Lorentz–Minkowski Plane

Abstract

Special relativity (also known as the special theory of relativity) is an experimentally well-confirmed and universally accepted physical theory that explains how space and time are linked. It was originally proposed by Albert Einstein . Today, special relativity is accepted as the most accurate theory of motion at any speed when gravitational forces are negligible. Special relativity leads to a wide range of consequences, which have been experimentally confirmed, including length contraction, time dilation, relativity of simultaneity, a universal speed limit, and the mass-energy equivalence. It has replaced the long-standing notion of an absolute universal time by the notion of a relative time that is not independent of a reference frame and spatial position. Rather than treating the invariant time and the invariant spatial intervals between two events separately, we must consider an invariant spacetime interval, which enables us to understand spacetime from a geometric view of distances and isometries. In this chapter, we study the simple two-dimensional case. The goal is to formulate the geometry of special relativity in a truly equal setting along with other classical geometries. We will also see that hyperbolic geometry and relativistic geometry are intrinsically related.

“The essence of mathematics is its freedom.”

Georg Cantor (1845–1918)

“Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.”

Bertrand Russell (1872–1970)