Four-dimensional spacetime

Geometry of Lorentz transformations

As part of his analysis Poincaré described invariants under the rotation, like the quantity $${\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}$$ where the ${\displaystyle x,y,z,t}$ are differences between points in the four dimensional space, as distances. (The sum of the first three differences would be the square of the Euclidean distance in 3-space; such distances are invariant under rotation in 3-space). The rotation and concept of invariants as distances was elaborated by Minkowski, in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". The paper introduced tensors to the mathematical representation of the physics and reformulated Maxwell equations as a symmetrical set of equations in the four variables (x, y, z, ict) combined with redefined vector variables for electromagnetic quantities. He was able to show their invariance under Lorentz transformation. From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum.