chapter 17

 Minkowski’s Four-Dimensional Space

 

The non-mathematician is seized by a mysterious shuddering¹ when he hears of “four-dimensional” things, by a feeling not unlike that awakened² by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.

 

Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (co-ordinates) x comma y comma z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by co-ordinates such as x 1 comma y 1 comma z 1, which may be as near as we choose to the respective values of the co-ordinates x comma y comma z, of the first point. In virtue³ of the latter property we speak of a “continuum,” and owing to the fact that there are three co-ordinates we speak of it as being “three-dimensional.”

 

Similarly, the world of physical phenomena⁴ which was briefly⁵ called “world” by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x comma y comma z, and a time co-ordinate, the time value t. The “world” is in this sense also a continuum; for to every event there are as many “neighbouring” events (realised or at least thinkable) as we care to choose, the co-ordinates x 1 comma y 1 comma z 1 comma t 1 of which differ by an indefinitely small amount from those of the event x comma y comma z comma t originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent⁶ of the theory of relativity, time played a different and more independent role, as compared with the space coordinates⁷. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the Galileian transformation⁸ (t prime equals t)

 

The four-dimensional mode of consideration of the “world” is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation:

t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction period

Moreover, according to this equation the time difference normal upper Delta⁹ t prime of two events with respect to K′ does not in general vanish, even when the time difference normal upper Delta t prime of the same events with reference to K vanishes. Pure “space-distance” of two events with respect to K results in “time-distance” of the same events with respect to K′. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.1 In order to give due prominence¹⁰ to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude StartRoot negative 1 EndRoot dot c t proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely¹¹ formal addition to our knowledge, the theory perforce gained clearness in no mean measure.

 

These inadequate¹² remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes. Minkowski’s work is doubtless difficult of access to anyone inexperienced in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall leave it here at present, and revert¹³ to it only towards the end of Part II.

 

1 Cf. the somewhat more detailed¹⁴ discussion in Appendix II.

1 shuddering   

v.战栗( shudder的现在分词 )；发抖；（机器、车辆等）突然震动；颤动

参考例句：

• 'I am afraid of it,'she answered, shuddering. “我害怕，”她发着抖，说。 来自英汉文学 - 双城记
• She drew a deep shuddering breath. 她不由得打了个寒噤，深深吸了口气。 来自飘（部分）

2 awakened   

v.（使）醒( awaken的过去式和过去分词 )；（使）觉醒；弄醒；（使）意识到

参考例句：

• She awakened to the sound of birds singing. 她醒来听到鸟的叫声。
• The public has been awakened to the full horror of the situation. 公众完全意识到了这一状况的可怕程度。 来自《简明英汉词典》

3 virtue   

n.德行，美德；贞操；优点；功效，效力

参考例句：

• He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
• You need to decorate your mind with virtue.你应该用德行美化心灵。

4 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。

5 briefly   

adv.简单地，简短地

参考例句：

• I want to touch briefly on another aspect of the problem.我想简单地谈一下这个问题的另一方面。
• He was kidnapped and briefly detained by a terrorist group.他被一个恐怖组织绑架并短暂拘禁。

6 advent   

n.(重要事件等的)到来，来临

参考例句：

• Swallows come by groups at the advent of spring. 春天来临时燕子成群飞来。
• The advent of the Euro will redefine Europe.欧元的出现将重新定义欧洲。

7 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》

8 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。

9 delta   

n.（流的）角洲

参考例句：

• He has been to the delta of the Nile.他曾去过尼罗河三角洲。
• The Nile divides at its mouth and forms a delta.尼罗河在河口分岔，形成了一个三角洲。

10 prominence   

n.突出；显著；杰出；重要

参考例句：

• He came to prominence during the World Cup in Italy.他在意大利的世界杯赛中声名鹊起。
• This young fashion designer is rising to prominence.这位年轻的时装设计师的声望越来越高。

11 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。

12 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。

13 revert   

v.恢复，复归，回到

参考例句：

• Let us revert to the earlier part of the chapter.让我们回到本章的前面部分。
• Shall we revert to the matter we talked about yesterday?我们接着昨天谈过的问题谈，好吗？

14 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。