相对速度、几何学和空间膨胀
Relative velocities, geometry, and expansion of space
摘要：
空间膨胀是什么意思？解决这个问题的一种方法是研究相对速度。在这种情况下，如果一个非定域的测试粒子的相对速度超过了观测者的定域光速，那么它就是“超光速的”。在一个特定的宇宙模型中，后退的试验粒子的超光速相对速度的存在，表明它本身是该模型中空间膨胀的一个可能的标准。从这个观点来看，遥远的后退星系团的超光速是观测者和星系团之间空间膨胀的结果。然而，在这一方法有意义之前，必须解决一个基本的模糊性。非局部物体的相对速度的概念取决于坐标的选择，这种模糊性表明需要坐标独立的定义。在这项工作中，我们审查四个（不等价）几何定义和普遍的相对速度的概念：费米，运动学，天体测量，光谱相对速度。我们应用这种形式主义来测试粒子进行径向运动相对于comoving观察员在扩大罗伯逊-沃克宇宙学，并包括以前未发表的费米坐标的一类膨胀宇宙学的结果。我们相互比较相对速度，并展示了它们如何决定时空的几何性质，包括有足够数据的标度因子。给出了一般Robertson-Walker宇宙学中存在超光速退缩费米速度的充要条件。最后，我们讨论一下空间的扩张。
英文摘要：
What does it mean to say that space expands? One approach to this question is the study of relative velocities. In this context, a non local test particle is "superluminal" if its relative velocity exceeds the local speed of light of the observer. The existence of superluminal relative velocities of receding test particles, in a particular cosmological model, suggests itself as a possible criterion for expansion of space in that model. In this point of view, superluminal velocities of distant receding galaxy clusters result from the expansion of space between the observer and the clusters. However, there is a fundamental ambiguity that must be resolved before this approach can be meaningful. The notion of relative velocity of a nonlocal object depends on the choice of coordinates, and this ambiguity suggests the need for coordinate independent definitions. In this work, we review four (inequivalent) geometrically defined and universal notions of relative velocity: Fermi, kinematic, astrometric, and spectroscopic relative velocities. We apply this formalism to test particles undergoing radial motion relative to comoving observers in expanding Robertson-Walker cosmologies, and include previously unpublished results on Fermi coordinates for a class of inflationary cosmologies. We compare relative velocities to each other, and show how pairs of them determine geometric properties of the spacetime, including the scale factor with sufficient data. Necessary and sufficient conditions are given for the existence of superluminal recessional Fermi speeds in general Robertson-Walker cosmologies. We conclude with a discussion of expansion of space.
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