Superphysics Superphysics
Section 17

Minkowski’s Four–dimensional Space

by Albert Einstein Icon
2 minutes  • 405 words

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.