chapter 11

 The Lorentz Transformation¹

 

The results of the last three sections show that the apparent incompatibility² of the law of propagation of light with the principle of relativity (Section VII) has been derived³ by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows:

 

(1) The time-interval (time) between two events is independent of the condition of motion of the body of reference.

(2) The space-interval (distance) between two points of a rigid⁴ body is independent of the condition of motion of the body of reference.

If we drop these hypotheses, then the dilemma⁵ of Section VII disappears, because the theorem of the addition of velocities⁶ derived in Section VI becomes invalid⁷. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of Section VI in order to remove the apparent disagreement between these two fundamental results of experience? This question leads to a general one. In the discussion of Section VI we have to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity? In other words: Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity⁸ of transmission c relative to the embankment and relative to the train? This question leads to a quite definite positive answer, and to a perfectly⁹ definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another.

 

 

Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in Section II we can imagine this reference-body supplemented laterally¹⁰ and in a vertical¹¹ direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework. Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere¹² with each other, owing to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular¹³ to each other marked out, and designated as “co-ordinate planes” (“co-ordinate system”). A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K′ to the train. An event, wherever it may have taken place, would be fixed¹⁴ in space with respect to K by the three perpendiculars¹⁵ x comma y comma z on the co-ordinate planes, and with regard to time by a time value t. Relative to K′, the same event would be fixed in respect of space and time by corresponding values x prime comma y prime comma z prime comma t prime, which of course are not identical with x comma y comma z comma t. It has already been set forth¹⁶ in detail how these magnitudes are to be regarded as results of physical measurements.

 

Obviously our problem can be exactly formulated¹⁷ in the following manner. What are the values x prime comma y prime comma z prime comma t prime, of an event with respect to K′, when the magnitudes x comma y comma z comma t, of the same event with respect to K are given? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every ray) with respect to K and K′. For the relative orientation¹⁸ in space of the co-ordinate systems indicated in the diagram (Fig 2), this problem is solved by means of the equations:

StartLayout 1st Row 1st Column x prime 2nd Column equals 3rd Column StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndFraction 2nd Row 1st Column y prime 2nd Column equals 3rd Column y 3rd Row 1st Column z prime 2nd Column equals 3rd Column z 4th Row 1st Column t prime 2nd Column equals 3rd Column StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction EndLayout

This system of equations is known as the “Lorentz transformation.”1

 

If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:

StartLayout 1st Row 1st Column x prime 2nd Column equals 3rd Column x minus v t 2nd Row 1st Column y prime 2nd Column equals 3rd Column y 3rd Row 1st Column z prime 2nd Column equals 3rd Column z 4th Row 1st Column t prime 2nd Column equals 3rd Column t EndLayout

This system of equations is often termed the “Galilei transformation.” The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely¹⁹ large value for the velocity of light c in the latter transformation.

 

Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body K and for the reference-body K′. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation

x equals c t comma

i.e. with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t involves a relation between x′ and t′. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:

StartLayout 1st Row 1st Column x prime 2nd Column equals 3rd Column StartFraction left-parenthesis c minus v right-parenthesis t Over StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndFraction 2nd Row 1st Column t prime 2nd Column equals 3rd Column StartStartFraction left-parenthesis 1 minus StartFraction v Over c EndFraction right-parenthesis t OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction EndLayout

from which, by division, the expression

x prime equals c t prime

immediately follows. If referred to the system K′, the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the reference-body K′ is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever²⁰. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.

 

1 A simple derivation of the Lorentz transformation is given in Appendix I.

1 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。

2 incompatibility   

n.不兼容

参考例句：

• One cause may be an Rh incompatibility causing kernicterus in the newborn. 一个原因可能是Rh因子不相配引起新生儿的脑核性黄疸。
• Sexual incompatibility is wide-spread in the apple. 性的不亲合性在苹果中很普遍。

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 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。

5 dilemma   

n.困境，进退两难的局面

参考例句：

• I am on the horns of a dilemma about the matter.这件事使我进退两难。
• He was thrown into a dilemma.他陷入困境。

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 invalid   

n.病人，伤残人；adj.有病的，伤残的；无效的

参考例句：

• He will visit an invalid.他将要去看望一个病人。
• A passport that is out of date is invalid.护照过期是无效的。

8 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。

9 perfectly   

adv.完美地，无可非议地，彻底地

参考例句：

• The witnesses were each perfectly certain of what they said.证人们个个对自己所说的话十分肯定。
• Everything that we're doing is all perfectly above board.我们做的每件事情都是光明正大的。

10 laterally   

ad.横向地；侧面地；旁边地

参考例句：

• Shafts were sunk, with tunnels dug laterally. 竖井已经打下，并且挖有横向矿道。
• When the plate becomes unstable, it buckles laterally. 当板失去稳定时，就发生横向屈曲。

11 vertical   

adj.垂直的，顶点的，纵向的；n.垂直物，垂直的位置

参考例句：

• The northern side of the mountain is almost vertical.这座山的北坡几乎是垂直的。
• Vertical air motions are not measured by this system.垂直气流的运动不用这种系统来测量。

12 interfere   

v.(in)干涉，干预；(with)妨碍，打扰

参考例句：

• If we interfere, it may do more harm than good.如果我们干预的话，可能弊多利少。
• When others interfere in the affair,it always makes troubles. 别人一卷入这一事件，棘手的事情就来了。

13 perpendicular   

adj.垂直的，直立的；n.垂直线，垂直的位置

参考例句：

• The two lines of bones are set perpendicular to one another.这两排骨头相互垂直。
• The wall is out of the perpendicular.这墙有些倾斜。

14 fixed   

adj.固定的，不变的，准备好的；(计算机)固定的

参考例句：

• Have you two fixed on a date for the wedding yet?你们俩选定婚期了吗？
• Once the aim is fixed,we should not change it arbitrarily.目标一旦确定，我们就不应该随意改变。

15 perpendiculars   

n.垂直的，成直角的( perpendicular的名词复数 )；直立的

参考例句：

• It is normal if two hands are in a line and perpendiculars to the floor. 标准姿势就是两只手在一条直线上与地面成直角。 来自互联网

16 forth   

adv.向前；向外，往外

参考例句：

• The wind moved the trees gently back and forth.风吹得树轻轻地来回摇晃。
• He gave forth a series of works in rapid succession.他很快连续发表了一系列的作品。

17 formulated   

v.构想出( formulate的过去式和过去分词 )；规划；确切地阐述；用公式表示

参考例句：

• He claims that the writer never consciously formulated his own theoretical position. 他声称该作家从未有意识地阐明他自己的理论见解。 来自《简明英汉词典》
• This idea can be formulated in two different ways. 这个意思可以有两种说法。 来自《现代汉英综合大词典》

18 orientation   

n.方向，目标；熟悉，适应，情况介绍

参考例句：

• Children need some orientation when they go to school.小孩子上学时需要适应。
• The traveller found his orientation with the aid of a good map.旅行者借助一幅好地图得知自己的方向。

19 infinitely   

adv.无限地，无穷地

参考例句：

• There is an infinitely bright future ahead of us.我们有无限光明的前途。
• The universe is infinitely large.宇宙是无限大的。

20 whatsoever   

adv.（用于否定句中以加强语气）任何；pron.无论什么

参考例句：

• There's no reason whatsoever to turn down this suggestion.没有任何理由拒绝这个建议。
• All things whatsoever ye would that men should do to you,do ye even so to them.你想别人对你怎样，你就怎样对人。