In this paper we investigate the relation between the potential, and the geometrical time delays in the gravitational lensing. In the original paper of Shapiro (1964), it is stated that there is a time delay in the radar signals between Earth and Venus, passing near a massive object (the Sun), relative to a case of the absence of such mass. The reason is connected with the influence of the gravity on the speed of a light rays in a gravitational potential. The contribution from the change of the path length, which happens to be of a second order, is considered as negligible. Nevertheless, in the gravitational lens theory the geometrical delay, related to the change of path length, is routinely taken into account along with the potential term. Inspired by this apparent discrepancy, in this work we address the contribution of the geometric part of the time delay in different situations. We introduce a unified treatment with two limiting regimes of lensing. One of them corresponds to the time delay experiments near the Sun where geometrical delay is shown to be negligible. The second corresponds to the typical gravitational lens scenario with multiple imaging where the geometrical delay is shown to be significant. In particular, it is shown that the geometrical delay is non-negligible in the case of good alignment between source, lens and observer, because in such a case it becomes a first order quantity.

中文翻译：

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引力透镜中几何和潜在时间延迟的检查

在本文中，我们研究了引力透镜中势能和几何时间延迟之间的关系。在 Shapiro (1964) 的原始论文中，指出在地球和金星之间经过一个大质量物体（太阳）附近的雷达信号相对于没有这种质量的情况存在时间延迟。原因与引力对引力势中光线速度的影响有关。来自路径长度变化的贡献，恰好是二阶的，被认为可以忽略不计。然而，在引力透镜理论中，与路径长度变化相关的几何延迟通常与潜在项一起考虑。受到这种明显差异的启发，在这项工作中，我们解决了时间延迟的几何部分在不同情况下的贡献。我们引入了具有两个限制性透镜机制的统一处理。其中之一对应于太阳附近的时间延迟实验，其中几何延迟可以忽略不计。第二个对应于具有多重成像的典型引力透镜场景，其中几何延迟被证明是显着的。特别是，在光源、透镜和观察者之间良好对准的情况下，几何延迟是不可忽略的，因为在这种情况下，它成为一阶量。第二个对应于具有多重成像的典型引力透镜场景，其中几何延迟被证明是显着的。特别是，在光源、透镜和观察者之间良好对准的情况下，几何延迟是不可忽略的，因为在这种情况下，它成为一阶量。第二个对应于具有多重成像的典型引力透镜场景，其中几何延迟被证明是显着的。特别是，在光源、透镜和观察者之间良好对准的情况下，几何延迟是不可忽略的，因为在这种情况下，它成为一阶量。