chapter 26

 The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum

 

We are now in a position to formulate¹ more exactly the idea of Minkowski, which was only vaguely² indicated in Section XVII. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these “Galileian co-ordinate systems.” For these systems, the four co-ordinates x comma y comma z comma t, which determine an event or—in other words, a point of the four-dimensional continuum—are defined physically³ in a simple manner, as set forth⁴ in detail in the first part of this book. For the transition from one Galileian system to another, which is moving uniformly with reference to the first, the equations of the Lorentz transformation⁵ are valid⁶. These last form the basis for the derivation of deductions⁷ from the special theory of relativity, and in themselves they are nothing more than the expression of the universal validity of the law of transmission of light for all Galileian systems of reference.

 

Minkowski found that the Lorentz transformations⁸ satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body K by the space co-ordinate differences d x comma d y comma d z and the time-difference dt. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are d x Superscript prime Baseline comma d y Superscript prime Baseline comma d z Superscript prime Baseline comma d t prime. Then these magnitudes always fulfil the condition1.

d x squared plus d y squared plus d z squared minus c squared d t squared equals d x Superscript prime 2 Baseline plus d y Superscript prime 2 Baseline plus d z Superscript prime 2 Baseline minus c squared d t Superscript prime 2 Baseline period

The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude

d s squared equals d x squared plus d y squared plus d z squared minus c squared d t squared comma

which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x comma y comma z, StartRoot negative 1 EndRoot c t, by x 1 comma x 2 comma x 3 comma x 4, we also obtain the result that

d s squared equals d x 1 squared plus d x 2 squared plus d x 3 squared plus d x 4 squared

is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or four-dimensional points.

 

Thus, if we choose as time-variable the imaginary variable StartRoot negative 1 EndRoot c t instead of the real quantity t, we can regard the space-time continium—accordance with the special theory of relativity—as a “Euclidean” four-dimensional continuum, a result which follows from the considerations of the preceding section.

 

1 Cf. Appendixes I and II. The relations which are derived⁹ there for the co-ordinates themselves are valid also for co-ordinate differences, and thus also for co-ordinate differentials (indefinitely small differences).

1 formulate   

v.用公式表示；规划；设计；系统地阐述

参考例句：

• He took care to formulate his reply very clearly.他字斟句酌，清楚地做了回答。
• I was impressed by the way he could formulate his ideas.他陈述观点的方式让我印象深刻。

2 vaguely   

adv.含糊地，暖昧地

参考例句：

• He had talked vaguely of going to work abroad.他含糊其词地说了到国外工作的事。
• He looked vaguely before him with unseeing eyes.他迷迷糊糊的望着前面，对一切都视而不见。

3 physically   

adj.物质上，体格上，身体上，按自然规律

参考例句：

• He was out of sorts physically,as well as disordered mentally.他浑身不舒服，心绪也很乱。
• Every time I think about it I feel physically sick.一想起那件事我就感到极恶心。

4 forth   

adv.向前；向外，往外

参考例句：

• The wind moved the trees gently back and forth.风吹得树轻轻地来回摇晃。
• He gave forth a series of works in rapid succession.他很快连续发表了一系列的作品。

5 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。

6 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？

7 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。

8 transformations   

n.变化( transformation的名词复数 )；转换；转换；变换

参考例句：

• Energy transformations go on constantly, all about us. 在我们周围，能量始终在不停地转换着。 来自辞典例句
• On the average, such transformations balance out. 平均起来，这种转化可以互相抵消。 来自辞典例句

9 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》