Accel-1D Time Dilation 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 clock running slow relative to his own. But I wondered, couldn't we then call the phenomenon "time compression" from the vantage point of the frame in which it runs fast?

In my desire to visualize this more concretely, I decided to illustrate the process with an x-ct diagram. Even though the sound byte "time dilation" was familiar to me, and even though my intuition usually told me how to solve time dilation 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, "time dilation" problems involve two events - which we might refer to as start and stop events that take place on one particular clock. As I looked at the diagrams I realized that in time dilation problems, one and only one inertial frame, namely the "undilated one", was accorded and deserved special status. This special frame, inherent to the statement of the problem, is the frame in which the clock is traveling. All other inertial frames see the clocks in this privileged frame as traveling slower. Thus when you look at any moving clock, or equivalently when any clock moves, it's time is "dilated" and thus slowed down. Moving a clock never causes its time to "contract" or move faster!

To pose a specific problem, imagine an inertial clock which times an interval of one year, and an inertial frame that observes the start and stop of this timing adventure from frame traveling at constant coordinate-velocity of one third of a lightyear per map 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 clock 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 to have the experiment over more quickly.)
• The red line denotes the world line (path through time) of the clock. The red boxes denote the start and stop events under consideration, spaced one year apart in the frame of our clock.
• The blue grid shows lines of constant inertial time b (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 map year (i.e. v=c/3). The start event occurs at "time zero" for each observer. The red dash illustrates the rightward moving observer's measurement of the time of the stop event. Since a year on the rightward frame's clock is marked by the dotted blue axis crossing, the stop event happens a bit more than a year after the start event in that frame. Since gamma for the rightward frame is 1/Sqrt[1-(v/c)^2] = 1/Sqrt[1-1/9] = Sqrt[9/8] = 1.06, the time interval measured in the blue frame between events is 1.06 years.

As mentioned on our x-ct plotting page, the same information could be obtained for a plot that uses orthogonal axes for the rightward moving frame. Then of course the clock 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!

• Check out also this table-of-contents of map-based anyspeed motion resources.
• Send ideas/ comments/ questions/ complaints, and register for updates to these pages with an email subject-field containing "accel1", to staff@newton.umsl.edu. This page might contain original material. Hence if you echo, in print or on the web, a citation would be cool. `{Thanks. :) /pf}`