chapter 29

 The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity

 

If the reader has followed all our previous considerations, he will have no further difficulty in understanding the methods leading to the solution of the problem of gravitation.

 

We start off on a consideration of a Galileian domain¹, i.e. a domain in which there is no gravitational field relative to the Galileian reference-body K. The behaviour of measuring-rods and clocks with reference to K is known from the special theory of relativity, likewise the behaviour of “isolated” material points; the latter move uniformly and in straight lines.

 

Now let us refer this domain to a random² Gauss coordinate³ system or to a “mollusc” as reference-body K′. Then with respect to K′ there is a gravitational field G (of a particular kind). We learn the behavior of measuring-rods and clocks and also of freely-moving material points with reference to K′ simply by mathematical transformation⁴. We interpret this behaviour as the behaviour of measuring-rods, docks and material points tinder the influence of the gravitational field G. Hereupon we introduce a hypothesis: that the influence of the gravitational field on measuring-rods, clocks and freely-moving material points continues to take place according to the same laws, even in the case where the prevailing⁵ gravitational field is not derivable⁶ from the Galileian special care, simply by means of a transformation of co-ordinates.

 

The next step is to investigate the space-time behaviour of the gravitational field G, which was derived⁷ from the Galileian special case simply by transformation of the coordinates⁸. This behaviour is formulated⁹ in a law, which is always valid¹⁰, no matter how the reference-body (mollusc) used in the description may be chosen.

 

This law is not yet the general law of the gravitational field, since the gravitational field under consideration is of a special kind. In order to find out the general law-of-field of gravitation we still require to obtain a generalisation of the law as found above. This can be obtained without caprice, however, by taking into consideration the following demands:

 

(a) The required generalisation must likewise satisfy the general postulate¹¹ of relativity.

(b) If there is any matter in the domain under consideration, only its inertial mass, and thus according to Section XV only its energy is of importance for its effect in exciting a field.

(c) Gravitational field and matter together must satisfy the law of the conservation of energy (and of impulse).

Finally, the general principle of relativity permits us to determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent i.e. which have already been fitted into the frame of the special theory of relativity. In this connection we proceed in principle according to the method which has already been explained for measuring-rods, clocks and freely moving material points.

 

The theory of gravitation derived in this way from the general postulate of relativity excels not only in its beauty; nor in removing the defect attaching to classical mechanics which was brought to light in Section XXI; nor in interpreting the empirical law of the equality of inertial and gravitational mass; but it has also already explained a result of observation in astronomy, against which classical mechanics is powerless.

 

If we confine the application of the theory to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the coordinate system with velocities¹² which are small compared with the velocity¹³ of light, we then obtain as a first approximation the Newtonian theory. Thus the latter theory is obtained here without any particular assumption, whereas Newton had to introduce the hypothesis that the force of attraction between mutually attracting material points is inversely¹⁴ proportional to the square of the distance between them. If we increase the accuracy of the calculation, deviations¹⁵ from the theory of Newton make their appearance, practically all of which must nevertheless escape the test of observation owing to their smallness.

 

We must draw attention here to one of these deviations. According to Newton’s theory, a planet moves round the sun in an ellipse, which would permanently¹⁶ maintain its position with respect to the fixed¹⁷ stars, if we could disregard the motion of the fixed stars themselves and the action of the other planets under consideration. Thus, if we correct the observed motion of the planets for these two influences, and if Newton’s theory be strictly¹⁸ correct, we ought to obtain for the orbit of the planet an ellipse, which is fixed with reference to the fixed stars. This deduction¹⁹, which can be tested with great accuracy, has been confirmed for all the planets save one, with the precision that is capable of being obtained by the delicacy²⁰ of observation attainable²¹ at the present time. The sole exception is Mercury, the planet which lies nearest the sun. Since the time of Leverrier, it has been known that the ellipse corresponding to the orbit of Mercury, after it has been corrected for the influences mentioned above, is not stationary²² with respect to the fixed stars, but that it rotates exceedingly slowly in the plane of the orbit and in the sense of the orbital motion. The value obtained for this rotary²³ movement of the orbital ellipse was 43 seconds of arc per century, an amount ensured to be correct to within a few seconds of arc. This effect can be explained by means of classical mechanics only on the assumption of hypotheses which have little probability, and which were devised solely²⁴ for this purpose.

 

On the basis of the general theory of relativity, it is found that the ellipse of every planet round the sun must necessarily rotate in the manner indicated above; that for all the planets, with the exception of Mercury, this rotation²⁵ is too small to be detected with the delicacy of observation possible at the present time; but that in the case of Mercury it must amount to 43 seconds of arc per century, a result which is strictly in agreement with observation.

 

Apart from this one, it has hitherto been possible to make only two deductions²⁶ from the theory which admit of being tested by observation, to wit, the curvature of light rays by the gravitational field of the sun1, and a displacement²⁷ of the spectral²⁸ lines of light reaching us from large stars, as compared with the corresponding lines for light produced in an analogous²⁹ manner terrestrially (i.e. by the same kind of atom)2. These two deductions from the theory have both been confirmed.

