These transformations define the Rindler observer as an observer that is "at rest" in Rindler coordinates, i.e., maintaining constant x, y, z and only varying t as time passes. The coordinates are valid in the region 0 < X < \infty,\; -X < T < X, which is often called the Rindler wedge, if \alpha represents the proper acceleration (along the hyperbola x=1 / \alpha) of the Rindler observer whose proper time is defined to be equal to Rindler coordinate time. To maintain this world line, the observer must accelerate with a constant proper acceleration, with Rindler observers closer to x=0 (the Rindler horizon) having greater proper acceleration. All the Rindler observers are instantaneously at rest at time T=0 in the inertial frame, and at this time a Rindler observer with proper acceleration \alpha_{i} will be at position X=1/\alpha_{i} (really X=c^{2}/\alpha_{i}, but we assume units where c=1), which is also that observer's constant distance from the Rindler horizon in Rindler coordinates. If all Rindler observers set their clocks to zero at T=0, then when defining a Rindler coordinate system we have a choice of which Rindler observer's proper time will be equal to the coordinate time t in Rindler coordinates, and this observer's proper acceleration defines the value of \alpha above (for other Rindler observers at different distances from the Rindler horizon, the coordinate time will equal some constant multiple of their own proper time). It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration \alpha=1, so that \alpha can be eliminated from the equations.
