chapter 22

 A Few Inferences from the General Principle of Relativity

 

The considerations of Section XX show that the general principle of relativity puts us in a position to derive¹ properties of the gravitational field in a purely² theoretical manner. Let us suppose, for instance, that we know the space-time “course” for any natural process whatsoever³, as regards the manner in which it takes place in the Galileian domain⁴ relative to a Galileian body of reference K. By means of purely theoretical operations (i.e. simply by calculation) we are then able to find how this known natural process appears, as seen from a reference-body K′ which is accelerated relatively⁵ to K. But since a gravitational field exists with respect to this new body of reference K, our consideration also teaches us how the gravitational field influences the process studied.

 

For example, we learn that a body which is in a state of uniform rectilinear motion with respect to K (in accordance with the law of Galilei) is executing an accelerated and in general curvilinear motion with respect to the accelerated reference-body K′ (chest). This acceleration⁶ or curvature corresponds to the influence on the moving body of the gravitational field prevailing⁷ relatively to K. It is known that a gravitational field influences the movement of bodies in this way, so that our consideration supplies us with nothing essentially⁸ new.

 

However, we obtain a new result of fundamental importance when we carry out the analogous⁹ consideration for a ray of light. With respect to the Galileian reference-body K, such a ray of light is transmitted rectilinearly with the velocity¹⁰ c. It can easily be shown that the path of the same ray of light is no longer a straight line when we consider it with reference to the accelerated chest (reference-body K′). From this we conclude, that, in general, rays of light are propagated curvilinearly in gravitational fields. In two respects this result is of great importance.

 

In the first place, it can be compared with the reality. Although a detailed¹¹ examination of the question shows that the curvature of light rays required by the general theory of relativity is only exceedingly small for the gravitational fields at our disposal in practice, its estimated magnitude for light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of arc. This ought to manifest itself in the following way. As seen from the earth, certain fixed¹² stars appear to be in the neighbourhood of the sun, and are thus capable of observation during a total eclipse of the sun. At such times, these stars ought to appear to be displaced outwards¹³ from the sun by an amount indicated above, as compared with their apparent position in the sky when the sun is situated¹⁴ at another part of the heavens. The examination of the correctness or otherwise of this deduction¹⁵ is a problem of the greatest importance, the early solution of which is to be expected of astronomers¹⁶.1

 

In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited¹⁷ validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena¹⁸ (e.g. of light).

 

Since it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown¹⁹ by the general theory of relativity, it is perhaps advisable to make the facts of the case clearer by means of an appropriate comparison. Before the development of electrodynamics the laws of electrostatics were looked upon as the laws of electricity. At the present time we know that electric fields can be derived²⁰ correctly from electrostatic considerations only for the case, which is never strictly²¹ realised, in which the electrical masses are quite at rest relatively to each other, and to the co-ordinate system. Should we be justified²² in saying that for this reason electrostatics is overthrown by the field-equations of Maxwell in electrodynamics? Not in the least. Electrostatics is contained in electrodynamics as a limiting case; the laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time. No fairer destiny could be allotted²³ to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.

 

In the example of the transmission of light just dealt with, we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the laws of which are already known when a gravitational field is absent. But the most attractive problem, to the solution of which the general theory of relativity supplies the key, concerns the investigation²⁴ of the laws satisfied by the gravitational field itself. Let us consider this for a moment.

 

We are acquainted with space-time domains²⁵ which behave (approximately) in a “Galileian” fashion under suitable choice of reference-body, i.e. domains in which gravitational fields are absent. If we now refer such a domain to a reference-body K′ possessing any kind of motion, then relative to K′ there exists a gravitational field which is variable with respect to space and time.2 The character of this field will of course depend on the motion chosen for K′. According to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. Even though by no means all gravitational fields can be produced in this way, yet we may entertain the hope that the general law of gravitation will be derivable²⁶ from such gravitational fields of a special kind. This hope has been realised in the most beautiful manner. But between the clear vision of this goal and its actual realisation it was necessary to surmount²⁷ a serious difficulty, and as this lies deep at the root of things, I dare not withhold²⁸ it from the reader. We require to extend our ideas of the space-time continuum still farther.

 

1 By means of the star photographs of two expeditions equipped by a Joint²⁹ Committee of the Royal and Royal Astronomical³⁰ Societies, the existence of the deflection of light demanded by theory was first confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix III.) ↑

 

2 This follows from a generalisation of the discussion in Section XX.

1 derive   

v.取得；导出；引申；来自；源自；出自

参考例句：

• We derive our sustenance from the land.我们从土地获取食物。
• We shall derive much benefit from reading good novels.我们将从优秀小说中获得很大好处。

2 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。

3 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。

4 domain   

n.(活动等)领域，范围；领地，势力范围

参考例句：

• This information should be in the public domain.这一消息应该为公众所知。
• This question comes into the domain of philosophy.这一问题属于哲学范畴。

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 acceleration   

n.加速，加速度

参考例句：

• All spacemen must be able to bear acceleration.所有太空人都应能承受加速度。
• He has also called for an acceleration of political reforms.他同时呼吁加快政治改革的步伐。

