光学等价与射影等价
Optical Metrics and Projective Equivalence
摘要：
光线在静态时空中的轨迹由与洛伦兹时空度规相关联的黎曼光学度规的非参数化测地线描述。我们研究了这种结构的独特性，并证明了两个不同的观察者，相对于彼此移动，谁都认为宇宙是静态的，可能会确定不同的光线的几何形状。更具体地说，我们分类洛伦兹度量承认一个以上的超表面正交的时间-类Killing向量，并分析所产生的光学度量的投影等价性。这些指标被证明是射影等价的，如果静态Killing向量生成一个组$SL(2, \R)$，但不是射影等价的一般。我们还考虑了宇宙学的$C$ --爱因斯坦-麦克斯韦理论中的度量，并证明了对应于不同宇宙学常数值的光学度量是射影等价的。
英文摘要：
Trajectories of light rays in a static spacetime are described by unparametrised geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the universe as static may determine the geometry of the light rays differently. More specifically, we classify Lorentzian metrics admitting more than one hyper--surface orthogonal time--like Killing vector and analyze the projective equivalence of the resulting optical metrics. These metrics are shown to be projectively equivalent up to diffeomorphism if the static Killing vectors generate a group $SL(2, \R)$, but not projectively equivalent in general. We also consider the cosmological $C$--metrics in Einstein--Maxwell theory and demonstrate that optical metrics corresponding to different values of the cosmological constant are projectively equivalent.
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