2.8 "Future", "Past", and 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.
2.8 "Future", "Past", and the Light Cone (Space-Time Diagrams)
For the later FTL discussions, it will be important to understand the
way different observers have different notions concerning the future and the
past. This difference comes about because of the way the different
coordinate systems of the two observers compare to one another.
First, let me note that with what we have discussed we cannot make a
complete comparison of the two observers' coordinate systems. You see, we
have not seen how the lengths which represents one unit of space and time in
the reference frame of O compare with the lengths representing the same
units in O'. This will be covered in the Part II: More on Special Relativity
(which is "optional" for those of you just interested in the faster than
light discussions). We can, however, compare the observers' notions of
future and past.
Back on Diagram 2-9, in addition to the O and O' space and time axes, I
also marked a particular event with a star, "*". Recall that for O, any
event on the x axis occurs at the same time as the origin (the place and
time that the two observers pass each other). Since the marked event appears
under the x axis, then O must find that the event occurs before the
observers pass each other in his frame. Also recall that for O', those
events on the x' axis are the ones that occur at the same time the observers
are passing. Since the marked event appears above the x' axis, O' must find
that the event occurs after the observers pass each other in his frame. So,
when and where events occur with respect to other events is completely
dependent on ones frame of reference. Note that this is not a question of
when the events are seen to happen in different frames of reference, but it
is a question of when they really do happen in the different frames (recall
our discussion of reference frames in Section 1.1). So, how can this make
sense? How can one event be both in the future for one observer and in the
past for another observer. To better understand why this situation doesn't
contradict itself, we need to look at one other construction typically shown
on a space-time diagram.
In Diagram 2-12 I have drawn two light rays, one which travels in the
+x direction and another which travels in the -x direction. At some negative
time, the two rays were headed towards x = 0. At t = 0, the two rays finally
get to x = 0 and cross paths (at the origin). As time progresses, the two
then speed away from x = 0. This construction is known as a light cone.
Diagram 2-12
t
^
| light
\ + /
\ inside /
\ + /
outside \ | / outside
---+---+---o---+---+---> x
/ | \
/ + \
/ inside \
/ + \
|
A light cone divides a space-time diagram into two major sections: the
area inside the cone and the area outside the cone (as shown in Diagram 2-12
). (Let me mention here that I will specifically call the cone I have drawn
"a light cone centered at the origin", because that is where the two beams
meet.) Now, consider an observer who has been sitting at x = 0 (like our O
observer) and is receiving and sending signals at the moment marked by x =
0, t = 0 (at the origin). Obviously, if he sends out a signal, it proceeds
away from x = 0 into the future, and the event marked by someone receiving
the signal would be above the x axis (in his future). Also, if he is
receiving signals at t = 0 , then the event marked by someone sending the
signal would have to be under the x axis (in his past). Now, if it is
impossible for anything to travel faster than light, then the only events
occurring before t = 0 that the observer can know about at the moment are
those that are inside the light cone. Also, the only future events (those
occurring after t = 0) that he can influence are, again, those inside the
light cone.
Now, one of the most important things to note about a light cone is
that its position is the same for all observers (because the speed of light
is the same for all observers). For example, picture taking the skewed
coordinate system of the moving observer and superimposing it on the light
cone I have drawn (note: a diagram which shows this view will be given in
Part II: More on Special Relativity). If you were to move one unit "down"
the x' axis (a distance that represents one light second for our moving
observer), and you move one unit "up" the t' axes (one second for our moving
observer), then the point you end up at should lie somewhere on the light
cone. In effect, a light cone will always look the same on our diagram
regardless of which observer is drawing the cone.
This fact has great importance. Consider different observers who are
all passing by one another at some point in space and time. In general, they
will disagree with each other on when and where different events had and
will occur. However, if you draw a light cone centered at the point where
they are passing each other, then they will ALL agree as to which events are
inside the light cone and which events are outside the light cone. So,
regardless of the coordinate system for any of these observers, the
following facts remain: The only events that any of these observers can ever
hope to influence are those which lie inside the upper half of the light
cone. Similarly, the only events that any of these observers can know about
as they pass by one another are those which lie inside the lower half of the
cone. Since the light cone is the same for all the observers, then they all
agree as to which events can be known about as they are passing and which
can be influenced at some point after they pass.
Now let's apply this to the observers and event in Diagram 2-9. As you
can see, the marked event is indeed outside the light cone. Because of this,
even though the event is in one observer's past at the time in question (t =
t'= 0), he cannot know about the event at the time. Also, even though the
event is in the other observer's future at the time, he can never have an
effect on the event after. In essence, the event (when it happens, where it
happens, how it happens, etc.) is of absolutely no consequence for these two
observers at the time in question. As it turns out, anytime you find two
observers who are passing by one another and an event which one observer's
coordinate system places in the past and the other observer's coordinate
system places in the future, then the event will always be outside of the
light cone centered at the point where the observers pass.
But doesn't this relativistic picture of the universe still present an
ambiguity in the concepts of past and future? Perhaps philosophically it
does, but not physically. You see, the only time you can see these
ambiguities is when you are looking at the whole space-time picture at once.
If you were one of the observers who is actually viewing space and time,
then as the other observer passes by you, your whole picture of space and
time can only be constructed from events that are inside the lower half of
the light cone. If you wait for a while, then eventually you can get all of
the information from all of the events that were happening around the time
you were passing the other observer. From this information, you can draw the
whole space-time diagram, and then you can see the ambiguity. But by that
time, the ambiguity that you are considering no longer exists. So the
ambiguity can never actually play a part in any physical situation. Finally,
remember that this is only true if nothing can travel faster than the speed
of light.
Well, that concludes our introduction to special relativity and
space-time diagrams. The next section deals with these concepts with more
detail; however, if the reader wishes to skip to the FTL discussion, the
information provided in the above sections should be enough to follow that
discussion.
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