Minkowski spacetime in the context of "Lorentz invariant"

Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, ETH Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, or Russian. He created and developed the geometry of numbers and elements of convex geometry, and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.

Minkowski is perhaps best known for his foundational work describing space and time as a four-dimensional space, now known as "Minkowski spacetime", which facilitated geometric interpretations of Albert Einstein's special theory of relativity (1905).

Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration (special relativity) and Proper reference frame (flat spacetime).

In this article, the speed of light is defined by c = 1, the inertial coordinates are (X, Y, Z, T), and the hyperbolic coordinates are (x, y, z, t). These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time T = 0 at position X = 1/α (with α as the constant proper acceleration measured by a comoving accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located at time T = 0 at position X = 0, then the hyperbolic coordinates are sometimes called Møller coordinates or Kottler–Møller coordinates with the corresponding Kottler–Møller metric. An alternative chart often related to observers in hyperbolic motion is obtained using Radar coordinates which are sometimes called Lass coordinates. Both the Kottler–Møller coordinates as well as Lass coordinates are denoted as Rindler coordinates as well.