双曲空间的相对性

摘要：

纵向多普勒频移是双曲线距离的量度。匀速运动的变换由多普勒频移确定，而其平方根变换为匀加速帧。通过用几何时间放大贝尔特拉米坐标，导出了一个时间-速度空间度规，这与弗里德曼使用爱因斯坦方程获得的时间-速度空间度规相似，其中质量张量描述了零压力下的尘埃宇宙。没有这样的假设，也没有坐标随时间增加的任何近似（即，恒定速度）。双曲线速度与朗伯四边形的边有关。在锐角成为理想点的极限情况下，出现了匀加速的情况。与哈勃定律的关系，和指数红移，进行了讨论。

英文摘要：

The longitudinal Doppler shift is a measure of hyperbolic distance. Transformations of uniform motion are determined by the Doppler shift, while its square root transforms to a uniformly accelerated frame. A time-velocity space metric is derived, by magnifying the Beltrami coordinates with the geometric time, which is similar to the one obtained by Friedmann using Einstein's equations in which the mass tensor describes a universe of dust at zero pressure. No such assumption nor any approximation in which the coordinates increase with time (i.e., constant velocities) need be made. The hyperbolic velocities are related to the sides of a Lambert quadrilateral. In the limit when the acute angle becomes an ideal point, the case of uniform acceleration arises. The relations to Hubble's law, and to the exponential red shift, are discussed.

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