## 1 Introduction

Postulate I.The laws of physics are the same in all inertial reference frames.
Postulate II.The speed of light in vacuum is a universal constant.
See Fig (2). Assume two inertial frames $S,S\text{'}$ with origins $O,O\text{'}$ where the frames are similarly oriented and $S\text{'}$ moves with a uniform velocity of $v$ in the $x$-direction of $S$. When the origins $O,O\text{'}$ coincide, let the times of the frames be $t\text{'}=t=0$. The Lorentz transform of the frames is given by the following coordinates and time transformations:

## 2 Relativistic Length Contraction

It is customary to represent a position at a certain time moment as an "event" in special relativity with a 4-vector combining the position coordinates with the time in the frame as: $E\left(x,y,z,t\right)$. When dealing with the common Lorentz transformation as in equation (1.1), we would only use the shortened notation with only the $x$ coordinates with $t$ as event: $E\left({x}_{1},{t}_{1}\right)$,etc.
Now, $L\text{'}$ in equation (2.5) is the difference in the space coordinates of the two events $A\text{'}\left({x}_{1}\text{'},{t}_{1}\text{'}\right)$ and $B\text{'}\left({x}_{2}\text{'},{t}_{2}\text{'}\right)$ in frame $S\text{'}$. Because the times ${t}_{2}\text{'}$ and ${t}_{1}\text{'}$ have been chosen to be the same, $L\text{'}$ has the meaning of the length of the rod $AB$ on the ground as viewed by a stationary observer in $S\text{'}$ at a specific moment in time of the frame $S\text{'}$. From equation (2.5), we can see that if $v, then $L\text{'}. It is presently well accepted that nothing has been empirically observed to be able to travel faster than the speed of light $c$; so it can be said the relative velocity between inertial frames $v$ can never exceed $c$. So $L\text{'} is considered a fact of the physical world.
In our derivation, although it is the observer which is moving observing the rod stationary on the ground, the convention of special relativity considers the rod to be moving as the "observation" in frame $S\text{'}$ has to be represented by events $B\text{'}\left({x}_{2}\text{'},{t}_{2}\text{'}\right)$ and $A\text{'}\left({x}_{1}\text{'},{t}_{1}\text{'}\right)$ at the same time moment in $S\text{'}$; this is satisfied as ${t}_{2}\text{'}={t}_{1}\text{'}$. We have here derived the formula for the Lorentz-FitzGerald contraction:
An object in motion has its length contracted in the direction of its motion.

## 3 The Lorentz Transformation Is Only Abstract

From equation (2.5), we have $L\text{'}=L\sqrt{1-{v}^{2}/{c}^{2}}$. So it seems we also have $L\text{'}$ in the same unit of length as associated with $L$. It may come as a great surprise to many that $L\text{'}$ is not in the unit of length - it is only a pure real number. A great part of modern physics theories today are founded on the Lorentz transformation and it is assumed that all variables in the Lorentz transformation are representing real physical quantities without anyone calling into question if there is any justification for such an assumption. Not one physicist has ever given an explanation why the pure scalars in the image space $S\text{'}$ too may take on the same physical units as in the domain space.
Mathematical mappings only relate abstract objects from the domain space to the image space. It is irrelevant in mathematics whether the objects in a mathematical treatment have any physical association. It is only in physics that mathematical objects used must have physical significance as physical theories deal only with quantities that may be measured and examined experimentally. So when the Lorentz transformation is applied to arrived at $L\text{'}=L\sqrt{1-{v}^{2}/{c}^{2}}$, any association of $L$ with a physical unit become irrelevant. All the mathematics in the above derivations work only on the pure mathematical objects stripped of any association of any with physical units. So the variable $L\text{'}$ in equation $L\text{'}=L\sqrt{1-{v}^{2}/{c}^{2}}$ is only a pure real number with no association whatever with any standard unit of measure.
There is no natural principle that mathematical linear transformations also carry over associations of scalars with real physical units from the domain to the image space of the transformation.

## 6 相对论长度收缩效应

$\begin{array}{cc}{t}_{2}\text{'}-{t}_{1}\text{'}=\gamma \left({t}_{2}-{t}_{1}\right)-\gamma v/{c}^{2}\left({x}_{2}-{x}_{1}\right)& \left(6.4\right)\end{array}$在上述等式中的 $t2$$t1$ 可以是任何两个随机時间值，因为读取 $AB$ 的结束坐标的时间并不重要,点 $A$$B$ 可以随时分开读取。它们的差异仍然给出杆 $AB$ 相同的长度。我们将在等式 (6.4) 中选择 ${t}_{2}$${t}_{1}$ 使得 ${t}_{2}\text{'}-{t}_{1}\text{'}=0$。 以这种方式，从等式 (6.3) 和 (6.4)，我们将得到： $\begin{array}{cc}\begin{array}{cc}& L\text{'}=\gamma L-\gamma {v}^{2}/{c}^{2}\left({x}_{2}-{x}_{1}\right);\\ & L\text{'}=\gamma L\left(1-{v}^{2}/{c}^{2}\right);\\ & L\text{'}=L\sqrt{1-{v}^{2}/{c}^{2}}\end{array}& \left(6.5\right)\end{array}$在等式 (6.5) 中的 $L\text{'}$ 是帧 $S\text{'}$ 中的两个事件 $A\text{'}$$B\text{'}$ 的空间坐标的差。 因为时间 ${t}_{2}\text{'}$${t}_{1}\text{'}$ 被选择为相同，所以 $L\text{'}$ 的意思是观察者在一个特定的时刻在 $S\text{'}$ 中所观察到在地面上的杆 $AB$ 的长度。 从等式 (6.5) 可以看出，如果 $v，则 $L\text{'}。 目前普遍接受的是，没有证据表明任何物体可能比光速更快地移动。所以惯性框架 $v$ 之间的相对速度永远不会超于 $c$$L\text{'} 被认为是物理世界的一个事实。