相对论长度收缩动力学与Escherfest粒子
The Dynamics of Relativistic Length Contraction and the Ehrenfest Paradox
摘要：
相对论运动学通常被认为是时空的伪欧几里德（洛伦兹）几何的表现。然而，正如广义相对论中明确指出的那样，几何本身取决于动力学，特别是能量-动量张量。我们讨论了几个例子，说明在狭义相对论的框架内的长度收缩效应的动力学方面。我们展示了一些陷阱与直接应用的长度收缩公式的情况下，当一个扩展的对象加速。我们的分析揭示了长度收缩和加速系统内的内力动力学之间的密切联系。所开发的方法是用来分析两个全等磁盘之间的相关性-一个固定的和一个旋转的（埃克塞特悖论）。具体地说，我们考虑的过渡盘从静止状态下施加的力旋转状态。它揭示了潜在的物理机制，在相应的过渡，从欧几里得几何的固定磁盘的罗巴切夫斯基（双曲）几何的旋转磁盘的过程中，其旋转助推。得出结论：定半径旋转圆盘或圆环的静止质量中必须含有一个附加项，表示其材料的非欧周向变形势能。可能的实验观察到的表现罗巴切夫斯基的几何旋转系统进行了讨论。
英文摘要：
Relativistic kinematics is usually considered only as a manifestation of pseudo-Euclidean (Lorentzian) geometry of space-time. However, as it is explicitly stated in General Relativity, the geometry itself depends on dynamics, specifically, on the energy-momentum tensor. We discuss a few examples, which illustrate the dynamical aspect of the length-contraction effect within the framework of Special Relativity. We show some pitfalls associated with direct application of the length contraction formula in cases when an extended object is accelerated. Our analysis reveals intimate connections between length contraction and the dynamics of internal forces within the accelerated system. The developed approach is used to analyze the correlation between two congruent disks - one stationary and one rotating (the Ehrenfest paradox). Specifically, we consider the transition of a disk from the state of rest to a spinning state under the applied forces. It reveals the underlying physical mechanism in the corresponding transition from Euclidean geometry of stationary disk to Lobachevsky's (hyperbolic) geometry of the spinning disk in the process of its rotational boost. A conclusion is made that the rest mass of a spinning disk or ring of a fixed radius must contain an additional term representing the potential energy of non-Euclidean circumferential deformation of its material. Possible experimentally observable manifestations of Lobachevsky's geometry of rotating systems are discussed.
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