论文标题
高斯河网定理应用于螺母度量的镜头
Application of the Gauss-Bonnet theorem to lensing in the NUT metric
论文作者
论文摘要
我们在费马特（Fermat）的原则的帮助下表明，在螺母指标项目中，每个灯光般的大地测量都可以归于二维Riemannian指标的地理位置，我们称之为光学指标。光学度量是在（坐标）锥上定义的，其开口角是由灯光般的大地测量参数确定的。我们表明，令人惊讶的是，具有不同开口角的锥体上的光学指标在局部（但不是全球）等距。借助高斯 - 骨网定理，我们证明了灯泡测量线的挠度角取决于光学指标的高斯曲率上的积分区域。众所周知，对于静态和球形对称的空间，已知类似的结果是正确的。对螺母时空的概括既不是静态的，也不是球形对称的（至少在通常的意义上不是）。
We show with the help of Fermat's principle that every lightlike geodesic in the NUT metric projects to a geodesic of a two-dimensional Riemannian metric which we call the optical metric. The optical metric is defined on a (coordinate) cone whose opening angle is determined by the impact parameter of the lightlike geodesic. We show that, surprisingly, the optical metrics on cones with different opening angles are locally (but not globally) isometric. With the help of the Gauss-Bonnet theorem we demonstrate that the deflection angle of a lightlike geodesic is determined by an area integral over the Gaussian curvature of the optical metric. A similar result is known to be true for static and spherically symmetric spacetimes. The generalisation to the NUT spacetime, which is neither static nor spherically symmetric (at least not in the usual sense), is rather non-trivial.
