A New Line-Integral Method for Gravitational Lensing by Black Holes

We introduce a path-based curvature method for the gravitational bending of light in black-hole (BH) spacetimes. The construction is structurally distinct from the two standard routes. The deflection is not read off from an asymptotic comparison of incoming and outgoing geodesic directions (Bozza-Tsukamoto), and it is not a two-dimensional Gibbons-Werner Gauss-Bonnet (GB) surface integral. It is built instead as a one-dimensional line integral of the optical Gaussian curvature $\Kopt$ along the photon trajectory, weighted by a geometric kernel $W(r,b)$. The framework itself is generic in scope. The closed-form simplification $W=\sqrt{r^{2}-b^{2}}$ delivers the weak-field regime: it reproduces $\hat{\alpha}=4M/b$ for every static, asymptotically flat metric (Theorem~1), and the curvature integral evaluates analytically for Schwarzschild, Reissner-Nordström (RN), and equatorial Kerr. Effectiveness is quantified: agreement with the exact geodesic is $\sim 4\%$ at $b/M=100$ and degrades smoothly as $b$ approaches the photon-sphere edge, locating exactly where path-deformation corrections matter. The strong-deflection regime enters through a winding-sum continuation that maps onto the Bozza-Tsukamoto logarithm. Finite source-observer distances are handled through the Ono-Ishihara-Asada (OIA) construction. The deflection becomes a directly plottable cumulative quantity along the path, a feature both standard routes hide.