A Line-Integral Representation of Gravitational Lensing by Black Holes

We present a path-based curvature representation of the gravitational bending of light in black-hole (BH) spacetimes. The bending angle is written as a one-dimensional line integral of the optical Gaussian curvature Kopt along the photon trajectory, weighted by a geometric kernel W(r,b). This representation sits within the Gibbons–Werner Gauss–Bonnet (GB) optical-geometry family rather than alongside it: the kernel is fixed by a co-area reduction of the GB surface integral along an undeflected reference path, and the single new computational object is the resulting radial integral together with its cumulative, directly plottable reading of how the deflection builds up along the ray. With the lever-arm choice W=r2−b2, the integral reproduces α^=4M/b for every static, asymptotically flat metric (Theorem 1) and evaluates in closed form for Schwarzschild, Reissner–Nordström (RN), and equatorial Kerr. The representation becomes reliable at a large impact parameter; at the small impact parameters relevant to horizon-scale imaging, it is not numerically competitive with the standard expansions, a limitation we quantify. Beyond leading order the kernel must import information from the bent geodesic, after which the scheme reconstructs the known perturbative series; the second-order mismatch in the lever-arm result therefore measures, rather than hides, the deformation of the photon path away from the straight-line reference. Finite source–observer distances enter through the Ono–Ishihara–Asada (OIA) construction, and a winding-sum continuation outlines the route toward the strong-deflection regime, whose closed-form reduction is left to future work.