标题： 施瓦茨希尔德引力透镜的时空视角 显示英文标题

标题： Spacetime perspective of Schwarzschild lensing

摘要： 我们提出了一种不参考背景时空的精确透镜方程的定义，并在施瓦茨希尔德时空的情况下显式构造了精确透镜方程。 对于施瓦茨希尔德情况，我们给出了到光源的角直径距离的精确表达式，以及图像的放大因子和到达时间。 我们将精确透镜方程与在薄透镜弱场假设下推导的标准透镜方程进行比较（其中光线是背景时空的测地线，在透镜平面上有急剧的偏转，且引力场较弱），并验证了标准弱场薄透镜方程在小碰撞参数下是不充分的。 我们表明，弱场薄透镜方程的二阶修正也是不准确的。 最后，我们将精确透镜方程与最近提出的强场薄透镜方程进行比较，该方程是在直线路径假设下得到的，但没有使用小角度近似，即允许大的偏转角度。 我们表明，即使对于在到达观测者之前绕透镜多次转动的光线，强场薄透镜方程也非常准确。 显示英文摘要

摘要： We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.