chapter 28

 Exact Formulation of the General Principle of Relativity

 

We are now in a position to replace the pro¹. visional formulation of the general principle of relativity given in Section XVIII by an exact formulation. The form there used, “All bodies of reference upper K comma upper K prime comma etc., are equivalent for the description of natural phenomena² (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid³ reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially⁴ equivalent for the formulation of the general laws of nature.”

 

We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible⁵ than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation⁶, we replace the space-time variables x comma y comma z comma t, of a (Galileian) reference-body K by the space-time variables x prime comma y prime comma z prime comma t prime, of a new reference-body K′. According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables x 1 comma x 2 comma x 3 comma x 4, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

 

If we desire to adhere to our “old-time” three-dimensional view of things, then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains⁷, i.e. to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively⁸ to it.

 

Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies (cf. Sections XX and XXIII).

 

In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious⁹ rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility¹⁰ as in the special theory of relativity.

 

For this reason non-rigid reference-bodies are used, which are as a whole not only moving in any way whatsoever¹¹, but which also suffer alterations¹² in form ad lib. during their motion. Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed¹³ at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously¹⁴ on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusc”, is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusc” a certain comprehensibility as compared with the Gauss co-ordinate system is the (really unjustified) formal retention¹⁵ of the separate existence of the space co-ordinates as opposed to the time co-ordinate. Every point on the mollusc is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusc is considered as reference-body. The general principle of relativity requires that all these molluscs can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusc.

 

The great power possessed¹⁶ by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above.

1 pro   

n.赞成，赞成的意见，赞成者

参考例句：

• The two debating teams argued the question pro and con.辩论的两组从赞成与反对两方面辩这一问题。
• Are you pro or con nuclear disarmament?你是赞成还是反对核裁军？

2 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。

3 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。

4 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。

5 intelligible   

adj.可理解的，明白易懂的，清楚的

参考例句：

• This report would be intelligible only to an expert in computing.只有计算机运算专家才能看懂这份报告。
• His argument was barely intelligible.他的论点不易理解。

6 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。

7 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

8 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。

9 fictitious   

adj.虚构的，假设的；空头的

参考例句：

• She invented a fictitious boyfriend to put him off.她虚构出一个男朋友来拒绝他。
• The story my mother told me when I was young is fictitious.小时候妈妈对我讲的那个故事是虚构的。

10 plausibility   

n. 似有道理, 能言善辩

参考例句：

• We can add further plausibility to the above argument. 我们可以在上述论据之外，再进一步增添一个合理的论据。
• Let us consider the charges she faces, and the legal plausibility of those charges. 让我们考虑一下她面临的指控以及这些指控在法律上的可信性。

11 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。

12 alterations   

n.改动( alteration的名词复数 )；更改；变化；改变

参考例句：

• Any alterations should be written in neatly to the left side. 改动部分应书写清晰，插在正文的左侧。 来自《简明英汉词典》
• Gene mutations are alterations in the DNA code. 基因突变是指DNA 密码的改变。 来自《简明英汉词典》

13 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。

14 simultaneously   

adv.同时发生地，同时进行地

参考例句：

• The radar beam can track a number of targets almost simultaneously.雷达波几乎可以同时追着多个目标。
• The Windows allow a computer user to execute multiple programs simultaneously.Windows允许计算机用户同时运行多个程序。

15 retention   

n.保留，保持，保持力，记忆力

参考例句：

• They advocate the retention of our nuclear power plants.他们主张保留我们的核电厂。
• His retention of energy at this hour is really surprising.人们惊叹他在这个时候还能保持如此旺盛的精力。

16 possessed   

adj.疯狂的；拥有的，占有的

参考例句：

• He flew out of the room like a man possessed.他像着了魔似地猛然冲出房门。
• He behaved like someone possessed.他行为举止像是魔怔了。