chapter 18

 Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference

 

Hitherto I have purposely refrained from speaking about the physical interpretation¹ of space- and time-data in the case of the general theory of relativity. As a consequence, I am guilty of a certain slovenliness² of treatment, which, as we know from the special theory of relativity, is far from being unimportant and pardonable. It is now high time that we remedy this defect; but I would mention at the outset, that this matter lays no small claims on the patience and on the power of abstraction of the reader.

 

We start off again from quite special cases, which we have frequently used before. Let us consider a space time domain³ in which no gravitational field exists relative to a reference-body K whose state of motion has been suitably chosen. K is then a Galileian reference-body as regards the domain considered, and the results of the special theory of relativity hold relative to K. Let us suppose the same domain referred to a second body of reference K′, which is rotating uniformly with respect to K. In order to fix our ideas, we shall imagine K′ to be in the form of a plane circular disc, which rotates uniformly in its own plane about its centre. An observer who is sitting eccentrically on the disc K′ is sensible of a force which acts outwards⁴ in a radial direction, and which would be interpreted as an effect of inertia⁵ (centrifugal force) by an observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a reference-body which is “at rest”; on the basis of the general principle of relativity he is justified⁶ in doing this. The force acting⁷ on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field. Nevertheless, the space-distribution of this gravitational field is of a kind that would not be possible on Newton’s theory of gravitation.1 But since the observer believes in the general theory of relativity, this does not disturb him; he is quite in the right when he believes that a general law of gravitation can be formulated—a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself.

 

The observer performs experiments on his circular disc with clocks and measuring-rods. In doing so, it is his intention to arrive at exact definitions for the signification of time- and space-data with reference to the circular disc K′, these definitions being based on his observations. What will be his experience in this enterprise?

 

To start with, he places one of two identically constructed clocks at the centre of the circular disc, and the other on the edge of the disc, so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the non-rotating Galileian reference-body K. As judged from this body, the clock at the centre of the disc has no velocity⁸, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation⁹. According to a result obtained in Section XII, it follows that the latter clock goes at a rate permanently¹⁰ slower than that of the clock at the centre of the circular disc, i.e. as observed from K. It is obvious that the same effect would be noted¹¹ by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated¹² (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case, but I do not wish to go any farther into this question.

 

Moreover, at this stage the definition of the space co-ordinates also presents insurmountable difficulties. If the observer applies his standard measuring-rod (a rod which is short as compared with the radius¹³ of the disc) tangentially¹⁴ to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than 1, since, according to Section XII, moving bodies suffer a shortening in the direction of the motion. On the other hand, the measuring-rod will not experience a shortening in length, as judged from K, if it is applied¹⁵ to the disc in the direction of the radius. If, then, the observer first measures the circumference¹⁶ of the disc with his measuring-rod and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the familiar number pi equals 3.14 period period period, but a larger number,2 whereas of course, for a disc which is at rest with respect to K, this operation would yield pi exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length 1 to the rod in all positions and in every orientation¹⁷. Hence the idea of a straight line also loses its meaning. We are therefore not in a position to define exactly the co-ordinates x comma y comma z relative to the disc by means of the method used in discussing the special theory, and as long as the co-ordinates and times of events have not been defined, we cannot assign an exact meaning to the natural laws in which these occur.

 

Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour¹⁸ in order to be able to apply the postulate¹⁹ of general relativity exactly. I shall prepare the reader for this in the following paragraphs.

 

1 The field disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards. ↑

 

2 Throughout this consideration we have to use the Galileian (non-rotating) system K as reference-body, since we may only assume the validity of the results of the special theory of relativity relative to K (relative to K′ a gravitational field prevails).

1 interpretation   

n.解释，说明，描述；艺术处理

参考例句：

• His statement admits of one interpretation only.他的话只有一种解释。
• Analysis and interpretation is a very personal thing.分析与说明是个很主观的事情。

2 slovenliness   

参考例句：

• Slovenliness is no part of religion. 邋遢并非宗教的一部分。 来自辞典例句
• Slovenliness no part of religion. “邋遢”并非宗教的一部分。 来自互联网

3 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。

4 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。

5 inertia   

adj.惰性，惯性，懒惰，迟钝

参考例句：

• We had a feeling of inertia in the afternoon.下午我们感觉很懒。
• Inertia carried the plane onto the ground.飞机靠惯性着陆。

6 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。

7 acting   

n.演戏，行为，假装；adj.代理的，临时的，演出用的

参考例句：

• Ignore her,she's just acting.别理她，她只是假装的。
• During the seventies,her acting career was in eclipse.在七十年代，她的表演生涯黯然失色。

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 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。

10 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。

11 noted   

adj.著名的，知名的

参考例句：

• The local hotel is noted for its good table.当地的那家酒店以餐食精美而著称。
• Jim is noted for arriving late for work.吉姆上班迟到出了名。

12 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。

13 radius   

n.半径，半径范围；有效航程，范围，界限

参考例句：

• He has visited every shop within a radius of two miles.周围两英里以内的店铺他都去过。
• We are measuring the radius of the circle.我们正在测量圆的半径。

14 tangentially   

adv.无关地

参考例句：

• The circumstances are delicate, involving their mother and, more tangentially, their father. 情况很微妙，事关他们的母亲也触及他们的父亲。 来自英汉文学 - 廊桥遗梦

15 applied   

adj.应用的；v.应用，适用

参考例句：

• She plans to take a course in applied linguistics.她打算学习应用语言学课程。
• This cream is best applied to the face at night.这种乳霜最好晚上擦脸用。

16 circumference   

n.圆周，周长，圆周线

参考例句：

• It's a mile round the circumference of the field.运动场周长一英里。
• The diameter and the circumference of a circle correlate.圆的直径与圆周有相互关系。

17 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。

18 detour   

n.绕行的路，迂回路；v.迂回，绕道

参考例句：

• We made a detour to avoid the heavy traffic.我们绕道走，避开繁忙的交通。
• He did not take the direct route to his home,but made a detour around the outskirts of the city.他没有直接回家，而是绕到市郊兜了个圈子。

19 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。