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, sigma). This approach offers a unique perspective on gravitational lensing and expands our understanding of black hole interactions with gravitational waves.