3.3 Once Again: The Light Cone (Space-Time Diagrams)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
3.3 Once Again: The Light Cone (Space-Time Diagrams)
Here I want to demonstrate how a light cone appears in the two
coordinate systems. In Section 2.8 I mentioned that the light cone is drawn
exactly the same for the two observers. Now that we understand how to draw
the two coordinate systems completely (i.e. we can now draw "tick" marks on
the x' and the t' axes as well as the x and t axes because of the discussion
above) we can make a diagram which clearly shows this. To start, in Diagram
3-4 I have shown the results of our discussion above in that I have
indicated approximately where the tick marks (+) would appear on the x' and
t' axes.
Diagram 3-4
t t'
| /
| +t'=2
| /
t=2+ /
| /
| / x'
| / '
| / '
| + t'=1 +'
t=1+ / ' x'=2
| / '
| / + '
| / ' x'=1
| / '
|/ '
--+-----------o-----------+-----------+---> x
' /| x=1 x=2
' / |
(Note: Again, x' and t' are rough approximation for v = 0.6 c)
Next, in Diagram 3-5 I have drawn the x and t axes along with lines of
simultaneity and lines of constant position (for O) at each tick mark. In
addition, the upper half of a light cone centered at the origin is shown
using # symbols. As you see (and as we would expect), it passes through the
points x = 1 light-second, t = 1 second; x = 2 light-seconds, t = 2 seconds;
etc.
Diagram 3-5
t
|
| # = Light
| | | | #
-------------+-----------------------#-
| | | # |
| | | # |
| | | # |
| | | # |
# | | | # |
-#-----------+-----------#------------
| # | # | |
| # | # | |
| # | # | |
| # | # | |
| # | # | |
--+-----------o-----------+-----------+----> x
| | | |
|
Continuing with the diagrams, Diagram 3-6 shows the x' and t' axes
along with lines of simultaneity and lines of constant position (for O') at
each tick mark along those axes. Again, the upper half of a light cone
centered at the origin is also shown. As you see (and as we would again
expect), it passes through the points x' = 1 light-second, t' = 1 second;
etc. Note that the point x' = 1, t' = 1 is marked with an "@" symbol and the
tick marks on the x' and t' axes are marked with "+" marks to help make it
clear how the coordinate system works. Also notice that the light cone
itself is drawn exactly the same as it is in Diagram 3-5.
Diagram 3-6
t'
/
/ ' / / ' / /
/ ' / + ' / # / '
'/ / '/ / # '/
' / / ' / / # ' /
/ / ' / /# ' / x'
/ ' / @ / '
/ ' / / ' #/ / '
' / / ' # / / '
# ' / +' # / +' /
# / ' / # / ' / /
# / ' / # / ' / /
# / ' / # + ' / / '
'/# / # '/ / '/
' / # / # ' / / ' /
/ # /# ' / / ' /
/ o / ' /
/ ' / / ' / / '
/ ' / / ' / / '
Finally, I want to superimpose Diagram 3-5 and Diagram 3-6 to some
extent onto Diagram 3-7. It would be quite cluttered to put all the lines
included in the two diagrams, but I want to include the lines which make up
x = 1, t = 1, x'= 1, and t'= 1. These lines are thus drawn on Diagram 3-7,
but they terminate where they meet the light cone which is also shown. You
should begin to see the relationship between the two different frames of
reference and the fact that the light cone itself is exactly the same in
both coordinate systems. This is a direct result from the fact that every
step we took in producing these diagrams used the assumption that the speed
of light is the same in all inertial frames of reference.
Diagram 3-7
t t'
| /
| /
| + #
| / #
+ / #
| / # x'
| / # '
| / ' #/ '
| / ' # / +'
| +' # / '
+-----/-----# / '
| / # | / '
| / # | + '
| / # '
| / # ' |
|/# ' |
--+-----------o-----------+-----------+-----------+--->x
' /|
' / |
Though this concludes our discussion of space-time diagrams, we will
continue to see them in the next section, because they can be vital tools
for understanding paradoxes in special relativity.
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