Travel in the Positive and Negative X Direction
Here we have graphed things slightly differently. We put the line that represents Δτ coincident with the τ-axis. Therefore, the origin represents the location of the event rather than the location of the observer. The velocity of the observer in the graph above is 80% the speed of light.
This time, instead of interpreting the horizontal axis as the absolute distance that the observer has traveled from the event, we interpret it as the directed distance traveled in the plus or minus x direction.
Travel
in negative x direction
This is travel at 50% the speed of light in the direction that we have designated as negative.
Altering our representation, as we have done in these last two graphs, can illuminate different aspects of our mental model--and, potentially, new aspects of the phenomenon itself. In this case, our alteration allows us to construct the graphs below.
The Hyperbolic Nature of Relativistic Spacetime
This is a juxtaposition of many graphs like the two above. The individual graphs represent observer speeds from - 0.97c to + 0.97c, although we don't have room for all the labels.
| QUESTION: | |
| Note that the observer traveling at 0.70c will measure that she has traveled 3 light-seconds. Why will she measure 3 light-seconds for a 3-second event at a time when she is not traveling at lightspeed? (Hint: look at the green line.) | |
So far, our graphs have been unconventional in that the observer times, represented by the green lines, have been plotted at a slant rather than vertically. Now, let us straighten the lines back up.
Scrolling through this sequence of graphs may give the viewer a sense of animation. What is lost is the striking acceleration that occurs within the animated version.
| QUESTION: | |
| The mathematically knowledgeable student will note that we have shown only the top half of the hyperbola, the positive half. Why do we do this? What meaning could be attributed to the negative half, had we shown it? | |