A futuristic spaceship is stationed at its base, which is static at
10 light years (ly) from Earth in free space. The ship and base clocks
are rate adjusted to run on Earth time.
The ship now departs from base and accelerates outward along the Earth radial at 1g on its
accelerometer, for one year on its clock. Relativistic dynamics tells us that it achieves a speed of
0.76c relative to base and that the
acceleration lasted for 1.175 base years. I also says that the
ship was then 0.54 ly from base and 10.54 ly from Earth, all in base
inertial coordinates.^{[1]}
However, as observed in the ship's (new) inertial frame, that 10.54 ly Earth distance will be Lorentz contracted to 10.54(10.76^{2})^{0.5} = 6.83
ly. This means that in ship coordinates, despite accelerating away from
Earth for one year, Earth is 106.83 = 3.17 ly closer to the ship. What
is more, this decrease in distance occurred at an average rate of 3.17
ly per year, which is effectively 3.17c.^{[2]}
Paradoxical? Maybe. What do you think?
Regards, Jorrie
Notes (for the relativistically minded):
[1] Acceleration of 1g is very close to 1 ly per y^{2}. Note
that c=1 in these units. For 1g, the relative speed change is given by a
simple hyperbolic tan function: Δv/c = tanh(Δt) = tanh(1) = 0.76, where
Δt is the acceleration time as measured by the ship's clock.
In the base frame (X,T), time and
distance traveled during the ship's acceleration is: ΔT = sinh(Δt) and ΔX
= cosh(Δt)1 respectively. It is obvious that we are here dealing with hyperbolic
spacetime movement of the ship in the base frame. It is commonly known as Rindler coordinates. See https://en.wikipedia.org/wiki/Rindler_coordinates
[2] While we know
that it isn't strange to find 'funny' speeds during acceleration, the
initial and final distances were at least measured when the ship was inertial. It is true that after the acceleration,
the ship could not have instantly measured the distance to Earth. But
there could have been a handy inertial frame around, one that also moves
at 0.76c away from Earth. The ship's captain could simply have asked a
nearby stationary observer: "how far is Earth now?" The answer would undoubtedly have been 6.85
ly.

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