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arXiv · 2510.01150

The bending of a straight line

Zonghai Li·Xiao-Jun Gao
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Abstract

In gravitational lensing under the weak-field approximation, the usual viewpoint is that light bending measures how a ray deviates from a straight line in Euclidean space. In this work, we take the opposite perspective: we ask how a straight line bends in a curved space, such as optical geometry--that is, how it deviates from geodesics. Using the Gauss-Bonnet theorem, we show that, at leading order, the deflection angle can be written as the integral of the geodesic curvature of a straight line in curved space. This reformulation emphasizes the global, coordinate-independent nature of the deflection angle and provides a complementary way of understanding the classical Gibbons-Werner method. To illustrate the idea, we apply it to three familiar spacetimes--Schwarzschild, Reissner-Nordstr\"om, and Kerr--and recover the well-known results. Furthermore, we extend the method to massive particles using the Jacobi metric, and illustrate it with the Reissner-Nordstr\"om spacetime.

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Zonghai Li, Xiao-Jun Gao. 2025-10-01. The bending of a straight line. https://arxiv.org/abs/2510.01150

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