静态球对称时空中弱引力偏转的相平面表述
Phase-plane formulation of weak gravitational deflection in static spherical spacetimes
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中文总结 AI 辅助
本文以史瓦西时空为案例，提出静态球对称黑洞引力光偏转的相平面表述，推导弱偏转系数公式，重现标准系数并扩展至一般静态球对称几何，明确相位映射奇点与临界光子轨道的关联。
中文摘要 AI 辅助
本文以史瓦西时空为主要案例，构建了静态球对称黑洞引力光偏转的相平面表述。我们不再对零测地线进行微扰并定位位移后的出射渐近线，而是通过振幅和本征相位来表示轨道。偏转角则由两个固定渐近端点之间本征相位推进时积累的多余物理方位角得出。对于史瓦西时空，精确的径向第一积分将振幅演化简化为三次代数关系，其物理分支通过单次逆代数映射生成本地相位因子；拉格朗日反演进而给出不变量比\uf04d/\uf062的幂次形式的显式全阶弱偏转系数公式，每个系数可拆分为代数相位贡献与通用三角矩，这解释了史瓦西级数交替出现的有理项与π相关项的结构。与精确径向散射积分及常规轨道微扰的独立比对重现了标准弱偏转系数。该相位框架可扩展至一般静态球对称几何，而史瓦西三次式是更广泛代数类中一个特别简单的成员；相位映射的分支奇点还与临界光子轨道重合，决定了弱偏转展开的收敛性。
英文摘要
This paper develops a phase-plane formulation of gravitational light deflection by static and spherically symmetric black holes, with Schwarzschild spacetime as the principal case. Instead of perturbing the null trajectory and locating the displaced outgoing asymptote, we represent the orbit through an amplitude and an intrinsic phase. The bending angle then follows from the excess physical azimuth accumulated while the intrinsic phase advances between two fixed asymptotic endpoints. For Schwarzschild spacetime, the exact radial first integral reduces the amplitude evolution to a cubic algebraic relation. Its physical branch generates the local phase factor through a single inverse algebraic map. Lagrange inversion then yields an explicit all-order weak-deflection coefficient formula in powers of the invariant ratio \(M/b\). Each coefficient separates into an algebraic phase contribution and a universal trigonometric moment, which explains the alternating rational and \(π\)-dependent structure of the Schwarzschild series. Independent comparison with the exact radial scattering integral and conventional orbit perturbation reproduces the standard weak-bending coefficients. The same phase framework extends to general static spherical geometries, while the Schwarzschild cubic represents an especially simple member of a broader algebraic class. The branch singularity of the phase map also coincides with the critical photon orbit and governs the convergence of the weak-deflection expansion.
