相对论性拉格朗日力学中的几何时间与因果时间
Geometric Time and Causal Time in Relativistic Lagrangian Mechanics
摘要：
在这篇文章中，我们认为两种不同类型的时间应该考虑在相对论物理学：几何时间，它源于时空的结构和它的度量，和因果时间，表明从过去到未来的流动。因果时代的一个特殊性是，它的价值没有内在的意义，因为它们的演变本身是有意义的。在相对论性拉格朗日力学的背景下，因果时间对应于允许的路径参数化，我们表明，为了一个拉格朗日不依赖于任何特定的因果时间（因为它的值没有内在的意义），它必须是均匀的速度参数。我们以势场中的自由粒子为例来说明这一性质。然后，使用几何拉格朗日量（即参数化独立的拉格朗日量，这也是明显的协变），我们介绍的概念，年龄的一条路径的措施，在何种程度上的路径是远离测地线，并显示如何使用的概念可以在双胞胎悖论区分两个双胞胎遵循的路径。
英文摘要：
In this article, we argue that two distinct types of time should be taken into account in relativistic physics: a geometric time, which emanates from the structure of spacetime and its metrics, and a causal time, indicating the flow from the past to the future. A particularity of causal times is that its values have no intrinsic meaning, as their evolution alone is meaningful. In the context of relativistic Lagrangian mechanics, causal times corresponds to admissible parameterizations of paths, and we show that in order for a langragian to not depend on any particular causal time (as its values have no intrinsic meaning), it has to be homogeneous in its velocity argument. We illustrate this property with the example of a free particle in a potential. Then, using a geometric Lagrangian (i.e. a parameterization independent Lagrangian which is also manifestly covariant), we introduce the notion of ageodesicity of a path which measures to what extent a path is far from being a geodesic, and show how the notion can be used in the twin paradox to differentiate the paths followed by the two twins.
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