Gravitational Lensing in a Kasner Background: Distinguishing Wormholes and Black Holes
We investigate gravitational lensing by compact objects embedded in anisotropic Bianchi-I cosmologies using directional Jacobi maps within the thin-lens approximation. The formalism is developed for a general diagonal Bianchi-I spacetime and specialized to the Kasner solution as an analytically tractable background. Using the Ellis--Bronnikov wormhole and the Schwarzschild black hole as representative lenses, we derive anisotropic lens equations, characteristic axis-aligned lensing scales, and the corresponding critical curves. We show that the directional splitting of the characteristic scales depends on the complete source--lens--observer optical propagation and provides a geometric probe of anisotropic expansion independent of the overall lens scale. By contrast, the exact critical curves exhibit a much weaker deformation, indicating that characteristic-scale splitting and critical-curve morphology probe distinct aspects of the lens mapping. The comparison between wormhole and black-hole lenses further reveals that identical anisotropic backgrounds are filtered differently by distinct weak-field deflection laws. These results provide a simple framework for disentangling cosmological anisotropy from the local geometry of compact lenses.