3.2 Comparing Space for O and O' (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.
3.2 Comparing Space for O and O' (Space-Time Diagrams)
So, we have found a correlation between the lengths which represent
certain times along the t axis for O and the lengths which represent certain
times along the t' axis for O'. We did this by using (1) the idea of time
dilation which was found earlier to be caused by the fact that light always
travels at c for all inertial observers and (2) the lines of simultaneity
for different observers which we learned how to draw by also using the fact
that light always travels at c for all inertial observers. Similarly, we can
find a correlation between lengths which represent certain distances along
the x axis for O and the lengths which represent certain distances along the
x' axis for O'. As an example, I have drawn a comparison of distances in
Diagram 3-3 which will be explained below.
Diagram 3-3
t t'
^ /
| /|< line of constant position
| / | for O at x = 1
| / |
| / | / x'
| / | / '
t = 1 + / | / ' * = point where
| / | / ' x' = 0.8
| / | #
| / * /< line of constant position
| / ' | / for O' at x' = 1.25
|/ ' |/
---------o-----------+-------> x
| |
| x = 1
|
(Note: The line for x' is a rough approximation)
Perhaps the best way to explain this diagram is as follows: Consider a
rod being held by the O observer such that one end of the rod follows the t
axis (and is thus always next to the O observer) while the other end follows
the vertical line drawn at x = 1. The rod then is obviously stationary in
the O observer's frame of reference. Second consider a rod being held by the
O' observer such that one end follows the t' axis and the other end follows
the line of constant position for O' which I have drawn.
Well, in the O observer's frame, his rod is obviously 1 light-second
long. But notice that in his frame the ends of the O' observer's rod are
next to the ends of the O observer's rod at t = 0. Thus, in the O observer's
frame, the O' observer's rod is also 1 light-second long. But length
contraction tells us that in the O observer's frame, the O' observer's rod
is shorter than its "rest length" by a factor of 1.25. Thus, in the O'
observer's frame (the frame in which his rod is at rest), his rod must
actually be 1.25 light-seconds long. That is how I know that the line of
constant position for O' I drew was for x'= 1.25.
Now, look at the distance along x' from the origin (o) to the point
marked "#". That distance represents the length of the O' observer's rod
from his own frame of reference (i.e. 1.25 light-seconds). Also, the
distance along x' from the origin to the point marked "*" represents the
length of the O observer's rod in the O' observer's frame of reference. That
distance must be 0.8 because in the O' frame, it is O and his rod which are
moving, and thus his rod seems length contracted by a factor of 1.25 from
its length in the frame of reference in which it is at rest (the O frame).
That number could have also been found by using the fact that the distance
from o to "#" was 1.25 light-seconds.
Finally, we again note the power of the space-time diagram. At one
glance of Diagram 3-3 we are able to see that in the O' observer's frame,
his rod is 1.25 light-seconds long, while in the O observer's frame it is
only 1 light-second long. At the same time we are able to see that in the O
observer's frame, his rod is 1 light-second long, while in the O' observer's
frame, it is only 0.8 light-seconds long. Thus, each observer believes that
the other observer's rod is shorter than it is in the frame of reference in
which the rod is at rest. They each believe that the other is experiencing
length contraction, and with a space-time diagram, we are able to see how
that is so.
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