A table of contents for these "frame-dependent relativity" pages.
The Problem
You board a spaceship which accelerates at 1 "gee" continuously
until it has traveled half of the 2.2 million lightyear distance to
the Andromeda Galaxy*. It then decelerates (also at 1 "gee")
over the remaining 1.1 million lightyears of distance to halt in
a star system with an earth-like planet in the Andromeda
galaxy itself! How much older are you at the end of the trip?
You may be surprised.
*Andromeda Galaxy is one of the most distant objects visible to the naked eye.
Figuring out the First Leg of the trip...
Maximum Traveler-Kinematic Velocity
First, calculate velocity at the end of the first (acceleration) leg
of the trip. Galilean velocity obeys
Vf2 - Vo2 =
2 α dx.
Initial velocity Vo is zero, α is
1.03 ly/yr2, and dx is 1.1 million lightyears. Hence the
Galilean final velocity at the end of the leg is
Vf = Sqrt[2×1.03{ly/yr2}×1,100,000{ly}]
= Sqrt[2.266×106{ly2/yr2}] = 1505{ly/yr}.
Of course, in high speed problems like this no physical clocks
tick with a Galilean beat, and hence this speed is of mathematical
significance only.
Using equations which were provided as a hint with the
original problem
(and derived elsewhere on these
pages), the final proper velocity follows from
Galilean velocity via the conversion w =
V Sqrt[1 + ¼( V/c )2],
so that wf = 1505{ly/yr} Sqrt[1 + ¼(1505)2] =
1,133,001 or {lightyears per traveler year} or {rodden-berries}. This is
quite a high speed, given that color TV's accelerate picture tube
electrons to a traveler velocity of c Sqrt[γ2-1]
where γ = E/mc2 = (K+mc2)/mc2
for kinetic energy K=30{keV} and electron mass-energy of
mc2=511{keV}, or a w of only around 0.35{rb}.
Even 100GeV electrons accelerated at FermiLab don't quite make 200,000{rb}!
Elapsed Times and Coordinate Velocity
For the first leg of the trip, traveler (or proper) time elapsed is
dτ =
(ArcSinh[wf/c] - ArcSinh[wo/c])
c/α = 14.2 {years}.
Hence the average proper-velocity during the first leg of
the trip is dx/dτ = 77425{rb}. Note that this average
velocity can be less than half of the final value because,
for constant proper-acceleration α, the derivative of
proper-velocity with respect to traveler-time (dw/dτ)
is not constant, even though for unidirectional motion the derivative
of Galilean-kinematic velocity with respect to Galilean chase-plane
time (dV/dT) is constant.
Earth (or map) time elapsed, for comparison to traveler (or proper) time,
is dt = (wf-wo)/α
= 1,100,001{yr}.
Hence the average coordinate-velocity for the first leg is
dx/dt = 0.99999912{c}, just less than the speed of light
as one might have guessed.
By comparison the final coordinate-velocity
vf = wf/Sqrt[1+(wf/c)2] = 0.999999999999611{c}
or {lightyears per map year}. This means that 14 years of 1 "gee"
acceleration gets you VERRRRRRRRRRRRY close to the speed of light!
In fact, this is only a 100 microns per second slower than the
speed of light, even though light travels 300 million meters in that
same second. As you might guess, however, it never gets you
to the speed of light. Here the average coordinate-velocity
can be more than half of its final value, since the derivative
of coordinate-velocity with respect to map time (dv/dt)
is not a constant either.
Figuring out the Whole Trip
The second leg of the trip is basically the same as the first leg with
time reversed, i.e. in which one begins at high speed and slows to a
stop at 1 gee of deceleration. Times elapsed will be the same on the
second leg of the trip as on the first leg. Final velocities will be
zero. Total times elapsed will be twice the times calculated for the
first leg. Hence traveler time elapsed is 2×14.2 = 28.4 years,
while bystanders waiting for the trip to be over will have to wait
2×1.1×106 = 2.2 million years from beginning to end!
The practical problems with actually doing this involve the amount of fuel
required, and the possibility near journey midpoint of running into a
molecule going fast enough to destroy your spacecraft. For more on
these issues, see Lagoute & Davost 1995 in Am. J. Phys. 63, 221.
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