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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.01150 |
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| _version_ | 1866914621068148736 |
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| author | Li, Zonghai Gao, Xiao-Jun |
| author_facet | Li, Zonghai Gao, Xiao-Jun |
| contents | 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-Nordström, 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-Nordström spacetime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The bending of a straight line Li, Zonghai Gao, Xiao-Jun General Relativity and Quantum Cosmology 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-Nordström, 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-Nordström spacetime. |
| title | The bending of a straight line |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2510.01150 |