Clicking
on this graph will halt the animation. In this non-interactive version,
the user will have to measure the angle θ manually to obtain
the values for the rest of the variables. In the interactive Graphing
Calculator version, clicking on the graph will cause the values of
the variables to display within the graph pane.
Here
we have graphed things slightly differently. We hold the line that
represents Δτ coincident with the τ-axis and let
the lines that represent the observer's time and space swing about
that axis with changing V. Therefore, the origin
represents the location of the event rather than the location of the
observer. We have also included both positive and negative X-directions.
Altering
our representation 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
We
have overlaid many shots from the animated graph above. The individual
shots 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.
(3
Megs download--This could take awhile, so you may as well ponder the
QUESTION.)
| 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? |