Fundamentals of Time and Relativity

Consider a Lorentz boost with velocity $v$ in the positive $x$-direction. For simplicity, we assume that the observer $O$ in the stationary frame and the observer $O^{\prime }$ in the boosted frame have set their clocks to zero at the common origin of the two coordinate systems; see the Minkowski diagram below.

In the diagram the lines of simultaneity for the moving observer $O^{\prime }$ are given by $t-vx=\text{const}$, labeled by $t^{\prime }$ , and the lines of same-locations by $x-vt=\text{const}$, labeled by $x^{\prime }$. Thus, the inertial coordinate systems set up by $O$ and $O^{\prime }$ are related by

with scale factor $\gamma (v)$. The inverse relation is

Since the transformations go in opposite directions, one deduces from the relativity principle, that each of the two matrices in (5.4),(5.5) must be the inverse of the other, and this inverse must be a Lorentz boost in the opposite direction, obtained by simply replacing the relative velocity by its negative:

The transformations (5.4),(5.5) were independently obtained by Hendrik Anton Lorentz (1899) and Joseph Larmor (1900) as the linear coordinate changes that leave the form of Maxwell’s wave equations invariant. In 1905 Henri Poincaré demonstrated that these transformations, which he named after Lorentz, form a group.

Poincaré also noticed that a Lorentz transformation is a rotation in four-dimensional spacetime about the origin. For the 2d case this is easily shown by introducing the so-called rapidity $\varphi$, or pseudo-velocity, through $\tanh \varphi (v)=v$. Then the transformation of the axes can be described as a hyperbolic rotation: