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the 4th dimension [Q&A]

Vinay

Gautham of Poornaprajna College asks, what is the definition of time in 4-dimensional space?

Typically, when people speak of the 4th dimension, they are talking of a spatial dimension. Any point in the 3-D space we are used to can be represented by 3 coordinates, say (x,y,z), with unit vectors all perpendicular to each other. The distance between 2 points is the familiar Euclidean distance, dr = √ (dx2 + dy2 + dz2).

In 4-D space, one would require 4 coordinates, say (x,y,z,w), again with all the unit vectors perpendicular to each other. While this is mathematically trivial to write down (the distance between two points, dr = √ (dx2 + dy2 + dz2 + dw2)), picturing such a thing is beyond the capability of our brains. In this context, time has exactly the same meaning as it does in 3-D space, and its passage is measured by all the usual methods.

When Einstein worked out the Special Theory of Relativity, the fact that the speed of light is a constant for everybody everywhere provided a means by which to tie spatial distance to temporal interval. Thus, events can be located in space-time with a 4-tuple of independent coordinates, (x,y,z,t). This has led to confusion in the popular science literature about identifying time as the “4th dimension”. You can say it is, and nobody will disagree, but it is important to keep in mind that it is not a spatial dimension.

Furthermore, the distance between two points in space-time is not the usual Euclidean measure. It is actually dr = √ (dx2 + dy2 + dz2 c2 dt2), where c is the speed of light. Notice the negative sign. Two events that are separated in 3-D space by a distance that can be covered at the speed of light have a distance measure of zero. Distance between events can even be complex. This in turn has led to more confusion about the nature of time, because the distance metric can be written in the form of a Euclidean distance if we write the time coordinate as ict, where i=sqrt(-1). Because i is called an imaginary number, it has led to some people thinking that time must be imaginary. It is not, any more than imaginary numbers are really imaginary; it is just a name for a mathematical concept and should not be over-interpreted in terms of its associated English meanings.