8.2 How FTL Travel Implies Violation of Causality
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
8.2 How FTL Travel Implies Violation of Causality
I refer you back to Diagram 2-9 (reproduced below as Diagram 8-1) so
that I can demonstrate the causality problem involved with FTL travel. There
you see two observers passing by one another.
Diagram 8-1
(Copy of Diagram 2-9)
t t'
| /
+ /
| / __--x'
+ / __C'-
|/__--
-+---+---+-__o---+---+---+- x
* __-- /|
__-- / +
-- / |
/ +
/ |
The origin marks the place and time where the two observers are right next
to one another. The x' and t' axes are said to represent the frame of
reference of O' (I'll use Op--for O-prime--so that I can easily indicate the
possessive form of O as O's and the possessive form of O' as Op's). The x
and t axes are then the reference frame of the O observer. We consider the O
system to be our rest system, while the Op observer passes by O at a
relativistic speed. As you can see from the two coordinate systems, the two
observers measure space and time in different ways. Now, consider again the
event marked "*". Cover up the x and t axis and look only at the Op system.
In this system, the event is above the x' axis. If the Op observer at the
origin could look left and right and see all the way down his space axis
instantaneously, then he would have to wait a while for the event "*" to
occur. Now cover up the Op system and look only at the O system. In this
system, the event is below the x axis. So to O, the event has already
occurred by the time the two observers are passing one another.
Normally, this fact gives us no trouble. If you draw a light cone (as
discussed in Section 2.8) through the origin, then the event will be outside
of the light cone. As long as no signal can travel faster than the speed of
light, then it will be impossible for either observer to know about or
influence the event. So even though it is in one observer's past, he cannot
know about it, and even though it is in the other observer's future, he
cannot have an effect on it. This is how relativity saves its own self from
violating causality.
However, consider the prospect of FTL travel with this diagram in mind.
As O and Op pass by one another, the event "*" has not happened yet in Op's
frame of reference. Thus, if he can send an FTL signal fast enough, then he
should be able to send a signal (from the origin) which could effect "*".
However, in O's frame, "*" has already occurred by the time O and Op pass by
one another. This means that the event "Op sends out the signal which
effects *" occurs after the event which it effects, "*", in O's frame. For
O, The effect precedes the cause. Thus, the signal which travels FTL in Op's
frame violates causality for O's frame. Similarly, since "*" has already
occurred in O's frame when O and Op pass one another, then in his frame an
FTL signal could be sent out from "*" which could reach O and tell him about
the event as the two observer's past. However, for Op, the event "O learns
about * as O and Op pass one another" comes before * itself. Thus, the
signal which is FTL in O's frame violates causality in Op's frame.
In short, for any signal sent FTL in one frame of reference, another
frame of reference can be found in which that signal actually traveled
backwards in time, thus violating causality in that frame.
Notice that in this example I never mentioned anything about how the
signal gets between the origin and *. I didn't even require that the signal
be "in our universe" when it was "traveling" (remember our definition of FTL
travel in Section 6.1). The only things I required were that (1) the
signal's "sending" and "receiving" were events in our universe and (2) the
space-time between the origin and "*" is flat (i.e. it is correctly
described by special relativity diagrams). Some FTL ideas may invalidate the
second assumption, but we will consider them a bit later. We will find,
however, that violation of causality still follows from all the FTL travel
concepts.
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