In this paper we use the Homotopy analysis method to obtain an analytic approximation for the entire photon trajectory in the Schwarzschild spacetime. This is usually expressed exactly in terms of an elliptic integral. We compare our approximation with other formulae found in the literature, which were specifically obtained for the Schwarzschild solution. Unlike some of these formulae, our approximation can be applied and maintains a good accuracy for emission point close to the event horizon and also for emission angles close to and greater than \(\pi /2\). We show that our method can easily be applied to other spherically symmetric solutions such as the Reissner-Nordström solution. Such an approximation would be useful when accurate determination of the light trajectories around compact objects is required without the need to revert to time consuming numerical integration of elliptic integrals.

在本文中，我们使用同伦分析方法来获得史瓦西时空中整个光子轨迹的解析近似。这通常用椭圆积分来精确表达。我们将我们的近似值与文献中找到的其他公式进行比较，这些公式是专门为史瓦西解获得的。与其中一些公式不同，我们的近似可以适用于接近事件视界的发射点，并且对于接近和大于 \(\pi /2\) 的发射角也可以保持良好的精度。我们证明我们的方法可以很容易地应用于其他球对称解，例如 Reissner-Nordström 解。当需要精确确定紧凑物体周围的光轨迹而不需要恢复到耗时的椭圆积分数值积分时，这种近似将很有用。