WAIT A MINUTE! Shouldn’t we switch
to general relativity when dealing with accelerated systems? Not
necessarily. If you accelerate and decelerate along a straight
trajectory that joins two star systems and ignore the curvature of
space caused by other objects, you are fine with special relativity.

This article tells you how to calculate
the time spent by a subluminal (i.e., no warp drives!) star ship
constantly accelerating half of the way towards its destination and
then constantly decelerating during the second half of its voyage.
Its purpose is to support Science Fiction writers who need to write
about interstellar travel. I adapted the formulae from an article from the University of California Riverside and,
for fun, I re-obtained them from the standard Lorentz transformations
for length and time. Initially, I thought I would also explain how it is done, but it would have been a bit too complicated for most
people. If you are curious, I found a paper from the University of Leipzig
and another article from UCR to
be useful.

First of all, let’s define some
terminology:

- ‘a’ is the acceleration of the ship measured on the ship itself. Technically called the
*proper acceleration*of the ship, which is the acceleration felt by the passengers. That is, what an accelerometer placed on the ship will measure.

- ‘D’ is the distance between the point of departure and point of arrival, measured when the ship is moving at a speed much lower than the speed of light and with its engines off. Basically, you can take it as the distance we would measure from Earth.

- ‘T’ is the time needed by the ship to make its journey, as measured on Earth. Earth orbits the Sun at 30km/s and the Sun moves at 370 km/s with respect to the cosmic microwave background. But we can ignore these speeds, because they only represent some 0.1% of the speed of light. On Earth, we are also subjected to its gravity and both Earth and the Sun move on curved trajectories, but these accelerations can also be ignored for our purposes. I just read that special-relativity effects slow down the clocks on GPS satellites (orbiting at 20,000km above sea level and travelling at one orbit per 12 hours, or 3.83km/s) by 7μs/day, while general-relativity effects (Earth’s gravitational force is much weaker up there) speed them up by 45μs/day.

- ‘t’ is the time needed by the ship to make its journey, as measured on the ship.

Here we go. Let’s start with the
time measured on Earth. This is given by:

T = 2 sqrt[(D/2/c)

^{ 2}+ D/a]
To make our life easier, we will
measure time in y (year), distances in ly (light year, the distance
light covers in one year), and speeds as fractions of c (the speed of
light ~300,000km/s). In this units, g (gravitational acceleration on
Earth’s surface, 9.81 m/s

^{2}) turns out to be 1.03 ly/y^{2}. With this choice of units, c disappears from the above formula, because c = 1.
For example, let’s suppose that we
want to reach Proxima Centauri, the nearest star to our solar system
(4.24 ly) and that our ship can sustain the acceleration of 0.1g. For
the people left back on Earth, the journey will take:

T = 2 * sqrt[(4.24/2)

^{ 2}+ 4.24/(0.1 * 1.03)] = 13.51y
With an acceleration of 1g (ten times
higher), still from the point of view of Earth-bound people, the journey
would take 5.87y (you only need to remove the 0.1 from the above
expression).

The time measured on the ship is given
by:

t = c / a * 2 * arcsinh[a*T/c/2]

With c = 1, the formula becomes:

t = 2 * arcsinh[a*T/2] / a

arcsinh is the inverse function of the
hyperbolic sine. You’ll probably find it in Excel (haven’t
checked). I have it in the calculator application on my Mac when I
set it to scientific mode.

So, how older do the passengers of our
ship become when they travel to Proxima at 0.1g and 1g? You only
need to plug a and T into the formula and obtain 12.61y and 3.55y.

Not a big deal, is it? In case you are
wondering, the top speed, when the ship is half a way to Proxima and
switches from 1g of acceleration to 1g of deceleration, is given by:

v = a* T / 2 / sqrt[1 + (a*T/2/c)

^{ 2}] = 1.03 * 5.87 / 2 / sqrt[1 + (1.03*5.87/2)^{ 2}] = 0.95c
If you go to Tau Ceti, a star similar
to ours that is 11.9 ly away, you get, for 1g acceleration:

T = 23.98y

t = 6.23y

v = 0.9967c

Now the differences become more
significant. Still, you would have imagined a more dramatic
difference, wouldn’t you? I did.

OK. Let’s look at the planet HD
40307g. It is the latest Earth-like planet discovered. It might
have a gravity twice as strong as Earth’s, but it orbits a star
slightly cooler than ours with a 200-day period. It also seems that
it rotates on its axis, which would imply a day-and-night cycle. It
could have liquid water and be able to sustain life. Its distance
from us is 42 ly.

T = 43.90y

t = 7.40y

v = 0.9990c

Well, here the twin paradox is
definitely dramatic. After a round trip, the twin on Earth would be
2 * (43.9 – 7.4) = 73y older.

Now, what type of propulsion could
possibly accelerate a ship at 1g for more than seven years? You tell
me!

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