A coordinate is a line that holds a particular value constant in the set of values used provide a location on a graph. As a more simple example, we might have an (x,y) graph and we can identify a line where all the points on that line have x=3. It might feel a little bit weird because the lines where x equals a constant are vertical and the lines where y equals a constant are horizontal.
That line doesn’t have to be a straight line.
If we have the point (4,3) we can say it has an x coordinate of 4 (a vertical line) and a y coordinate of 3 (a horizontal line) and the point we are describing is the intersection of the those two lines.
There is still more that is “strange” (hopefully “fun”) about the Rindler picture. In this story we are working in only one dimension, the x-dimension. What you would normally call the y-axis is actually the time access.
Time flows in the “upward” or “north” direction. If you chose to not move in space then your path is “north” and if you like you can ride one of those blue lines (in which case you are on a coordinate that is an integer).
If you look at the red curvy-line you see that as we go upwards it moves westward, indicating a change of location in space in the positive x-direction.
Here’s where it gets tricky/fun. The Red line could be a Rindler spatial coordinate.
Now, we just said that if you are riding a coordinate in the north direction you were not changing position.
Everywhere on that red curvy-line is the same location in space, yet you can see the red line is moving in the x-direction positively as time moves forward.
That red line could be the flight path of a spaceship. While on board the spaceship you are moving in the positive x-direction relative to the ground, but relative to the ship you are not moving at all, which is why you say on the same “red” number.
Rob Sterling 