 

1 First observed by Eddington and others in 1919. (Cf. Appendix III, pp. 126–129). ↑

 

2 Established by Adams in 1924. (Cf. p. 132)

1 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。

2 random   

adj.随机的；任意的；n.偶然的(或随便的)行动

参考例句：

• The list is arranged in a random order.名单排列不分先后。
• On random inspection the meat was found to be bad.经抽查，发现肉变质了。

3 coordinate   

adj.同等的，协调的；n.同等者；vt.协作，协调

参考例句：

• You must coordinate what you said with what you did.你必须使你的言行一致。
• Maybe we can coordinate the relation of them.或许我们可以调和他们之间的关系。

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 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。

6 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

7 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》

8 coordinates   

n.相配之衣物；坐标( coordinate的名词复数 )；（颜色协调的）配套服装；[复数]女套服；同等重要的人（或物）v.使协调，使调和( coordinate的第三人称单数 )；协调；协同；成为同等

参考例句：

• The town coordinates on this map are 695037. 该镇在这幅地图上的坐标是695037。 来自《简明英汉词典》
• The UN Office for the Coordination of Humanitarian Affairs, headed by the Emergency Relief Coordinator, coordinates all UN emergency relief. 联合国人道主义事务协调厅在紧急救济协调员领导下，负责协调联合国的所有紧急救济工作。 来自《简明英汉词典》

9 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》

10 valid   

adj.有确实根据的；有效的；正当的，合法的

参考例句：

• His claim to own the house is valid.他主张对此屋的所有权有效。
• Do you have valid reasons for your absence?你的缺席有正当理由吗？

11 postulate   

n.假定，基本条件；vt.要求，假定

参考例句：

• Let's postulate that she is a cook.我们假定她是一位厨师。
• Freud postulated that we all have a death instinct as well as a life instinct.弗洛伊德曾假定我们所有人都有生存本能和死亡本能。

12 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

13 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。

14 inversely   

adj.相反的

参考例句：

• Pressure varies directly with temperature and inversely with volume. 压力随温度成正比例变化，与容积成反比例变化。 来自《简明英汉词典》
• The amount of force needed is inversely proportional to the rigidity of the material. 需要的力度与材料的硬度成反比。 来自《简明英汉词典》

15 deviations   

背离，偏离( deviation的名词复数 )； 离经叛道的行为

参考例句：

• Local deviations depend strongly on the local geometry of the solid matrix. 局部偏离严格地依赖于固体矩阵的局部几何形状。
• They were a series of tactical day-to-day deviations from White House policy. 它们是一系列策略上一天天摆脱白宫政策的偏向。

16 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。

17 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。

18 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。

19 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。

20 delicacy   

n.精致，细微，微妙，精良；美味，佳肴

参考例句：

• We admired the delicacy of the craftsmanship.我们佩服工艺师精巧的手艺。
• He sensed the delicacy of the situation.他感觉到了形势的微妙。

21 attainable   

a.可达到的，可获得的

参考例句：

• They set the limits of performance attainable. 它们确定着可达到的运行限度。
• If objectives are to be meaningful to people, they must be clear, attainable, actionable, and verifiable. 如果目标对人们是具有意义的，则目标必须是清晰的，能达到的，可以行动的，以及可供检验的。

22 stationary   

adj.固定的，静止不动的

参考例句：

• A stationary object is easy to be aimed at.一个静止不动的物体是容易瞄准的。
• Wait until the bus is stationary before you get off.你要等公共汽车停稳了再下车。

23 rotary   

adj.(运动等)旋转的；轮转的；转动的

参考例句：

• The central unit is a rotary drum.核心设备是一个旋转的滚筒。
• A rotary table helps to optimize the beam incidence angle.一张旋转的桌子有助于将光线影响之方式角最佳化。

24 solely   

adv.仅仅，唯一地

参考例句：

• Success should not be measured solely by educational achievement.成功与否不应只用学业成绩来衡量。
• The town depends almost solely on the tourist trade.这座城市几乎完全靠旅游业维持。

25 rotation   

n.旋转；循环，轮流

参考例句：

• Crop rotation helps prevent soil erosion.农作物轮作有助于防止水土流失。
• The workers in this workshop do day and night shifts in weekly rotation.这个车间的工人上白班和上夜班每周轮换一次。

26 deductions   

扣除( deduction的名词复数 )； 结论； 扣除的量； 推演

参考例句：

• Many of the older officers trusted agents sightings more than cryptanalysts'deductions. 许多年纪比较大的军官往往相信特务的发现，而不怎么相信密码分析员的推断。
• You know how you rush at things,jump to conclusions without proper deductions. 你知道你处理问题是多么仓促，毫无合适的演绎就仓促下结论。

27 displacement   

n.移置，取代，位移，排水量

参考例句：

• They said that time is the feeling of spatial displacement.他们说时间是空间位移的感觉。
• The displacement of all my energy into caring for the baby.我所有精力都放在了照顾宝宝上。

28 spectral   

adj.幽灵的，鬼魂的

参考例句：

• At times he seems rather ordinary.At other times ethereal,perhaps even spectral.有时他好像很正常，有时又难以捉摸，甚至像个幽灵。
• She is compelling,spectral fascinating,an unforgettably unique performer.她极具吸引力，清幽如鬼魅，令人着迷，令人难忘，是个独具特色的演员。

29 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。