Boundary-only weak deflection angles from isothermal optical geometry

TL;DR

This paper develops a boundary-only formulation of finite-distance gravitational lensing using Gauss-Bonnet theory on the equatorial optical geometry. By casting the 2D optical metric in isothermal coordinates, the Gaussian curvature becomes $K=-\Delta\varphi$, enabling the bulk curvature term to be replaced by a boundary integral of the conformal factor and its normal derivative; the deflection angle $\alpha$ is then expressed purely in terms of boundary data and a controlled closing term. The authors implement this in a practical weak-field setup, showing that leading bending reduces to elementary 1D integrals along a flat reference ray, with finite-distance dependence entering only through endpoint data. They validate the method with Schwarzschild, Reissner-Nordström, and Kottler geometries, reproducing known finite-distance results and obtaining explicit $O(\Lambda)$ and $O(\Lambda M)$ contributions without orbit-dependent calibrations. The framework promises straightforward extensions to stationary spacetimes and media-modified environments, offering a geometrically transparent and computationally efficient approach to finite-distance lensing observables.$

Abstract

We develop a boundary only method for computing weak gravitational deflection angles at finite source and receiver distances within the Gauss-Bonnet theorem formulation of optical geometry. Exploiting the fact that the relevant equatorial optical manifold is two dimensional, we introduce isothermal (conformal) coordinates in which the optical metric is locally conformal to a flat reference metric and the Gaussian curvature reduces to a Laplacian of the conformal factor. Such an identity converts the curvature area term in the Gauss-Bonnet theorem into a pure boundary contribution via Green/Stokes-type relations, yielding a deflection formula that depends only on boundary data and controlled closure terms. The residual normalization freedom of the isothermal radius is isolated as an additive freedom in the conformal factor and is shown to leave physical observables invariant, eliminating the need for orbit dependent calibration prescriptions. We explicitly implement the boundary only formalism in weak deflection, where the leading bending reduces to elementary one-dimensional integrals evaluated on a flat reference ray in the conformal plane, with finite distance dependence entering solely through endpoint data. We validate the construction by reproducing finite distance weak deflection for Schwarzschild, deriving the leading finite distance charge correction for Reissner-Nordström, and applying the same boundary only framework to the Kottler (Schwarzschild-de Sitter) geometry as a representative non-asymptotically flat test case, recovering the standard finite distance expansion including the explicit $\mathcal{O}(Λ)$ and mixed $\mathcal{O}(ΛM)$ contributions to the total deflection angle.