Lorentz Transformations

Space-Time and Minkowski Space

In classical Newtonian mechanics space the three-dimensional "world" is a place where all the events occur and time is absolute and the same for everybody. Space and time are separate and independent of each other and they cannot be mixed in any way.

In special relativity, however, space and time are just different coordinates of the so-called space-time, i.e., space and time merge together into a four-dimensional "world". The space coordinates and the time coordinate are mixed up together by the so-called Lorentz transformations. Every event corresponds to a point in space-time. In the words of Minkowski: "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."

In classical Newtonian mechanics, we measure distances between two points by the formula

² = (x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²,

where (x₁, y₁, z₁) and (x₂, y₂, z₂) are the coordinates of the two points. This is possible, since we have an absolute concept of simultaneity. In special relativity, however, the interval between two events is defined by

s² = c² (t₁ - t₂)² - (x₁ - x₂)² - (y₁ - y₂)² - (z₁ - z₂)²,

where (t₁, x₁, y₁, z₁) and (t₂, x₂, y₂, z₂) are the coordinates of the two events. Note that s² can in fact be negative! If we consider two events separated infinitesimally, (t, x, y, z) and (t + dt, x + dx, y + dy, z + dz), then the interval becomes

ds² = c² dt² - dx² - dy² - dz².

This "distance" formula is called the Minkowski metric and the corresponding four-dimensional "world" is called the Minkowski space-time or just the Minkowski space.