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This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by
others.
2.5 Constructing One for a "Moving" Observer (Space-Time Diagrams)
Now comes an important addition to our discussion of space-time
diagrams. The coordinate system we have drawn will work fine for any
observer who is not moving with respect to the O observer. Now we want to
construct a coordinate system for an observer who IS traveling with respect
to the O observer. The trajectories of two such observers have been drawn in
Diagram 2-6 and Diagram 2-7. Notice that in our discussion we will usually
consider moving observers who pass by the O observer at the time t = 0 and
at the position x = 0. Thus, the origin will mark the event "the two
observers pass by one another".
Now, the traveler in Diagram 2-6 is moving slower than the one in
Diagram 2-7. You can see this because in a given amount of time (distance
along the t axis), the Diagram 2-7 traveler has moved further away from the
time axis than the Diagram 2-6 traveler. So the faster a traveler moves, the
more slanted this line becomes.
Diagram 2-6 Diagram 2-7
t t
| / | /
+ + /
| / | /
+ + /
|` |/
-+---+---o---+---+--- x -+---+---o---+---+- x
,| /|
+ / +
/ | / |
+ / +
/ | / |
What does this line actually represent? Well, remember that the line
marks the position of our observer at different times on our diagram. But,
also, consider an object sitting right next to our moving observer. If a few
seconds later the object is still sitting right next to him (practically on
that line), then, in his point of view, the object has not moved. So, the
line is a line of constant position for the moving observer. Nothing on that
line is moving with respect to him. But that means that this line represents
the same thing for the moving observer as the t axis represented for the O
observer; and in fact, this line becomes the moving observer's new time
axis. We will mark this new time axis as t' (t-prime). All lines parallel to
this slanted line will also be lines of constant position for our moving
observer.
Now, just as we did for the O observer, we want to construct lines of
constant time for our traveling observer. To do this, we will use the same
method that we did for the O observer. The moving observer will send out a
light beam at some time t'= -T, and the beam will bounce off some mirror so
that it returns to him at time t'= +T. Now remember, light travels at the
same speed in any direction for ALL observers, so our traveling observer
must conclude that the light beam took the same amount of time traveling out
as it did coming back in his frame of reference. If in his frame the light
left at t'= -T and returned at t'= +T, then the point at which the beam
bounces off the mirror must have occurred simultaneously with the origin,
where t'= t = 0, in the frame of reference of our moving observer.
There is a very important point to note here. What if instead of light,
we wanted to throw a ball at 0.5 c, have it bounce off some wall, and then
return at the same speed (0.5 c). The problem with this is that to find a
line of constant time for the moving observer, the ball must travel at 0.5 c
BOTH WAYS in the reference frame of the MOVING observer. But we have not yet
defined the coordinate system for the moving observer, so we do not know
what a ball moving at 0.5 c with respect to him will look like on our
diagram. However, because of relativity, we know that the speed of light
itself CANNOT change from one observer to the next. In that case, a beam of
light traveling at c in the frame of the moving observer will also be
traveling at c for the O observer. So, a line which makes a 45 degree angle
with respect to the x and t axes will ALWAYS represent a beam of light
traveling at speed c for ANY observer in ANY frame of reference.
In Diagram 2-8, I have labeled a point A' on the t' axes which occurs
some amount of time before t'= 0 and a point B' which occurs the same amount
of time after t'= 0. I then drew the two light rays (remember, these are "45
degree angle" lines) as before--one leaving from A and going to the right,
and one moving to the left and coming in to B. I then found the point where
they would meet (C') which marks the point where the ray from A' would have
had to bounce in order to get back to the moving observer at B'. Thus, C'
and o occur at the same time in the frame of the moving observer. Notice
that for the O observer, C' is above his line of simultaneity at o (the x
axis). So while the moving O' observer says that C' occurs when the two
observers pass (at the origin), the O observer says that C' occurs after the
two observers have passed by one another. We will further discuss this
difference in the concepts of future and past in Section 2.8.
In Diagram 2-9, I have drawn a line passing through C' and o. This line
represents the same thing for our moving observer as the x axis did for the
O observer. So we label this line x'.
Diagram 2-8 Diagram 2-9
t t t'
| / | /
+ B' + /
| / \ | / __--x'
+ / C' + / __C'-
|/ / |/__--
-+---+---+---o---/---+---+- x -+---+---+-__o---+---+---+- x
/| / * __-- /|
/ / __-- / +
// | -- / |
A' + / +
/ | / |
From the geometry involved in finding this x' axis, we can state a
general rule for finding the x' axis for any moving observer. First recall
that the t' axis is the line that represents the moving observer's position
on the space-time diagram. The faster O' is moving with respect to O, the
greater the angle between the t axis and the t' axis. So the t' axis is
rotated away from the t axis at some angle (either clockwise or
counterclockwise, depending on the direction O' is going--right or left).
The x' axis is then a line rotated at the same angle away from the x axis,
but in the opposite direction (counterclockwise or clockwise).
Now, x' is a line of constant time for O', and any line drawn parallel
to x' is also a line of constant time. Such lines, along with the lines of
constant position, form a grid of the space-time coordinates for the O'
observer. I have tried my best to draw such a grid in Diagram 2-10. If you
squint your eyes while looking at that diagram, you can see the skewed
squares of the coordinate grid. You can see that if you pick a point on the
space-time diagram, the two observers with their two different coordinate
systems will disagree on when and where the event occurs.
Diagram 2-10
t'
+-----------------/-------+
| / /_-/""/ /__/-"/ / _|
|/-"/ / _/--/" / /_-/""/|
| /_-/""/ /__/-"/ / _/-->x'
|"/ / _/--/" / /_-/""/ |
|/_-/""/ /__o-"/ / _/--/|
| / _/--/" / /_-/""/ /_|
|-/""/ /__/-"/ / _/--/" |
|/ _/--/" / /_-/""/ /__/|
|""/ /__/-"/ / _/--/" / |
+-------------------------+
As a final note about this procedure, think back to what really made
these two coordinate systems look differently. Well, the only thing we
assumed in creating these systems is that the speed of light is the same for
all observers. In fact, this is the only reason that the two coordinate
systems look the way they do.
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