7 prevailing   

adj.盛行的；占优势的；主要的

参考例句：

• She wears a fashionable hair style prevailing in the city.她的发型是这个城市流行的款式。
• This reflects attitudes and values prevailing in society.这反映了社会上盛行的态度和价值观。

8 essentially   

adv.本质上，实质上，基本上

参考例句：

• Really great men are essentially modest.真正的伟人大都很谦虚。
• She is an essentially selfish person.她本质上是个自私自利的人。

9 analogous   

adj.相似的；类似的

参考例句：

• The two situations are roughly analogous.两种情況大致相似。
• The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。

10 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。

11 detailed   

adj.详细的，详尽的，极注意细节的，完全的

参考例句：

• He had made a detailed study of the terrain.他对地形作了缜密的研究。
• A detailed list of our publications is available on request.我们的出版物有一份详细的目录备索。

12 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。

13 outwards   

adj.外面的，公开的，向外的；adv.向外；n.外形

参考例句：

• Does this door open inwards or outwards?这门朝里开还是朝外开？
• In lapping up a fur,they always put the inner side outwards.卷毛皮时，他们总是让内层朝外。

14 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。

15 deduction   

n.减除，扣除，减除额；推论，推理，演绎

参考例句：

• No deduction in pay is made for absence due to illness.因病请假不扣工资。
• His deduction led him to the correct conclusion.他的推断使他得出正确的结论。

16 astronomers   

n.天文学者，天文学家( astronomer的名词复数 )

参考例句：

• Astronomers can accurately foretell the date,time,and length of future eclipses. 天文学家能精确地预告未来日食月食的日期、时刻和时长。 来自《简明英汉词典》
• Astronomers used to ask why only Saturn has rings. 天文学家们过去一直感到奇怪，为什么只有土星有光环。 来自《简明英汉词典》

17 unlimited   

adj.无限的，不受控制的，无条件的

参考例句：

• They flew over the unlimited reaches of the Arctic.他们飞过了茫茫无边的北极上空。
• There is no safety in unlimited technological hubris.在技术方面自以为是会很危险。

18 phenomena   

n.现象

参考例句：

• Ade couldn't relate the phenomena with any theory he knew.艾德无法用他所知道的任何理论来解释这种现象。
• The object of these experiments was to find the connection,if any,between the two phenomena.这些实验的目的就是探索这两种现象之间的联系，如果存在着任何联系的话。

19 overthrown   

adj. 打翻的，推倒的，倾覆的 动词overthrow的过去分词

参考例句：

• The president was overthrown in a military coup. 总统在军事政变中被赶下台。
• He has overthrown the basic standards of morality. 他已摒弃了基本的道德标准。

20 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》

21 strictly   

adv.严厉地，严格地；严密地

参考例句：

• His doctor is dieting him strictly.他的医生严格规定他的饮食。
• The guests were seated strictly in order of precedence.客人严格按照地位高低就座。

22 justified   

a.正当的，有理的

参考例句：

• She felt fully justified in asking for her money back. 她认为有充分的理由要求退款。
• The prisoner has certainly justified his claims by his actions. 那个囚犯确实已用自己的行动表明他的要求是正当的。

23 allotted   

分配，拨给，摊派( allot的过去式和过去分词 )

参考例句：

• I completed the test within the time allotted . 我在限定的时间内完成了试验。
• Each passenger slept on the berth allotted to him. 每个旅客都睡在分配给他的铺位上。

24 investigation   

n.调查，调查研究

参考例句：

• In an investigation,a new fact became known, which told against him.在调查中新发现了一件对他不利的事实。
• He drew the conclusion by building on his own investigation.他根据自己的调查研究作出结论。

25 domains   

n.范围( domain的名词复数 )；领域；版图；地产

参考例句：

• The theory of thermodynamics links the macroscopic and submicroscopic domains. 热力学把宏观世界同亚微观世界联系起来。 来自辞典例句
• All three flow domains are indicated by shading. 所有三个流动区域都是用阴影部分表示的。 来自辞典例句

26 derivable   

adj.可引出的，可推论的，可诱导的

参考例句：

• These results are derivable from the theory of quantum mechanics. 这些结果是根据量子力学理论推导出来的。 来自辞典例句

27 surmount   

vt.克服；置于…顶上

参考例句：

• We have many problems to surmount before we can start the project.我们得克服许多困难才能著手做这项工作。
• We are fully confident that we can surmount these difficulties.我们完全相信我们能够克服这些困难。

28 withhold   

v.拒绝，不给；使停止，阻挡

参考例句：

• It was unscrupulous of their lawyer to withhold evidence.他们的律师隐瞒证据是不道德的。
• I couldn't withhold giving some loose to my indignation.我忍不住要发泄一点我的愤怒。

29 joint   

adj.联合的，共同的；n.关节，接合处；v.连接，贴合

参考例句：

• I had a bad fall,which put my shoulder out of joint.我重重地摔了一跤，肩膀脫臼了。
• We wrote a letter in joint names.我们联名写了封信。

30 astronomical   

adj.天文学的，（数字）极大的

参考例句：

• He was an expert on ancient Chinese astronomical literature.他是研究中国古代天文学文献的专家。
• Houses in the village are selling for astronomical prices.乡村的房价正在飙升。