Fundamentals of Time and Relativity

Minkowski space

  • Einstein’s principles and definition of spacetime coordinates bestow a specific geometric structure on the spacetime point set.
  • The distance between two nearby points of Minkowski space is an invariant.

Henceforth space by itself, and time by itself,
are doomed to fade away into mere shadows,
and only a kind of union of the two will preserve an independent reality.
Hermann Minkowski, 1908

Any space’s metric prescribes how to compute the value of the distance between any two points in that space. In the case of spacetime it includes ‘distance in time’ which is another word for duration. Indeed, in special relativity, the four-dimensional abstract manifold that represents space and time consist of a 3-d spatial and a 1-d temporal part that are both Euclidean. However, as a whole, the spatial-temporal geometry of Einstein spacetime is not Euclidean, but has a particular Minkowskian metric.

In famous lectures delivered in the period 1907-8, the mathematician Hermann Minkowski argued that, by adopting Einstein's principles, one bestows a specific geometric structure on the spacetime point set { x μ ;μ=0,1,2,3 } . He stated that spacetime is the fundamental entity and that space and time are just two separate aspects of spacetime. Minkowski also understood that different inertial reference frames will divide spacetime differently into their time part and space part.

Like Euclidean space, Minkowski space is characterized by its invariance properties. From the relativity postulate, in combination with the light postulate, it follows that the equation describing a spherically propagating light wave, c 2 (Δt) 2 − (Δx) 2 − (Δy) 2 − (Δz) 2 =0 , should look the same in all inertial frames.

In fact, with homogeneity of spacetime and isotropy of space as a given, it is the defining property of Minkowski spacetime that the quadratic expression

4.1

(Δs) 2 := (Δ x 0 ) 2 − (Δ x 1 ) 2 − (Δ x 2 ) 2 − (Δ x 3 ) 2

is an invariant, independent of the choice of inertial coordinates Δ x μ , μ=0,1,2,3 . This guarantees that the speed of light is the same in every inertial frame. Note the negative signs in (4.1); if they were plus signs, the metric would be Euclidean.

Minkowski interpreted the quantity Δs squared, as the the invariant four-dimensional distance between two nearby points in spacetime.

  • The above statement about the invariance of the four-dimensional distance (4.1) between all inertial frames of reference is the core of relativity theory as presently understood.
  • A set of points endowed with the metric structure (4.1) is called a Minkowski space.