Geodesics of both lightrays and timelike particles with nonzero mass are deflected in a gravitational field. In this work we apply the perturbative method developed in Ref. \cite{Jia:2020dap} to compute the deflection angle of both null and timelike rays in the weak field limit for four spacetimes. We obtained the deflection angles for the Bardeen spacetime to the eleventh order of $m/b$ where $m$ is the ADM mass and $b$ is the impact parameter, and for the Hayward, Janis-Newman-Winicour and Einstein-Born-Infeld spacetimes to the ninth, seventh and eleventh order respectively. The effect of the impact parameter $b$, velocity $v$ and spacetime parameters on the deflection angle are analyzed in each of the four spacetimes. It is found that in general, the perturbative deflection angle depends on and only on the asymptotic behavior of the metric functions, and in an order-correlated way. Moreover, it is shown that although these deflection angles are calculated in the large $b/m$ limit, their minimal valid $b$ can be as small as a few $m$'s as long as the order is high enough. At these impact parameters, the deflection angle itself is also found large. As velocity decreases, the deflection angle in all spacetime studied increases. For a given $b$, if the spacetime parameters allows a critical velocity $v_c$, then the perturbative deflection angle will deviate from its true value as $v$ decreases to $v_c$. It is also found that if the variation of spacetime parameters can only change the spacetime qualitatively at small but not large radius, then these spacetime parameter will not cause a qualitative change of the deflection angle, although its value is still quantitatively affected. The application and possible extension of the work are discussed.

中文翻译：

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类时光线的微扰偏转角

光线和具有非零质量的类时粒子的测地线在引力场中偏转。在这项工作中，我们应用了参考文献中开发的微扰方法。\cite{Jia:2020dap} 计算四个时空的弱场限制中零射线和类时射线的偏转角。我们获得了 Bardeen 时空的 11 阶 $m/b$ 的偏转角，其中 $m$ 是 ADM 质量，$b$ 是撞击参数，对于 Hayward、Janis-Newman-Winicour 和 Einstein-Born -Infeld 时空分别达到九阶、七阶和十一阶。在四个时空的每一个中分析了撞击参数$b$、速度$v$和时空参数对偏转角的影响。可以发现，一般来说，微扰偏转角取决于且仅取决于度量函数的渐近行为，并且以阶相关的方式。此外，结果表明，虽然这些偏转角是在大 $b/m$ 限制下计算的，但只要阶数足够高，它们的最小有效 $b$ 可以小到几个 $m$。在这些冲击参数下，偏转角本身也很大。随着速度的降低，所研究的所有时空中的偏转角都会增加。对于给定的$b$，如果时空参数允许临界速度$v_c$，则随着$v$ 减小到$v_c$，扰动偏转角将偏离其真实值。还发现如果时空参数的变化只能在小半径而不是大半径下定性地改变时空，那么这些时空参数不会引起偏转角的质变，尽管其值仍然受到定量影响。讨论了这项工作的应用和可能的扩展。