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arXiv 2008.10093gr-qcastro-ph.GAmath-phmath.MP
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Gauss-Bonnet定理在NUT度量中的应用

Application of the Gauss-Bonnet theorem to lensing in the NUT metric

Mourad Halla, Volker Perlick

发表信息：
Gen. Rel. Grav. 52, 112 (2020)
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摘要：

我们证明了在费马原理的帮助下，NUT度量中的每一条类光测地线都投影到一个二维黎曼度量的测地线上，我们称之为光学度量。光学度规定义在一个（坐标）圆锥上，圆锥的张角由类光测地线的碰撞参量决定。我们发现，令人惊讶的是，不同的开度角锥的光学指标是局部（但不是全球）等距。借助于高斯-博内定理，我们证明了类光测地线的偏转角是由光学度规的高斯曲率上的面积积分决定的。对于静态和球对称的时空，也有类似的结果。NUT时空既不是静态的，也不是球对称的（至少不是通常意义上的），它的推广是相当重要的。

英文摘要：

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.

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