black-hole alternatives.