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The bending of a straight line

Zonghai Li, Xiao-Jun Gao

TL;DR

This work reframes gravitational light bending by asking how a straight line bends in curved optical geometry, rather than how a light ray deviates from a straight Euclidean line. Using Gauss–Bonnet, the deflection angle at leading order is shown to be the integral of the geodesic curvature along the straight-line path in the curved optical space, ${}^{L}\delta = -\int_{S}^{O_b} {}^{L}k_g(\gamma_b)\, dt$, connecting the Gibbons–Werner approach to a global, coordinate-independent interpretation. The method is applied to Schwarzschild, Reissner–Nordström, and Kerr spacetimes, yielding the standard leading-order results ${}^{L}\delta = 4M/b$, ${}^{L}\delta = 4M/b - 3\pi Q^2/(4b^2)$, and ${}^{L}\delta = 4M/b \pm 4Ma/b^2$, respectively. The formulation highlights a conceptual shift with potential extensions to the Jacobi metric for massive particles and to asymptotically non-flat spacetimes, offering a complementary geometric understanding of light deflection and lensing phenomena.

Abstract

In gravitational lensing, 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.

The bending of a straight line

TL;DR

This work reframes gravitational light bending by asking how a straight line bends in curved optical geometry, rather than how a light ray deviates from a straight Euclidean line. Using Gauss–Bonnet, the deflection angle at leading order is shown to be the integral of the geodesic curvature along the straight-line path in the curved optical space, , connecting the Gibbons–Werner approach to a global, coordinate-independent interpretation. The method is applied to Schwarzschild, Reissner–Nordström, and Kerr spacetimes, yielding the standard leading-order results , , and , respectively. The formulation highlights a conceptual shift with potential extensions to the Jacobi metric for massive particles and to asymptotically non-flat spacetimes, offering a complementary geometric understanding of light deflection and lensing phenomena.

Abstract

In gravitational lensing, 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.

Paper Structure

This paper contains 14 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Review of the Gibbons-Werner Method
  3. The Gauss--Bonnet Theorem
  4. The Optical Metric
  5. The Gibbons--Werner Formula
  6. The bending of the straight line
  7. The deflection angle from the bending of a straight line
  8. Geodesic curvature of $r=\frac{b}{\sin\phi}$
  9. Stationary spacetime
  10. Examples
  11. Schwarzschild spacetime
  12. Reissner--Nordström spacetime
  13. Kerr spacetime
  14. Conclusion

Figures (3)

  • Figure 1: Region $D_R\subset(\mathcal{M}^{2},\alpha_{ij})$ bounded by the photon trajectory $\gamma$ and the circular arc $C_R$. A light ray from the source $S$ is deflected by the lens and received by the observer $O$, with impact parameter $b$ and deflection angle $\delta$.
  • Figure 2: Region $D_R^b \subset (\mathcal{M}^{2},\alpha_{ij})$ bounded by the straight line $\gamma_b$ and the circular arc $C_R$. The straight line from the source $S$ to the observer $O_b$ has nonvanishing geodesic curvature due to the lens.
  • Figure 3: Region $D_R^b \subset (\mathcal{M}^{2},\bar{g}_{ij})$ with coordinates $(X,Y)$. The straight line $\gamma_b$ is described by $Y=b$. Note that the coordinates $X$ and $Y$ are not necessarily orthogonal.