chapter 12

 The Behaviour of Measuring-Rods and Clocks in Motion

 

Place a metre-rod in the x′-axis of K′ in such a manner that one end (the beginning) coincides with the point x prime equals 0 whilst the other end (the end of the rod) coincides with the point x prime equals 1. What is the length of the metre-rod relatively¹ to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K. By means of the first equation of the Lorentz transformation² the values of these two points at the time t equals 0 can be shown to be

StartLayout 1st Row 1st Column x Subscript left-parenthesis beginning of rod right-parenthesis 2nd Column equals 3rd Column 0 dot StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot 2nd Row 1st Column x Subscript left-parenthesis end of rod right-parenthesis 2nd Column equals 3rd Column 1 dot StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndLayout

the distance between the points being StartRoot 1 minus v squared slash³ c squared EndRoot.

 

But the metre-rod is moving with the velocity⁴ v relative to K. It therefore follows that the length of a rigid⁵ metre-rod moving in the direction of its length with a velocity v is StartRoot 1 minus v squared slash c squared EndRoot of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity v equals c we should have StartRoot 1 minus v squared slash c squared EndRoot equals 0, and for still greater velocities⁶ the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.

 

Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these became meaningless if we choose values of v greater than c.

 

If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to K, then we should have found that the length of the rod as judged from K′ would have been StartRoot 1 minus v squared slash c squared EndRoot; this is quite in accordance with the principle of relativity which forms the basis of our considerations.

 

A Priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes z comma y comma x comma t, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galileian transformation we should not have obtained a contraction⁷ of the rod as a consequence of its motion.

 

Let us now consider a seconds-clock which is permanently⁸ situated⁹ at the origin (x prime equals 0) of K′. t prime equals 0 and t prime equals 1 are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:

t equals 0

and

t prime equals StartFraction 1 Over StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndFraction period

As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but StartFraction 1 Over StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndFraction seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.

1 relatively   

adv.比较...地，相对地

参考例句：

• The rabbit is a relatively recent introduction in Australia.兔子是相对较新引入澳大利亚的物种。
• The operation was relatively painless.手术相对来说不痛。

2 transformation   

n.变化；改造；转变

参考例句：

• Going to college brought about a dramatic transformation in her outlook.上大学使她的观念发生了巨大的变化。
• He was struggling to make the transformation from single man to responsible husband.他正在努力使自己由单身汉变为可靠的丈夫。

3 slash   

vi.大幅度削减；vt.猛砍，尖锐抨击，大幅减少；n.猛砍，斜线，长切口，衣衩

参考例句：

• The shop plans to slash fur prices after Spring Festival.该店计划在春节之后把皮货降价。
• Don't slash your horse in that cruel way.不要那样残忍地鞭打你的马。

4 velocity   

n.速度，速率

参考例句：

• Einstein's theory links energy with mass and velocity of light.爱因斯坦的理论把能量同质量和光速联系起来。
• The velocity of light is about 300000 kilometres per second.光速约为每秒300000公里。

5 rigid   

adj.严格的，死板的；刚硬的，僵硬的

参考例句：

• She became as rigid as adamant.她变得如顽石般的固执。
• The examination was so rigid that nearly all aspirants were ruled out.考试很严，几乎所有的考生都被淘汰了。

6 velocities   

n.速度( velocity的名词复数 )；高速，快速

参考例句：

• In experimenting we find out that sound travels with different velocities through different substances. 在实验中，我们发现声音以不同的速度通过不同的物质而传播。 来自《现代汉英综合大词典》
• A gas in thermal equilibrium has particles of all velocities. 处于热平衡的气体，其粒子有一切速度。 来自辞典例句

7 contraction   

n.缩略词，缩写式，害病

参考例句：

• The contraction of this muscle raises the lower arm.肌肉的收缩使前臂抬起。
• The forces of expansion are balanced by forces of contraction.扩张力和收缩力相互平衡。

8 permanently   

adv.永恒地，永久地，固定不变地

参考例句：

• The accident left him permanently scarred.那次事故给他留下了永久的伤疤。
• The ship is now permanently moored on the Thames in London.该船现在永久地停泊在伦敦泰晤士河边。

9 situated   

adj.坐落在...的，处于某种境地的

参考例句：

• The village is situated at the margin of a forest.村子位于森林的边缘。
• She is awkwardly situated.她的处境困难。