2.4 Constructing One for a "Stationary" Observer (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.
2.4 Constructing One for a "Stationary" Observer (Space-Time Diagrams)
At this point, we want to decide exactly how to represent events on
this coordinate system for a particular observer. First note that it is
convenient to think of any particular space-time diagram as being
specifically drawn for one particular observer. For Diagram 2-2, that
particular observer (let's call him the O observer) is the one whose
coordinate system has the vertical time axis and horizontal space axis shown
in that diagram. Now, other frames of reference (which don't follow those
axes) can also be represented on this same diagram (as we will see).
However, because we are used to seeing coordinate systems with horizontal
and vertical axes, it is natural to think of this space-time diagram as
being drawn specifically with the O observer in mind. In fact, we could say
that in this space-time diagram, the O observer is considered to be "at
rest".
So if the O observer starts at the origin, then one second later he is
still at x = 0 (because he isn't moving in this coordinate system). Two
seconds later he is still at x = 0, etc. If we look at the diagram, we see
that this means he is always on the time axis in our representation.
Similarly, any lines drawn parallel to the t axis (in this case, vertical
lines) will represent lines of constant position. If a second observer is
not moving with respect to the first, and this second observer starts at a
position two light seconds away to the right of the first, then as time
progresses he will stay on the vertical line that runs through x = 2.
Next we want to figure out how to represent lines of constant time. We
might first find a point on our diagram that represents an event which
occurs at the same time as, say, the origin (t = 0). To do this we will use
a method that Einstein used. First we choose a point on the t axis which
occurred prior to t = 0. Let's use an example where this point occurs at t =
-3 seconds. At that time we send out a beam of light in the positive x
direction. If the beam bounces off of a distant mirror at t = 0 and heads
back toward the t axis, then it will come back to the us at t = 3 seconds.
We know this because (1) it will have traveled for three seconds away from
us, (2) it will have the same distance to travel back to us in our frame of
reference, and (3) according to relativity it must travel at the same speed,
c, going AND coming back. Thus, it must take three seconds to get back to us
as well which means it reaches as at the time t = 3 seconds. So, if we send
out a beam at t = -3 seconds and it returns at t = 3 seconds, then the event
"it bounced off the mirror" occurred simultaneously with the time t = 0 at
the origin.
To use this on our diagram, we first pick the two points on the t axis
that mark t = -3 and t = 3 (let's call these points A and B respectively).
We then draw one light beam leaving from A in the positive x direction. Next
we draw a light beam coming to B in the negative x direction. Where these
two beams meet (let's call this point C) marks the point where the original
beam bounces off the mirror. Thus the event marked by C is simultaneous with
t = 0 (the origin). A line drawn through C and o will thus be a line of
constant time. All lines parallel to this line will also be lines of
constant time. So any two events that lie along one of these lines truly
occur at the same time in this frame of reference. I have drawn this
procedure in Diagram 2-4, and you can see that the x axis is the line
through both o and C which is a line of simultaneity (as one might have
expected).
Note that the event marked by C is not seen by the O observer (who,
remember, is represented by the t axes because he sits at x = 0) at the
moment it happens (t = 0) but it is seen once light from C reaches the O
observer (which is the point marked B). However, because of the way we did
the experiment, we know that in this frame of reference, C truly did happen
simultaneously with the origin, o. This just goes to illustrate, as
discussed in Section 1.1, that when I say that two events happened
simultaneously in some frame of reference, I am not talking about when they
are seen by some observer in that frame. Rather, I am talking about when
they actually occur in that frame of reference. On our diagrams, events are
represented at their actual space-time locations relative to one another,
and in a particular frame of reference that means that we show exactly when
and where the event occurred (not "observed" but truly occurred) in that
frame.
Now, by constructing a set of simultaneous time lines and constant
position lines we will have a grid on our space-time diagram. Any event has
a specific location on the grid which tells where and when it occurs in this
frame of reference. In Diagram 2-5 I have drawn one of these grids and
marked an event (@) that occurred 3 light seconds away to the left of the
origin (x = -3) and 1 second before the origin (t = -1).
Diagram 2-4 Diagram 2-5
t
|
B t
| \ | | | | | |
+ \ ---+---+---+---+---+---+---
| \ | | | | | |
+ \ ---+---+---+---+---+---+---
| \ | | | | | |
-+---+---o---+---+---C- x ---+---+---+---o---+---+--- x
| / | | | | | |
+ / ---@---+---+---+---+---+---
| / | | | | | |
+ / ---+---+---+---+---+---+---
| / | | | | | |
A
|
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