注：多普勒因子和相关相对论定律的公理化推导
Note: Axiomatic Derivation of the Doppler Factor and Related Relativistic Laws
摘要：
在两个强不变性公理的背景下研究了相对论性多普勒效应的公式。公理表示在一个抽象的操作概括的相对论性的速度增加。我们证明了以下结果。(1)如果不预先假定该操作的标准表示，则这两个公理中的每一个都与相对论性多普勒效应公式和洛伦兹-菲茨杰拉德收缩一致。(2)如果假设操作的标准表示，则这两个公理彼此等价，并且等价于相对论性多普勒效应公式。因此，在这种情况下，公理与洛伦兹-菲茨杰拉德收缩不一致。(3)如果假设洛伦兹-菲茨杰拉德收缩，那么这两个公理彼此等价，并且等价于适用于垂直运动情况的运算的不同数学表示。相对论性多普勒效应推导出一个正指数参数（代替平方根）。我们证明这些事实的规律性和其他合理的背景条件下。
英文摘要：
The formula for the relativistic Doppler effect is investigated in the context of two compelling invariance axioms. The axioms are expressed in terms of an abstract operation generalizing the relativistic addition of velocities. We prove the following results. (1) If the standard representation for the operation is not assumed a priori, then each of the two axioms is consistent with both the relativistic Doppler effect formula and the Lorentz-Fitzgerald Contraction. (2) If the standard representation for the operation is assumed, then the two axioms are equivalent to each other and to the relativistic Doppler effect formula. Thus, the axioms are inconsistent with the Lorentz-FitzGerald Contraction in this case. (3) If the Lorentz-FitzGerald Contraction is assumed, then the two axioms are equivalent to each other and to a different mathematical representation for the operation which applies in the case of perpendicular motions. The relativistic Doppler effect is derived up to one positive exponent parameter (replacing the square root). We prove these facts under regularity and other reasonable background conditions.
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