Minkowski Space

Minkowski Space, defined simply and quickly, is a name for the fourth dimensional space in which our universe resides. This concept was used by Albert Einstein in his paper Relativity: The Special and General Theory, and was central to his core argument. Einstein said, "Without it (Minkowski's work) the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes."

We live in a three dimensional universe, in which a point can represented by coordinates (x, y, z), which is embedded in a four dimensional universe which adds a fourth coordinate, t, representing time. Now, a point can be represented as (t, x, y, z). Perhaps an hour later, whatever was in that point has moved, and something else occupies that space. However, the point is not the same; time has passed, and we now call the point (t', x, y, z). Without this fourth dimension, our lives would be like taking every frame on a movie reel and stacking them on top of each other, a huge jumble of every moment occurring at the same time, with no sequential movement. This 4-D representation of the universe is often called space-time. Einstein used this idea, in the form of the fourth equation of the Lorentz transformation, to prove that time was not independent of space.

**snip all the stuff about curved space... turns out Minkowski Space isn't curved. I've moved the description of space curved by gravity to the curved space node, minus the globe thing, since there's a great description of that sort of thing there already**

And many thanks to cjeris and Miles_Dirac for pointing out what I didn't know, and then enlightening me (and all of us).

Minkowski space is not curved.

Contrary to a common misunderstanding, Minkowski space refers to a four dimensional flat space, the one consistent with the special theory of relativity.

Space, that is length, breadth, and depth are Euclidean dimensions, which means when we rotate something from one 3D direction to another we use sines and cosines in our formulae. When we add the dimension of time, we add it into our "metric" with a factor of i (the square root of -1). This means that in addition to rotations in 3 dimensions, we can talk about 4D rotations, which encompasses the usual rotations plus "Lorentz boosts". The latter is what leads, through special relativity, to phenomena such as length contraction and time dilation. When a rotation is made in all 4 dimensions, the Minkowski metric means that some of the sines and cosines become hyperbolic: sinh and cosh.

To summarize in mathematical terms: when we measure the norm (length) of a position vector in 3D: it is given by

sqrt( x² + y² + z² )

If our 4 dimensions were Euclidean, a postion vector's norm would be

sqrt( x² + y² + z² + (ct)² )

However, special relativity says that our 4 dimensions are Minkowskian, so a position vector's norm is

sqrt( x² + y² + z² - (ct)² )

That minus sign makes a big difference! But this is still a flat space. I.e. the metric (and so the formula for the norm) is the same, independent of where you are. In curved space, there are position-depedent coefficients in the formula for the norm.