# Accel-1D Length Contraction Plots

When I began to think about teaching special relativity for my first modern physics course, I could usually guess which observer in a given problem would see a ruler shrink relative to his own. But I wondered, couldn't we then call the phenomenon "length expansion" from the vantage point of the frame in which it is shrunk? For the answer, continue reading...

In my desire to visualize this more concretely, I decided to illustrate the process with an x-ct diagram. Even though the sound byte "length contraction" was familiar to me, and even though my intuition usually told me how to solve length contraction problems, I didn't really understand the concept until this was done. As Jonathon Swift's Gulliver might have said, "When I could hold the fruit in my hand and taste it, the inadequacy of my prior wisdom became apparent." This was one of my first object lessons in the sloppiness of our own language for dealing with relativistic spacetime effects.

In special relativity terms, "length contraction" problems involve THREE events - which we might refer to as left measure event common to both frames, and the right measure at rest and right measure moving events which take place simultaneously on the other end of the ruler from the viewpoint of one observer or another. As I looked at the diagrams I realized that in length contraction problems, one and only one inertial frame, namely the "uncontracted one", was accorded and deserved special status. This special frame, inherent to the statement of the problem, is the frame in which the ruler is traveling. All other inertial frames see the rulers in this priviledged frame as contracted. Thus when you look at any moving ruler, or equivalently when any yardstick moves, it's length is contracted in the direction of motion. Moving a ruler never causes its length to increase or "expand"!

To pose a specific problem, imagine a large ruler, one light-year in length, and an inertial frame which measures at one of its "instants" the length of this ruler while traveling at a constant coordinate velocity of one third of a lightyear per inertial year (i.e. v = c/3) to the right.

To draw the x-ct plot below, the four steps are:

• The rest frame of the ruler is chosen as the one with orthogonal axes.
• Imagine that the tick marks on the green grid denote times (vertical) of one year, and distances (horizontal) of one light-year. (This would work as well with seconds & light-seconds, respectively, if we wanted the ruler to be shorter.)
• The red lines denote the world line (path through time) of the left and right ends of the ruler. The largest red box denotes the left side measurement event (i.e. point in time and space when a frame yardstick is used to measure our ruler). The other two boxes denote the position of the other end of the ruler at the instant of the measurement, from the point of view of the frames in which the ruler is at rest (medium square), and in which it is moving (smallest square).
• The blue grid shows lines of constant maptime t (about 18 degrees up from the horizontal right) and constant position (about 18 degrees rightward from the vertical) for an inertial observer moving to the right at one third of a lightyear per mapyear (i.e. v=c/3). The measurement occurs at "time zero" for each observer. Note that the point where the world-line of the right edge of the ruler intersects the blue (rightward moving observer) time-zero line (denoted by the smallest red square) is inside of the dotted blue axis crossing -- hence the measured length of the ruler is less its resting length of one light-year. Since gamma for the rightward frame is 1/Sqrt[1-(v/c)^2] = 1/Sqrt[1-1/9] = Sqrt[9/8] = 1.06, the length measured in the blue frame for the ruler is only 1/1.06 = 0.94 lightyears.

As mentioned on our x-ct plotting page, the same information could be obtained for a plot which uses orthogonal axes for the rightward moving frame. Then of course the ruler is moving to the left. As in other examples elsewhere on these pages, the picture looks quite different even though the information it provides is the same!