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Re: special relativity: accelerated frames



Regarding John Denker's homehork assignment:

Suppose an interstellar spaceship starts from rest, and
accelerates such that the passengers feel one Gee (980 gal)
for one year. How fast are they going at the end of the year?

Huge hint: V=tanh(rho) where rho is called the "rapidity". Find rho.

This *is* a huge hint. A related tidbit is that for such a uniformly
(w.r.t. on board experiments) accelerating spaceship, its average speed
(measured in the same inertial coordinate system that the spaceship
was initially at rest w.r.t at the commencement of the trip) has a
simple relationship w.r.t. its peak speed. If we use John's hint
formula: V = tanh(rho) as yielding the peak speed, we find that the
average speed: is V_avg = tanh(rho/2).

A eye opening exercise is to use the above problem to come up with a
modified twin paradox problem, one where the rocket twin leaves Earth in
a uniformly accelerating spaceship which has a constant acceleration 'a'
(as measured by the experimental inertial forces experienced by the
ship's inhabitant). This acceleration is kept up for a time interval T
at which point the craft decelerates with the same constant negative
acceleration 'a' it had when it was positively accelerating. This
deceleration lasts for another (on board) time interval of T. At this
point the craft is at rest w.r.t. Earth. The travelling twin takes a
quick photo of the destination planet/star system, whatever, and again
fires up the craft for the return flight which also has a uniform
acceleration 'a' for time interval T, followed by a similar deceleration
'a' phase of another length T. So after a total interval of 4*T (of on
board time) the travelling twin returns home to Earth.

Questions:
a) How much time elapsed on Earth while the travelling twin was away?
b) How far was the destination planet/star system from Earth, and how
far was the round trip distance covered as measured in the Earth's
rest frame?
c) What was the spaceship's peak speed w.r.t. Earth? What was its
average speed?

If we make the acceleration 'a' to be close to 1 g and let T be a few
years some startling results obtain. In particular, suppose that we
take T = 20 years (so the whole trip takes one typical human lifetime of
80 years). Then if the rocket acceleration is a = g = 9.81 m/s^2, then
the round trip distance covered is 1.79 Gly. If we up the acceleration
a to 1.25 g then the round trip distance covered jumps to 250.2 Gly. And
if we up the acceleration further to 1.5 g and recalculate the round
trip distance we find it is 36370 Gly! (Note that the current proper
horizon distance to the observable edge of the observable universe is no
more than a few ten's of Gly.) Clearly for such a long term flight
covering such cosmological distances we *would* have to use GR to
properly solve this problem, because while the travelling twin was gone
the universe's expansion would become a significant factor.

In any event, what this exercise shows if that *if* we could invent a
rocket propulsion technology that would allow accelerations comparable
to 1 g for a few tens of years, then there is no place in the currently
observable universe which would not be unaccessible to human visitation
over the lifespan of a single human being (all the while no traveller
would be subjected to any unphysically large g-forces). Using on board
colonys of human populations which have only desendents of the original
travellers survive when the rocket arrives at the destination would be
unnecessary for making the trip. Even using suspended animation during
the trip would not be necessary either.

David Bowman
David_Bowman@georgetowncollege.edu