chapter 27

 The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum

 

In the first part of this book we were able to make use of space-time co-ordinates which allowed of a simple and direct physical interpretation¹, and which, according to Section XXVI, can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity² of tight. But according to Section XXI the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. In connection with a specific illustration in Section XXIII, we found that the presence of a gravitational field invalidates the definition of the coordinates³ and the time, which led us to our objective in the special theory of relativity.

 

In view of the results of these considerations we are led to the conviction that, according to the general principle of relativity, the space-time continuum cannot be regarded as a Euclidean one, but that here we have the general case, corresponding to the marble slab⁴ with local variations of temperature, and with which we made acquaintance as an example of a two-dimensional continuum. Just as it was there impossible to construct a Cartesian co-ordinate system from equal rods, so here it is impossible to build up a system (reference-body) from rigid⁵ bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly⁶ with respect to one another, shall indicate position and time directly. Such was the essence of the difficulty with which we were confronted in Section XXIII.

 

But the considerations of Sections XXV and XXVI show us the way to surmount⁷ this difficulty. We refer the four-dimensional space-time continuum in an arbitrary manner to Gauss co-ordinates. We assign to every point of the continuum (event) four numbers, x 1 comma x 2 comma x 3 comma x 4 (co-ordinates), which have not the least direct physical significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement does not even need to be of such a kind that we must regard x 1 comma x 2 comma x 3, as “space” co-ordinates and x 4, as a “time” co-ordinate.

 

The reader may think that such a description of the world would be quite inadequate⁸. What does it mean to assign to an event the particular co-ordinates x 1 comma x 2 comma x 3 comma x 4, if in themselves these co-ordinates have no significance? More careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary⁹ existence without duration, then it would to described in space-time by a single system of values x 1 comma x 2 comma x 3 comma x 4. Thus its permanent existence must be characterised by an infinitely¹⁰ large number of such systems of values, the co-ordinate values of which are so close together as to give continuity; corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional continuum. In the same way, any such lines in our continuum correspond to many points in motion. The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of co-ordinate values, x 1 comma x 2 comma x 3 comma x 4, in common. After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements.

 

When we were describing the motion of a material point relative to a body of reference, we stated nothing more than the encounters of this point with particular points of the reference-body. We can also determine the corresponding values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials. It is just the same in the case of space-measurements by means of measuring-rods, as a little consideration will show.

 

The following statements hold generally: Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x 1 comma x 2 comma x 3 comma x 4. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.

 

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 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。

3 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》

4 slab   

n.平板，厚的切片；v.切成厚板，以平板盖上

参考例句：

• This heavy slab of oak now stood between the bomb and Hitler.这时笨重的橡木厚板就横在炸弹和希特勒之间了。
• The monument consists of two vertical pillars supporting a horizontal slab.这座纪念碑由两根垂直的柱体构成，它们共同支撑着一块平板。

5 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。

6 rigidly   

adv.刻板地，僵化地

参考例句：

• Life today is rigidly compartmentalized into work and leisure. 当今的生活被严格划分为工作和休闲两部分。
• The curriculum is rigidly prescribed from an early age. 自儿童时起即已开始有严格的课程设置。

7 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。

8 inadequate   

adj.(for,to)不充足的，不适当的

参考例句：

• The supply is inadequate to meet the demand.供不应求。
• She was inadequate to the demands that were made on her.她还无力满足对她提出的各项要求。

9 momentary   

adj.片刻的，瞬息的；短暂的

参考例句：

• We are in momentary expectation of the arrival of you.我们无时无刻不在盼望你的到来。
• I caught a momentary glimpse of them.我瞥了他们一眼。

10 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。