Closed timelike curves in 𝒫𝒯-symmetric wormholes

In this paper, we investigate a modified Einstein–Rosen wormhole model, made unidirectionally traversable through a bimetric geometry defined by two regular metrics, [Formula: see text] and [Formula: see text] and characterized by [Formula: see text] symmetry combining time reversal ([Formula: see text]) and spatial inversion ([Formula: see text]). In this framework, two distinct spacetime regions are identified at the wormhole throat ([Formula: see text]) via [Formula: see text] symmetry, forming a single spacetime sheet. This model employs Eddington–Finkelstein coordinates to eliminate coordinate singularities at the throat, enabling traversability with a lightlike membrane of exotic matter at the junction to satisfy the Einstein field equations, similar to other traversable wormhole models. We extend this model by coupling two such wormholes to generate closed timelike curves (CTCs), made possible by the opposing causal orientations defined by the two metrics, while adhering to Novikov’s self-consistency principle. An effective theory is developed for a scalar field crossing the wormhole, yielding [Formula: see text]-symmetric Klein–Gordon equations with a real energy spectrum ensured by pseudo-unitarity, consistent with quantum mechanical dynamics. These results open new avenues for exploring the effects of [Formula: see text] symmetry on causality and the quantization of scalar fields in traversable geometries.