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From: throopw@rtp47.UUCP (Wayne Throop)
Newsgroups: net.physics
Subject: Re: FTL and time-travel -- exercise for the reader
Message-ID: <100@rtp47.UUCP>
Date: Sun, 21-Jul-85 21:22:40 EDT
Article-I.D.: rtp47.100
Posted: Sun Jul 21 21:22:40 1985
Date-Received: Thu, 25-Jul-85 00:44:20 EDT
References: <375@sri-arpa.ARPA> <851@oddjob.UUCP> <860@oddjob.UUCP> <1013@mhuxt.UUCP>
Organization: Data General, RTP, NC
Lines: 129
I'd like to be able to reference a text for those who don't see how FTL
implies time-travel (or at least signaling into the past). However, I
don't know of such a text. What I know of the subject, I found out in a
physics class word-of-mouth from the teacher so to speak (as the result
of a wise-guy in the class posing special relativity "paradoxes", and
the teacher eventually spending an entire class explaining things).
SO: does anybody else know of a text readable by laycritters that covers
this ground? I have read the one by C.P. Steinmetz, and have heard of
the one by Einstein himself, but neither of these seem to address how
(enter doubletalk mode) spacelike worldlines imply time-travel
(end doubletalk mode).
In any event, let me add a few long-winded points to the discussion.
> > Suppose that at time t=0 person A emits a signal with velocity
> > u > c in A's own frame of reference. This signal is received
> > by B who at that instant is at a distance d from A (as measured
> > by A) and is moving away from A at speed v, with c^2/u < v < c.
> > B immediately replies by sending back a signal at speed u in
> > B's own reference frame.
> >
> > At what time does the reply signal from B reach A?
>
> Well, the answer I got (which evidently was wrong) was that the time
> for the first signal to get to B would be t1=d/(u-v), and that the time
> for the second signal to get back to A would be t2=du/(u-v)^2. Both of
> these quantities are positive for uc and v > Answer: In A's frame, the reply arrives at time
> >
> > t = d/u - d*(uv/c^2 - 1)/(u-v).
> >
> > Under the assumption c^2/u < v < c, this is negative.
>
> I can see that. What I can't see is how you went about deriving this.
> I would appreciate seeing that, since *if I could be convinced about the
> correctness of the above formula*, I would then understand how FTL signalling
> violates causality
Now these formulas are more familiar, just a simple d/u going out, and a
Lorentz transform on the velocity coming back (so that it is velocity in
A's frame rather than in B's). "Look" more familiar. I still don't
quite follow the exact transform on the second term there (the negative
one). Can anybody clarify this for me? (Matt?)
The most convincing (to me) method of demonstrating the equivalence of
FTL with time-travel is with space-time diagrams and assuming "u"
infinite (that is, "instant messages"). I have several times tried to
make these diagrams using character graphics so that they could be
posted to the net, but they just don't look good, and are confusing as a
result. Let me try this anyhow, and y'all can let me know if it is of
any use.
Consider two world lines, A and B (in stars), the three events x, y and
z, and the "lines of simultenaity" drawn in A's frame from x and in B's
frame from z (in dots). (It so happens that B crosses A at z.)
| . . *
| . . *
| . . *
| . . *
^ | . . *
| | A *******************x****z*******************************
space | .* .
| * . .
| * . .
| * ..
| * y
| B * ..
| . .
| . .
+---------------------------------------------------------------
time -->
Now then, the vertical "line of simultaneity" is with respect to A's
world-line, and the "tilted" one is with respect to B's. The exact
angles and so on can be worked out (see below), but this diagram is just
to give you the idea. Note that A considers events x and y to be
simultaneous, while B considers events y and z simultaneous.
Thus, if somebody in B's frame sends an instant message from z, it can
go to y. Now, if somebody at y *in A's frame* sends an instant message
from y, it can go to x. Thus a message relay can send messages from a
"later" point in a timeline to an "earlier" point in that timeline.
Note also that if a message can be sent by any ammount faster than light
in A's frame, there exists a frame such that that message is seen as an
"instant message".
If there is a "preferred frame" in which instant messages are possible
(and they are possible in no other frames), then this problem can be
avoided. This may be so, but no such preferred frames are known, nor is
there (strong) reason to suppose that they exist.
Note that a relay is not required for time-travel to be involved.
The message in B's frame from z to y "looks like" it is traveling
backwards in time when somebody in A's frame looks at it. The only
reason a relay is needed is to loop back on a single world-line.
Now, about the "lines of simultenaity". We can find two events that B
will consider simultaneous by following B's world-line forward x
A-seconds (or x/(1-v^2/c^2)^-2 B-seconds) forward from event e.1, then
follow A's world line back x/(1-v^2/c^2)^-2 B-seconds
(or (x/(1-v^2/c^2)^-2) / (1-v^2/c^2)^-2 A-seconds) to find event e.2. B
will consider e.1 and e.2 simultaneous (and all events on the line of
which e.1 and e.2 are members). (I think this double-transform
is how we get the second term in Matt's answer, above, but I haven't
worked it all out to be sure of this).
I suggest actually drawing several Ls of S for various values of v, and
following the consequences of sending messages along any space-like
line. It turns out that *somebody* will see a message on *any*
space-like line as going "pastward".
I hope this helps rather than hinders. (I also hope I haven't grossly
distorted the concepts involved... this is, after all, remembered from a
class 10 years ago.)
> > Matt University crawford@anl-mcs.arpa
> > Crawford of Chicago ihnp4!oddjob!matt
> --
> Jeff Sonntag
> ihnp4!mhuxt!js2j
--
Wayne Throop at Data General, RTP, NC
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