Reference-renormalized curvature-primitive Gauss-Bonnet formalism for finite-distance weak gravitational lensing in static spherical spacetimes

TL;DR

This paper introduces a new reference-renormalized formalism for finite-distance weak gravitational lensing in static spherical spacetimes, providing a unified and geometrically transparent approach that aligns with existing methods when applicable.

Contribution

It develops a gauge-fixed, reference-based renormalization scheme for Gauss-Bonnet lensing, enabling accurate deflection angle calculations without circular orbit assumptions.

Findings

Reproduces Ishihara's weak-deflection formulas for various spacetimes.

Provides a method applicable even when no circular null orbit exists.

Clarifies the choice of reference geometries in different backgrounds.

Abstract

We develop a reference-renormalized (photon-sphere-free) normalization scheme for Gauss-Bonnet gravitational lensing at finite distance in static, spherically symmetric spacetimes. The method treats the curvature primitive used to reduce the Gauss-Bonnet curvature-area integral as a quantity defined only modulo an additive constant (an additive gauge freedom). We fix this gauge by matching to a physically chosen reference optical geometry in an outer regime where the physical geometry approaches that reference, thereby defining a unique renormalized discrepancy primitive $\mathcal{P}_e(r)$ by reference subtraction. The resulting master formula yields the Ishihara-Li finite-distance deflection angle without invoking any circular null orbit, while remaining fully compatible with orbit-normalized prescriptions whenever a suitable photon sphere exists (the two gauges differ only by a constant shift and give identical $\alpha$). In asymptotically flat settings the canonical reference is Minkowski, while in Kottler-type backgrounds the canonical reference is de Sitter within the static patch, making the operational fiducial explicit. We validate the method by reproducing Ishihara's finite-distance weak-deflection formulas for Schwarzschild, Reissner-Nordstr\"om, and Kottler spacetimes, including the mixed $r_g\Lambda$ term in the Kottler case within the static-patch fiducial. We also present a demonstrative example in which orbit normalization is genuinely inapplicable because no circular null orbit exists in the physical optical region (the Janis-Newman-Winicour spacetime for $\gamma\le \tfrac12$). The result is a unified, geometrically transparent route to finite-distance lensing that preserves compatibility with orbit-normalized prescriptions whenever those apply.