Imprints of a gravitational wave through the weak field deflection of photons

In this study,we investigate the novel phenomenon of gravitational lensing experienced by gravitational waves traveling past a Schwarzschild black hole perturbed by a specific,first-order,polar gravitational wave.We apply the Gauss-Bonnet theorem,finding a topological contribution to the deflection of light rays passing near the black hole.We demonstrate that the deflection angle can be determined by analyzing a region entirely outside the path of the light ray,leading to a calculation based solely on the parameters of the perturbing wave(Legendre polynomial order,l;frequency,σ).This approach offers a unique perspective on gravitational lensing and expands our understanding of black hole interactions with gravitational waves.