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.

中文翻译：

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Gauss-Bonnet 定理在 NUT 度量中的透镜应用

我们在费马原理的帮助下表明，NUT 度量中的每个类似光的测地线都投影到二维黎曼度量的测地线上，我们称之为光学度量。光学度量定义在（坐标）锥体上，锥体的张角由类光测地线的冲击参数确定。我们发现，令人惊讶的是，具有不同张角的锥体的光学度量是局部（但不是全局）等距的。在 Gauss-Bonnet 定理的帮助下，我们证明了类光测地线的偏转角由光学度量的高斯曲率上的面积积分决定。已知类似的结果适用于静态和球对称时空。对 NUT 时空的推广，