Section 17

# Minkowski’s Four–dimensional Space

March 31, 2022

Space is a 3D continuum wherein:

• an object exists in 3 co-ordinates `x, y, z` as Point 1
• there are infinite points around Point 1
• These are designated as `x1, y1, z1`, `x2, y2, z2`, `x3, y3, z3`, etc.
• These points create a “continuum”

Our universe is a 4D space-time continuum, as described by Minkowski. It has the additional coordinate of time-value `t`.

Every event has infinite “neighbouring” events (realised or imaginary). The leads to co-ordinates `x1, y1, z1, t1`.

Before my theory of Relativity, time played a different and more independent rôle to the space co-ordinates. This is why we usually treat time as an independent continuum*.

*Superphysics Note= The time-coordinate was not important in the past because electromagnetism was not so important in the 18th century. Even in daily life today, we rarely encounter motions that are faster than a jet or racecar as to bother about time-coordinates. Einstein is like a lobbyist for light that promotes the 4D spacetime over the 3D one

Time is absolute in classical mechanics – it is independent of the position and the condition of motion of the system of coordinates. This is expressed in the last equation of the Galileian transformation as `t' = t`.

The 4D spacetime is natural to the theory of relativity which robs time of its independence. This is shown by the 4th equation of the Lorentz transformation=

``````t' = (t - (v/c2) x) / √ 1-(v2/c2)
``````

The time difference `∆ t'` of two events with respect to `K'` does not vanish in general, even when the time difference `∆ t` of the same events with reference to non-moving `K` vanishes. Pure “space-distance” of two events with respect to non-moving `K` results in “time-distance” of the same events with respect to the moving `K'`.

Minkowski’s discovery was important for the development of my theory of relativity.

The 4D space-time continuum of the theory of relativity has a pronounced relationship with the 3D continuum of Euclidean geometrical space. To emphasize this relationship, we must replace the usual time coordinate `t` by an imaginary magnitude `√ −1ct` proportional to it. This allows the natural laws that satisfy the demands of Special Relativity to assume mathematical forms wherein the time coordinate plays exactly the same rôle as the `x, y, z` coordinates.

These 4 coordinates correspond exactly to the 3 space coordinates in Euclidean geometry.

Without Minkowski’s 4D spacetime, General Relativity could not be developed.