From: Hal Finney (hal@rain.org)
Date: Sun Jul 05 1998 - 11:29:23 MDT
>> I guess my real question is, according to the people on the
>> ship, does it appear to take more and more energy to maintain a
>> constant 1 g acceleration experience or is this only true in the
>> space/time frame of the people on earth? So, according to the people
>> on the ship, you can accelerate at 1 g forever right?
By the principle of relativity (which says that you can't tell if you're
moving, i.e. all motion is relative), it can't make any difference how
fast someone else thinks you are going. So it will take the same amount
of power, from your point of view, to accelerate at 1 g, no matter how
fast someone else thinks you are going.
As you go faster relative to the universe, it looks to you like the
universe (stars, galaxies, etc.) is going faster relative to you. As the
stars approach the speed of light, they don't speed up as much as you
would expect given the 1 g acceleration that you feel. They approach
the speed of light slower and slower, but do not exceed it. What happens
instead is that they undergo Lorentz contraction, flattening out in the
direction of motion.
The net result is that you go past stars, galaxies, etc., faster and
faster, not because they are moving past you more quickly, but because
they are becoming thinner and flatter, so even though they never move
faster than the speed of light, you move past more of them in a given
amount of time.
In relativity, given constant acceleration "a", the equations of motion
are as follows:
Let t be the time measured by the moving observer. Then the distance
he has travelled as measured by a stationary observer is:
X = 1/g * sinh (g*t).
The time it has taken for him to travel this far, as measured by a
stationary observer, is:
T = 1/g * cosh (g*t).
sinh and cosh are "hyperbolic sine" and "hyperbolic cosine", defined as
sinh(x) == 1/2 * (e^(x) - e^(-x))
cosh(x) == 1/2 * (e^(x) + e^(-x))
Here, x, t, and a must be measured in relativistic units where c = 1
e.g. x is in light years, t is in years. In these units, 1 g is
approximately 1/light-year. So after 1 subjective year at 1 g, t = 1
and
X = 1/2 * (e - 1/e) = 1.18 light years
T = 1/2 * (e + 1/e) = 1.54 years
After a large number of subjective years at 1 g, the equation can be
simplified to approximately X = T = e^t. You are moving at close to the
speed of light, hence X = T in these units (e.g. it takes 100 years for
you to travel 100 light years as measured by a stationary observer).
But with time as measured by you, the distance you cross in terms of
galaxy units is exponential.
Hal
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