Relativity 03: Space and Time in Classical Mechanics

The purpose of mechanics is to describe how bodies change their position in space with "time." I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.

It is not clear what is to be understood here by "position" and "space." I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space" ? From the considerations of the previous section the answer is self-evident. In the first place we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of " body of reference " we insert " system of co-ordinates," which is a useful idea for mathematical description, we are in a position to say : The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (lit. "path-curve" ¹⁾), but only a trajectory relative to a particular body of reference.

In order to have a complete description of the motion, we must specify how the body alters its position with time ; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction ; the man at the railway-carriage window is holding one of them, and the man on the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.

 

力学的目的是描述物体如何随着“时间”在空间中的位置发生变化，如果我用这种方式来表述力学的目的，我的良心就会背上严重的罪恶，违背清醒的神圣精神。让我们继续揭露这些罪恶。

 

“我站在这没有石子的车厢里，是什么意思？”。然后，不管空气阻力的影响，我看到石头是直线下降的。一个行人从人行道上观察到了这一错误行为，他注意到石头以抛物线的形式落在地上。我现在要问：石头经过的“位置”是在直线上还是在抛物线上？此外，这里所说的“空间运动”是什么意思？从上一节的考虑来看，答案是不言而喻的。首先，我们完全回避“空间”这个模糊的词，我们必须诚实地承认，我们不能形成一点概念，我们将其替换为“相对于实际刚性参考体的运动”。相对于参考体（铁路车厢或路堤）的位置已在上一节中详细定义。如果我们插入“坐标系”而不是“参考体”，这是数学描述的一个有用的想法，那么我们可以说：石头穿过一条直线，而这条直线是与牢牢固定在马车上的坐标系有关的，但相对于严格附着在地面（路堤）上的坐标系统，它描述了一条抛物线。在这个例子的帮助下，我们可以清楚地看到，不存在独立存在的轨迹（lit）路径曲线“1”），但仅是相对于特定参考体的轨迹。

 

为了对运动有一个完整的描述，我们必须说明物体是如何随时间改变其位置的；也就是说，对于轨迹上的每个点，都必须说明物体在什么时候处于那里。这些数据必须由这样一个时间定义来补充，根据这个定义，这些时间值基本上可以被视为能够观测的震级（测量结果）。如果我们站在经典力学的基础上，我们可以用下面的方式来满足这个要求。我们想象两个结构相同的钟，火车车厢窗口的人拿着一个，人行道上的人拿着另一个。每一个观察者都会在他手中的时钟每一次滴答声中确定自己的参考物体上被石头占据的位置。在这方面，我们没有考虑到光传播速度的有限性所涉及的不精确性。有了这一点，再加上这里普遍存在的第二个困难，我们以后将不得不详细讨论。