From: KELSEY@SLACVM.SLAC.STANFORD.EDU (KELSEY, Michael H.)
Newsgroups: rec.arts.startrek.tech
Subject: Re: Twin Paradox
Date: 22 Aug 92 07:37:56 GMT
 <1992Aug21.190829.26406@erg.sri.com>
Organization: Stanford Linear Accelerator Center

In article <1992Aug21.190829.26406@erg.sri.com>, rat@erg.sri.com (Ray Trent)
says:
>
>Does this have anything to do with the fact that it is inherently
>impossible for a 1g acceleration to continue indefinitely because
>velocity can never increase beyond c? Or am I forgetting something
>having to do with how the acceleration is perceived by the person
>accelerating?

Where did you get this information?  A 1g acceleration, produced for
example by a Bussard ramjet (so all the nit-pickers out there won't
argue with me about carrying infinite fuel :-), can certainly be
continued indefinitely.  What will happen is that your kinetic energy
will increase indefinitely.  Now, the total energy (rest mass plus
kinetic energy) is given by E = gamma*m*c^2, where gamma is the Lorentz
factor, 1 / sqrt(1 - beta^2), beta = v/c.  As gamma increases smoothly
to infinity, the velocity, as measured by an external observer, becomes
arbitrarily close to c without exceeding it (i.e. beta -> 1 as gamma ->
infinity).

This is how a modern particle accelerator, like the proposed SSC, works.
You pump energy into the particles at a constant rate (using microwave
electromagnetic fields), which constantly accelerates them, and their
energy increases indefinitely -- an electron, with a rest mass of 0.5
million electron volts, can be accelerated to an energy of 50 billion
electron volts with little more effort than paying a HUGE electric bill ;-)
and there is no reason to stop there, except that our accelerator here is
only 4 km long.  At that energy, the gamma factor for the electrons is
about 100,000, and their velocity is (supposing gamma = EXACTLY 100,000)
0.99999999995 c, which is awfully close to c.  If we had an accelerator
twice as long, we could accelerate the electrons to 100 GeV (billion
electron volts), with gamma = 200,000, and beta = 0.999999999999999999995 c.

[ Note, I am using beta = SQRT(1 - 1/gamma^2), and approximating this
by beta ~= 1 - 0.5/gamma^2, to count the number of 9's.  For gamma > 100
this approximation is good to 1 part in 10^8, and the accuracy improves
as 1 / gamma^4, so don't quibble :-) ]

As you can see, in the relativistic regime what acceleration does is to
increase your _energy_ (and your _momentum_) drastically, while your
velocity approaches a limiting velocity (c) very slowly.  This is the
opposite of the low energy (car and billiard ball) regime, where accel-
eration increases your _velocity_ (and your _momentum_), while your TOTAL
energy (mass + kinetic), remains essentially constant.  A good rule of
thumb for the dividing line is when your total energy exceeds about twice
your rest energy, you are relativistic.
                                                  -- Mike Kelsey

[  My opinions are not endorsed by SLAC, Caltech, or the US government  ]
What is your _name_?    "kelsey@slacvm.slac.stanford.edu"
What is your _quest_?   "To get a Ph.D. in high-energy physics"
When will you _finish_? "I don't know.   Waaaaaaaaahhhhhhhh..."
Newsgroups: rec.arts.startrek.tech
From: chris@keris.demon.co.uk (Chris Croughton)