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From: matt@oddjob.UUCP (Matt Crawford)
Newsgroups: net.physics
Subject: Re: FTL and time-travel -- exercise for the reader
Message-ID: <889@oddjob.UUCP>
Date: Sun, 28-Jul-85 17:00:38 EDT
Article-I.D.: oddjob.889
Posted: Sun Jul 28 17:00:38 1985
Date-Received: Mon, 29-Jul-85 08:03:00 EDT
References: <375@sri-arpa.ARPA> <851@oddjob.UUCP> <860@oddjob.UUCP> <1013@mhuxt.UUCP> <100@rtp47.UUCP>
Reply-To: matt@oddjob.UUCP (Matt Crawford)
Organization: U. Chicago, Astronomy & Astrophysics
Lines: 87

I have received a lot of mail and seen a few followups which indicate
to me that posting a solution would be appreciated.  Several people
showed that they could write equations without understanding the con-
cepts of relativity, while one person showed that he could understand
the concepts without being able to do all the arithmetic.  (A gold star
to you, W.T.)  I will not attempt to draw figures on the CRT but I will
try to make the reasoning clear.  (I'll even leave in the c's!)

The question:

	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?


The solution:          (gee, this is like being a TA again!)

Outline: Find where the signal is received by B in A's coordinates.
Transform this to B's coordinates.  Then trace the path of the reply
in B's coordinates and transform the result to A's coordinate system.

Let the origins of both A's and B's coordinates be at the event of
A sending the initial signal.  Points in spacetime will be denoted
as P = [t, x] in A's coordinates or P' = [t', x'] in B's.  (One
spatial dimension is enough for this problem, so I won't write the
y's and z's.)  The quantity usually denoted by gamma will be

	G = (1-v^2/c^2)^(-1/2).

The Lorentz transformation between frames is given by:

	x' = G (x-vt)			x = G (x' + vt')
	t' = G (t-xv/c^2)		t = G (t' + vx'/c^2)

The reception by B of A's signal will occur at point P = [ d/u, d ],
In B's frame this is P' = [ Gd/u-Gdv/c^2, Gd-Gdv/u ].  (You can
already see the effects of the faster-than light velocity u here.
In B's coordinates the signal was emitted at time t' = 0 but was
received at an earlier time.  The time component of P' is negative.)

The reply signal from B will be emitted at P' in B's frame and will
travel to a point R' (to be determined) at speed u relative to B.
The distance travelled by the reply signal, in B's system, will be
some length r'.  The reception of the reply is at:

	R' = P' + [ r'/u, -r' ]
	   = [ Gd/u - Gdv/c^2 + r'/u, Gd - Gdv/u - r' ].

Transforming back to A's frame and simplifying gives:

	R = [ d/u - Gr'(v/c^2-1/u), d - Gr'(1-v/u) ],

but r' is still unknown.  Because A's position is at x = 0 for all
times and R is the event of the reception of the reply by A, set
the x-component of R equal to zero to solve for r'.  This yields

	r' = d / G(1-v/u) = du / G(u-v)

Substitute this into the t-component of R and find:

	R = [ d/u - d(vu/c^2 - 1)/(u-v), 0 ].

The condition uv > c^2 in my original posting is not strong enough
to ensure that the time component of R is negative.  The slightly
stronger condition uv > (1 + (1-v^2/c^2)^(1/2)) c^2 is needed to make
the reception of the reply precede the sending of the original signal.
If the relative speed of A and B of v = 0.5c, then the condition is
u > (2 + 3^(1/2))c.  (Note that u > 2c^2/v will always suffice.)


To anyone who wants to learn about relativity I enthusiastically
recommend the book _Spacetime_Physics_, by Taylor and Wheeler
[W.H. Freeman & Co., San Francisco].  The math needed does not go
beyond the v = dx/dt level, yet the content is sufficient for first-
year physics students at the best departments in the country.  The
book is accurately described on the cover as "A Brief, Readable
Exposition of Modern RELATIVITY THEORY Illustrated and Amplified
by a Wealth of PROBLEMS, PUZZLES, and PARADOXES and their Detailed
Solutions."
_____________________________________________________
Matt		University	crawford@anl-mcs.arpa
Crawford	of Chicago	ihnp4!oddjob!matt