chapter 13

 Theorem of the Addition of Velocities¹.

The Experiment of Fizeau

 

Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity² of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived³ from the foregoing considerations, and which has been most elegantly confirmed by experiment.

 

In Section VI we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics—This theorem can also be deduced readily horn the Galilei transformation⁴ (Section XI). In place of the man walking inside the carriage, we introduce a point moving relatively⁵ to the co-ordinate system K′ in accordance with the equation

x prime equals w t Superscript prime Baseline period

By means of the first and fourth equations of the Galilei transformation we can express x′ and t′ in terms of x and t, and we then obtain

x equals left-parenthesis v plus w right-parenthesis t period

This equation expresses nothing else than the law of motion of the point with reference to the system K (of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section VI,

upper W equals v plus w (A)

But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation

x prime equals w t prime

we must then express x′ and t′ in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

upper W equals StartStartFraction v plus w OverOver 1 plus StartFraction v w Over c squared EndFraction EndEndFraction (B)

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we axe⁶ enlightened by a most important experiment which the brilliant physicist⁷ Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists⁸, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Figure 3) when the liquid above mentioned is flowing through the tube with a velocity v?

 

Fig⁹. 3

Fig. 3

 

In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube.

 

It is clear that we have the problem of Section VI again before us. The tube plays the part of the railway embankment or of the co-ordinate system K, the liquid plays the part of the carriage or of the co-ordinate system K′, and finally, the light plays the part of the man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by W, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment1 decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow v on the propagation of light is represented by formula (B) to within one per cent.

 

Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely¹⁰ electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness¹¹ of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of Maxwell-Lorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly¹² independent of each other, on which electrodynamics was built.

 

1 Fizeau found upper W equals w plus v left-parenthesis 1 minus StartFraction 1 Over n squared EndFraction right-parenthesis, where n equals StartFraction c Over w EndFraction is the index of refraction of the liquid. On the other hand, owing to the smallness of StartFraction v w Over c squared EndFraction as compared with 1, we can replace (B) in the first place by upper W equals left-parenthesis w plus v right-parenthesis left-parenthesis 1 minus StartFraction v w Over c squared EndFraction right-parenthesis, or to the same order of approximation by w plus v left-parenthesis 1 minus StartFraction 1 Over n squared EndFraction right-parenthesis, which agrees with Fizeau’s result. 

1 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

2 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。

3 derived   

vi.起源；由来；衍生；导出v.得到( derive的过去式和过去分词 )；（从…中）得到获得；源于；（从…中）提取

参考例句：

• Many English words are derived from Latin and Greek. 英语很多词源出于拉丁文和希腊文。 来自《简明英汉词典》
• He derived his enthusiasm for literature from his father. 他对文学的爱好是受他父亲的影响。 来自《简明英汉词典》

4 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。

5 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。

6 axe   

n.斧子；v.用斧头砍，削减

参考例句：

• Be careful with that sharp axe.那把斧子很锋利，你要当心。
• The edge of this axe has turned.这把斧子卷了刃了。

7 physicist   

n.物理学家，研究物理学的人

参考例句：

• He is a physicist of the first rank.他是一流的物理学家。
• The successful physicist never puts on airs.这位卓有成就的物理学家从不摆架子。

8 physicists   

物理学家( physicist的名词复数 )

参考例句：

• For many particle physicists, however, it was a year of frustration. 对于许多粒子物理学家来说，这是受挫折的一年。 来自英汉非文学 - 科技
• Physicists seek rules or patterns to provide a framework. 物理学家寻求用法则或图式来构成一个框架。

9 fig   

n.无花果(树)

参考例句：

• The doctor finished the fig he had been eating and selected another.这位医生吃完了嘴里的无花果，又挑了一个。
• You can't find a person who doesn't know fig in the United States.你找不到任何一个在美国的人不知道无花果的。

10 purely   

adv.纯粹地，完全地

参考例句：

• I helped him purely and simply out of friendship.我帮他纯粹是出于友情。
• This disproves the theory that children are purely imitative.这证明认为儿童只会单纯地模仿的理论是站不住脚的。

11 conclusiveness   

n.最后； 释疑； 确定性； 结论性

参考例句：

12 formerly   

adv.从前，以前

参考例句：

• We now enjoy these comforts of which formerly we had only heard.我们现在享受到了过去只是听说过的那些舒适条件。
• This boat was formerly used on the rivers of China.这船从前航行在中国内河里。