CHAPTER XI 
FOUR-DIMENSIONAL TUBES OR CALAMOIDS AS 
QUANTA 
Absolute, true, and mathematical time, of itself, and from its 
own nature, flows equably without regard to anything external, and 
by another name is called duration. 
Absolute space, in its own nature, without regard to anything 
external, remains always similar and immovable. 
Sir Isaac Newton 
Henceforth space by itself and time by itself shall sink to mere 
shadows, and only a union of the two shall preserve reality. 
Minkowski 
1. The Four-dimensional World of Minkowski 
I N ordinary geometry the position of a point in space is fixed 
by means of three co-ordinates. But in relativity, time as 
well as position must be taken into consideration. An event, 
like the flashing of a lamp or the emission of an electron from 
an atom, which takes place at a certain point of space and at 
a certain instant of time may be called a “ point-event.” One 
measurement is required to determine the time of the event, 
and three measurements to determine the place. These are the 
four co-ordinates of the point-event. In other words, in the 
“ space-time world ” of Minkowski, four co-ordinates are required 
to fix a point-event. Thus the point-event is an element of the 
four-dimensional world as a point is an element of three-dimen 
sional space. 
The “ interval ” between two point-events involves both time 
and space, and when its measurement is defined in a certain 
way, all observers will agree as to the magnitude of an interval, 
or in mathematical language the expression for the interval is 
" covariant.” In the world of Minkowski, as de Sitter has said, 
“ a geometry of four dimensions is used, not a mere combina 
tion of a three-dimensional space and a one-dimensional time, 
but a continuum of truly fourfold order. This time-space is not 
Euclidean, since the time-component and the three space-com 
ponents are not on the same footing, but its fundamental formula 
has a great resemblance to that of Euclidean geometry.” 
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