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 < ∞ ∞ , − − X < T < X , which is often called the Rindler wedge, if α α represents the proper acceleration (along the hyperbola x = 1 / α α ) 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 α α i will be at position X = 1 / α α i (really X = c 2 / α α 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 α α 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 α α = 1 , so that α α can be eliminated from the equations.