## Exercise 2: Space-time Diagrams

 Animation 1 Animation 2 Animation 3

Einstein showed that the sharing of motion between the dimensions of space and time leads to all of the remarkable physics of special relativity.  In this view, all objects in the universe are always traveling through unified space-time at the speed of light.  We can explore this unusual idea with space-time diagrams.

Restart.  First consider the above animation of a woman walking (with position given in meters and time given in seconds).  The motion of the woman in Animation 1 is rather ordinary and is documented in a data table and position versus time graph.  What is the speed of the woman?

Now look carefully at Animation 2How does this representation of the motion differ from that of Animation 1?  We haven't changed much, but this picture of the motion is almost what physicists would call a space-time diagram.  Two things are still missing: we want to treat the time on the same footing as the position and we need to take into account the universal speed limit c.

Animation 3 puts time on equal footing with position.  How is this done?  What do we mean by putting time on the same footing as position?

v / c =

Finally, we need to take into account the universal speed limit cRestart.  In Animation 3 it made some sense to multiply the time by the speed of the woman.  For a true space-time diagram, we multiply the time by the speed of light.  The unit of the y axis now becomes the amount of time it takes for light to travel one meter or 3.33 X 10-9 seconds.  Select  v / c to be zero and then press the set value and play button.  Describe the motion of the woman through space-time.  Next, try a v / c of 0.9.  What does her "trajectory" or "world-line" on the space-time diagram look like now?  Finally, try a v / c of -0.9.  What has changed in this case?  Note that as the magnitude of v / c gets bigger (approaches 1) the trajectory of the woman on the space-time diagram approaches one of the 45 degree lines of slope +/-1 that appear on the graph.  What do these lines represent?  Can objects have world-lines below these lines?  Explain.

With space-time diagrams, we see that time slows down for an object moving relative to us because some of its motion through time is diverted into motion through space.  Note that something traveling through space at light speed will have no speed leftover for the passage of time.  Hence, light does not age in our universe!

Authored by Mario Belloni, and Wolfgang Christian, and Tim Gfroerer.
© 2004 by Mario Belloni and Wolfgang Christian.