9.5.3 "Producing" Restricted Space-Time Areas
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
9.5.3 "Producing" Restricted Space-Time Areas
This provision is sort of an extension on the previous one, but its
mechanism specifically targets the FTL travel so as to restrict one of the
FTL trips or messages one must use to produce an unsolvable paradox.
Remember that in the list of events for our FTL bullet example, there were
two different FTL portions (the FTL bullet and the FTL message from the
third observer to O). This provision would cause the sending or receiving of
one of these "messages" to strictly prohibit the sending or receiving of the
other. I will try to illustrate the basic way in which such restrictions
could work to always prevent unsolvable paradoxes. I will then give an
example where this provision is implemented with a particular FTL concept.
For the illustration, we need to consider each of two possibilities
within our FTL bullet example. In the first possibility, the Op observer is
allowed to send his FTL bullet which strikes the victim, but that FTL trip
must then restrict the third observer's ability to send the FTL message to
O. In the second example, the third observer happens to decide to send some
FTL signals to O at some point before the event "*" (which is the event in
our example that usually marked the victim's death). Now, we let the third
observer continue to send those FTL signals until some point after "*".
Then, if the victim dies at "*" because of the FTL bullet, then since the
third observer is sending FTL signals to O at that point, he would be able
to tell O about the victim's death, and the paradox would still be possible.
Thus, in this second case, the FTL bullet must not be allowed to strike the
victim (the FTL travel of the bullet is restricted because the third
observer sends FTL signals to O).
So, how would these restrictions work in these two possible cases?
Well, as it turns out, if all unsolvable paradoxes are going to be averted
while only placing restrictions on particular FTL trips, then there must be
a very specific provision in place. To explain this, we will look at both
possible situations, and consider diagrams which explain each one. (Note
that these diagrams are drawn a little differently from Diagram 8-1 so as to
better show the point I am trying to make here.)
Diagram 9-2
t t'
. | /
. + /
. | / __--x'
. . + / __C'-
. . |/__--
+---+.--+---+---+.--+---+---+-__o---+---+---+- x
. . __--./| .
. . __-- . / + .
* __-- . / | .
__-- . / + .
__-- . / | .
(Case 1--The FTL bullet is allowed to strike at the event "*")
In this diagram we mean to illustrate case one in which the FTL bullet
leaves the "passing event" (i.e. the origin, "o") and is "received" by the
victim who immediately dies at event "*". Now, I have also drawn parts of
two light cones (marked with dots). One part is the "upper half light cone
of the event '*'," and the other is the "lower half light cone of the
passing event, 'o'". The upper half light cone of "*" contains all events
which an observer at "*" (like the third observer in our bullet example) can
influence without having to travel FTL. All observers agree that all events
in this area occur some time after "*" (as discussed in Section 2.8). Also,
the lower half light cone of "o" contains all the events which could effect
"o" (which, remember, is the event at which the FTL bullet is sent) through
non-FTL means. Thus, as long as no FTL signal/traveler can leave as an event
in the upper half light cone of "*" and be received as an event in the lower
half light cone of "o", then all unsolvable paradoxes will be averted. There
would be no way for the third observer to witness the death of the victim
and afterwards get a signal to O before the bullet is fired.
Now, that seems to be straight forward. We just need to make this
provision: When an FTL signal is transmitted as event T, and it is received
as event R, then it must be impossible for any information to be sent as an
event in R's upper ("future") light cone and end up being received as an
event in T's lower ("past") light cone. If the universe restricted FTL
travel in this way, it would be impossible to produce unsolvable paradoxes.
However, we can see that the matter can get a little complicated when
we consider things from O's frame of reference (which is also the frame of
the third observer). In this frame, after the third observer witnesses the
victim's death at "*", the event "the bullet leaves" hasn't occurred yet. He
might then argue that no FTL signal has yet been sent which would keep him
from sending a FTL message to O. The problem with his argument is that he
has already witnessed the result of the FTL bullet being sent (even if it
hasn't occurred in his frame yet). Thus, any FTL signal he tries to send to
O (in the lower half light cone of the origin/passing
event/bullet-being-fired event) must be prevented from being received by O.
Ah, but what if he (the third observer) just happened to decide to
start sending FTL signals to O (just to chat) before the bullet strikes the
victim? That leads to our second case. Here, then, is a diagram we will use
to describe this second case.
Diagram 9-3
t t'
. | /
. + / .
. | / . __--x'
. + / ._C'-
. |/__.-
+---+---+---+---+---+---+---+-._o-.-+---+---+- x
__-- /R
T __-- / |
. * . __-- / |
. s _.-- / +
. __-- . / |
(Case 2--The FTL bullet may not be allowed to strike at the event "*")
Now, there are a few extra events here. The point "s" marks the point
where the third observer starts sending FTL signals to O while "T" marks the
point where he finishes sending those FTL signals. The point "R" marks the
point where O receives the last message which was sent at "T". Now, here we
have drawn the upper and lower half light cones of interest, and according
to our discussion above, it would be impossible for Op to send his bullet at
the origin, "o" (which is in the upper half light cone of R) and have it
"received" by the victim at "*" (which is in the lower half light cone of
T). So, according to that argument, the bullet doesn't strike while the
third observer is sending FTL signals to O, and so the third observer never
tells O about the victim's death.
However, this doesn't HAVE to be what happens, and we might just end up
back at the first case. You see, either (1) the signals sent by the third
observer are all successful, and the FTL bullet is restricted from striking
the victim at "*" (that's the second case); or (2) the FTL bullet does
strike the victim at "*" and any FTL signals that the third observer sends
after "*" are restricted from reaching the O observer before the bullet is
fired (this is the first case, even though the third observer was sending
signals to O just before the bullet hit). The obvious question, then, is
"which one of these two cases actually occurs?" The answer happens to be,
"it really doesn't matter." You see, as long as one or the other does occur,
the situation remains self consistent and no self inconsistent paradoxes are
produced. Roll some dice and pick one, if you like, or let some unknown
force decide which happens. It really doesn't matter for our argument. Is
that a bit odd? Yes. Is it self-inconsistent so as to produce unsolvable
paradoxes? No.
Finally, as example to show this provision in action with a particular
FTL concept, let's consider a case where space-time manipulation is used via
a wormhole. Recall that in our discussion of this FTL concept in Section
9.4, we showed that one can still produce unsolvable paradoxes. Notice, that
there still must be two FTL parts (we discussed one FTL "trip"--the
bullet--from A to B and another--an FTL message--from C to D). Now, to
prevent the paradox, the existence of the wormhole that allows the bullet to
travel from A to B could forbid the existence of the wormhole that allows
the FTL message to go from C to D. This is a situation where case 1 applies,
and here the way the provision is satisfied comes from the conceptual
ability of one wormhole's existence to forbid the existence of another
wormhole.
And so, we have a provision which simply restricts (in a very
particular way) certain FTL trips because of other FTL trips. We have found
that there doesn't have to be a discernible answer to the question of
whether trip A disallows trip B or trip B disallows trip A, but as long as
it is one case or the other, this provision will keep all situations self
consistent and thus avoid unsolvable paradoxes.
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