http://www.relativitybook.com/w/index.php?title=Einstein:Book_chapter_17_-_Minkowski%27s_Four-Dimensional_Space&feed=atom&action=historyEinstein:Book chapter 17 - Minkowski's Four-Dimensional Space - Revision history2022-08-17T01:58:29ZRevision history for this page on the wikiMediaWiki 1.26.3http://www.relativitybook.com/w/index.php?title=Einstein:Book_chapter_17_-_Minkowski%27s_Four-Dimensional_Space&diff=552&oldid=prevEric Baird: Eric Baird moved page Einstein:Book chapter 17 to Einstein:Book chapter 17 - Minkowski's Four-Dimensional Space2016-07-18T01:28:30Z<p>Eric Baird moved page <a href="/wiki/Einstein:Book_chapter_17" class="mw-redirect" title="Einstein:Book chapter 17">Einstein:Book chapter 17</a> to <a href="/wiki/Einstein:Book_chapter_17_-_Minkowski%27s_Four-Dimensional_Space" title="Einstein:Book chapter 17 - Minkowski's Four-Dimensional Space">Einstein:Book chapter 17 - Minkowski's Four-Dimensional Space</a></p>
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</td></tr></table>Eric Bairdhttp://www.relativitybook.com/w/index.php?title=Einstein:Book_chapter_17_-_Minkowski%27s_Four-Dimensional_Space&diff=522&oldid=prevEric Baird: Created page with "{{Bookblock|17}} <div id="PAGEBLOCK" > ==17: Minkowski's Four-Dimensional Space== <p class="NOINDENT">{{Boldword|T|HE}} non-mathematician is seized by a mysterious shuddering..."2016-07-18T00:54:06Z<p>Created page with "{{Bookblock|17}} <div id="PAGEBLOCK" > ==17: Minkowski's Four-Dimensional Space== <p class="NOINDENT">{{Boldword|T|HE}} non-mathematician is seized by a mysterious shuddering..."</p>
<p><b>New page</b></p><div>{{Bookblock|17}}<br />
<div id="PAGEBLOCK" ><br />
==17: Minkowski's Four-Dimensional Space==<br />
<p class="NOINDENT">{{Boldword|T|HE}} non-mathematician is seized by a mysterious shuddering when he hears of "four-<br />
dimensional" things, by a feeling not unlike that awakened by thoughts of the occult.<br />
And yet there is no more common-place statement than that the world in which we live<br />
is a four-dimensional space-time continuum. </p><br />
<br />
Space is a three-dimensional continuum. By this we mean that it is possible to describe<br />
the position of a point (at rest) by means of three numbers (co-ordinates) <math>x</math>, <math>y</math>, <math>z</math>, and that<br />
there is an indefinite number of points in the neighbourhood of this one, the position of which<br />
can be described by co-ordinates such as <math>x_1</math>, <math>y_1</math>, <math>z_1</math>, which may be as near as we choose to<br />
the respective values of the co-ordinates <math>x</math>, <math>y</math>, <math>z</math> of the first point. In virtue of the latter<br />
property we speak of a "continuum," and owing to the fact that there are three co-ordinates<br />
we speak of it as being "three-dimensional."<br />
<br />
Similarly, the world of physical phenomena which was briefly called "world" by Minkowski<br />
is naturally four-dimensional in the space-time sense. For it is composed of individual events,<br />
each of which is described by four numbers, namely, three space co-ordinates <math>x</math>, <math>y</math>, <math>z</math>, and a<br />
time co-ordinate, the time-value <math>t</math>. The "world" is in this sense also a continuum; for to every<br />
event there are as many "neighbouring" events (realised or at least thinkable) as we care to<br />
choose, the co-ordinates <math>x_1</math>, <math>y_1</math>, <math>z_1</math>, <math>t_1</math>, of which differ by an indefinitely small amount from<br />
those of the event <math>x</math>, <math>y</math>, <math>z</math>, <math>t</math> originally considered. That we have not been accustomed to<br />
regard the world in this sense as a four-dimensional continuum is due to the fact that in<br />
physics, before the advent of the theory of relativity, time played a different and more<br />
independent role, as compared with the space co-ordinates. It is for this reason that we have<br />
been in the habit of treating time as an independent continuum. As a matter of fact, according<br />
to classical mechanics, time is absolute, ''i.e.'' it is independent of the position and the condition<br />
of motion of the system of co-ordinates. We see this expressed in the last equation of the<br />
Galileian transformation (<math>t' = t</math>).<br />
<br />
The four-dimensional mode of consideration of the "world" is natural on the theory of<br />
relativity, since according to this theory time is robbed of its independence. This is shown by<br />
the fourth equation of the Lorentz transformation:<br />
<br />
<math>t'=\frac{t=\frac{v}{c^2} x}{\sqrt {1-\frac{v^2}{c^2}}}</math><br />
<br />
Moreover, according to this equation the time difference <math>∆t'</math> of two events with respect to<br />
<math>K'</math> does not in general vanish, even when the time difference <math>∆t</math> of the same events with<br />
reference to <math>K</math> vanishes. Pure "space-distance" of two events with respect to <math>K</math> results in<br />
"time-distance" of the same events with respect to <math>K'</math>.<br />
<br />
But the discovery of Minkowski, which was of importance for the formal development of<br />
the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition<br />
that the four-dimensional space-time continuum of the theory of relativity, in its most essential<br />
formal properties, shows a pronounced relationship to the three-dimensional continuum of<br />
Euclidean geometrical space.<br />
15 In order to give due prominence to this relationship, however,<br />
we must replace the usual time co-ordinate <math>t</math> by an imaginary magnitude <math>\sqrt{-1}.ct</math> proportional to<br />
it. Under these conditions, the natural laws satisfying the demands of the (special) theory of<br />
relativity assume mathematical forms, in which the time co-ordinate plays exactly the same<br />
role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to<br />
the three space co-ordinates in Euclidean geometry. It must be clear even to the non-<br />
mathematician that, as a consequence of this purely formal addition to our knowledge, the<br />
theory perforce gained clearness in no mean measure.<br />
<br />
These inadequate remarks can give the reader only a vague notion of the important idea<br />
contributed by Minkowski. Without it the general theory of relativity, of which the fundamental<br />
ideas are developed in the following pages, would perhaps have got no farther than its long<br />
clothes. Minkowski's work is doubtless difficult of access to anyone inexperienced in<br />
mathematics, but since it is not necessary to have a very exact grasp of this work in order to<br />
understand the fundamental ideas of either the special or the general theory of relativity, I<br />
shall leave it here at present, and revert to it only towards the end of Part II. <br />
<br />
{{BookNotes<br />
|# Cf. the somewhat more detailed discussion in Appendix II.}}<br />
<br />
</div></div>Eric Baird