Weyl Tensor and the Possibility of Exotic-Matter-Free Wormholes in Conformal Gravity

We compute the Weyl conformal tensor and its quadratic invariant C² = Cμνρσ Cμνρσ for the general static spherically symmetric Morris–Thorne wormhole metric parametrised by the redshift function Φ(r) and the shape function b(r). The calculation is performed symbolically using SymPy. We find that all independent components of the Weyl tensor are proportional to a single master function Q(r), and C² = Q(r)²/(3r⁶). The condition C² = 0 (conformal flatness) reduces to a single second-order ordinary differential equation linking Φ(r) and b(r). For the zero‑tidal‑force case Φ′ = 0 we obtain b(r) ∝ r³, which violates the flare‑out condition and does not describe a traversable wormhole. The question of whether a non‑trivial conformally flat traversable wormhole exists remains open and requires further analysis of the derived Riccati equation. The Python (SymPy) source code is provided as supplementary material.