8.3 How We Get Unsolvable Paradoxes (Relativity and FTL Travel)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
8.3 How We Get Unsolvable Paradoxes (Relativity and FTL Travel)
As I mentioned before, violations of causality (as strange as they may
be) do not have to truly, logically contradict themselves. However, it isn't
too difficult to show (starting with the above arguments) that FTL travel
can be used to produce an unsolvable paradox (a situation which contradicts
its own existence). As a note, in the past I have called such situations
"gross" violations of causality.
I'll illustrate the point with an example (again referring to Diagram
8-1) Remember we said that as O and Op pass, Op can send an FTL message out
(from his frame of reference) which effects "*". However, rather than having
him send a message out, let's say that Op sends out a bullet that travels
faster than the speed of light. This bullet can go out and kill someone
light-years away in only a few hours (for example) in Op's frame of
reference. So, say he fires this bullet just as he passes by O. Then the
death of the victim can be the event (*). Now, in O's frame of reference,
the victim is already dead ("*" has occurred) when Op passes by. This means
that another observer (stationary in O's frame) who was at the position of
the victim when the victim was shot could have sent an FTL signal just after
the victim's death, and that signal could reach O before Op passed by him.
So O can know that Op will shoot his gun as they pass each other.
To intensify the point I will make, we can let the signal which was
sent to O be a picture of the victim, or even an ongoing video signal of the
victim's body. Thus, O has evidence of the victim's death before Op has
fired the weapon (a plain ol' violation of causality). However, at this
point O can decide to stop Op from firing the gun. But if the bullet doesn't
go out, and the victim never dies, then why (and how) would a video
signal/picture of the victim's dead body ever be sent to O? And yet, O has
that video/picture.
In the end, it is the death of the victim which causes O to prevent the
victim's death, and that is a self contradicting situation. Thus, if there
are no special provisions (which we will discuss later ) FTL travel will not
only allow violation of causality, but it can also produce unsolvable
paradoxes.
At this point, I want to clearly list the various events which must
happen to produce an unsolvable paradox in our "FTL bullet" example. Through
the rest of our FTL discussion, this will be helpful as a reference listing.
Event Listing and Comments:
1. As observers O and Op pass by one another (as they are shown in Diagram
8-1) Op uses some method to send out an FTL bullet from his reference
frame. The event "O and Op pass one another" will be called the
"passing event" from here on.
2. The bullet strikes and kills a victim who's death is the event marked
"*" in Diagram 8-1. This event occurs after the passing event in Op's
frame of reference, but it occurs before the passing event in O's
frame.
3. A third observer is at the victim's side as he dies and thus he
witnesses the death. This third observer is stationary in O's frame of
reference (i.e. his frame is the same as O's), so the victims death
("*") occurs before the passing event (when the bullet was fired) in
this third observer's frame. Thus, the third observer has witnessed a
result which comes from an event in his future--he has information
about a future event in his frame of reference.
4. The third observer sends this information about the future to O using
an FTL signal, and in the third observer's frame of reference, O can
receive this information before the passing event occurs (and thus
before the bullet is fired).
5. O receives the message and learns of the victims death before the
bullet is fired. He thus knows about the bullet being fired--an event
in his own future which will occur at his very location.
6. O uses this information to prevent Op from firing the bullet, thus
causing a self-inconsistent situation--an unsolvable paradox.
It is important to note that the real crux of this problem does not
come from the form of the FTL travel used, but from the relationship between
the two, ordinary frames of reference for observers (O and Op) who never
themselves travel FTL. This ordinary relationship (determined by relativity)
can be demonstrated through experiments today, and as long as the exact same
experiments can be performed in the future to yield the same results, then
this argument must still hold. This is the power of this problem, and we
will see that the special provisions we will discuss later must concern
themselves with the ability of the observers to use the relationship between
themselves in order to produce unsolvable paradoxes. Thus, the provisions
will not be specifically concerned with the form of FTL travel used or the
future theories which might suggest FTL travel, because the problem we have
discussed here will be present regardless of either of these considerations.
And so, we have discussed the two problems which arise with FTL travel.
Our next job is to consider various, specific FTL concepts in light of these
problems. If your not interested in the discussion of the various forms of
FTL travel, and you want to take my word for it that they will all suffer
from the problem discussed above, then you may want to skip to the "Special
Provisions" section.. I'll leave that to the reader.
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