Interstellar Travel Below The Speed of Light - I Do The Math For You

Introduction

The distances to the stars are vast, to say the least, and the technology for travel faster than the speed of light is nowhere in sight.

Does this mean that a human astronaut will never be able to get to the stars before he or she dies of old age? With the help of special relativity and the benefits of time dilation it just may be possible.

The nearest star system which contains Alpha Centauri is about 4.3 light years (ly) away and as we know a light year is the distance traveled by light in one year. Since the speed of light is about 300,000,000 meters per second (m/s) this means Alpha Centauri is 4.07 x 10¹⁶ meters away from us. That's a lot of meters.

Time Dilation

So do we need to go faster than the speed of light to get there within a reasonable time? No, as long as we can build a space ship that can accelerate at 1 g all the time. The term 1 'g' is physics short hand for the acceleration we feel due to the pull of gravity and it is measured to be about 9.8 meters per second per second (m/s²). That is, if you fell from an airplane, after 1 second your speed would be 9.8 m/s. The next second you would add another 9.8 m/s to your speed, and on and on (until you hit the ground, and this example also ignores air friction)

So let's assume that we have the technology where we can accelerate a space ship at 1 g without end. The speed will build up and eventually you will start approaching the speed of light. Due to the effects of special relativity, more specifically the effect of time dilation the time on board the ship will slow down relative to us here on Earth. The closer you get to the speed of light the more it slows down.

The other effect that needs to be remembered is that the speed of light cannot be exceeded. To us on Earth the ship will still be accelerating but at a continuously reduced rate such that it will never break light speed. To the astronauts on the ship it will appear to them that they are continuing to accelerate at 1 g but the effect of length contraction will make the distance to Alpha Centauri appear to be shorter and shorter.

The ship will therefore have to accelerate at 1 g for one-half of the distance to the star, then turn around and decelerate at 1 g for the remaining half of the distance reaching a full stop at their destination.

The Calculation

I plugged all of this into an Excel spreadsheet taking into account both time dilation and the summation of velocities at relativistic speeds. The variable t is time, d is distance, v is velocity and a is acceleration. If there is a parameter with a prime on it that means it is in the ship's frame of reference (and without a prime it is in Earth's frame of reference). Gamma is the time dilation factor.

After doing all that work I arrive at an answer of about t' = 354 days (this is ship time) to reach the half way point to Alpha Centauri which is 2.15 light years (ly). They would then turn the ship around and decelerate at 1 g for the next half of the trip giving a total of about 700 days (or 1.9 years) ship time to get there.

On Earth it would look to us like it took them t = 2.7 years to get to the half way point. This mean that it would take 2.7 years times 2, or about 5.4 years, to get to Alpha Centauri (ignoring signal propagation time of course).

Closing Words

Not being able to travel faster than the speed of light is not a barrier to interstellar travel. Getting a large ship up to a decent acceleration and finding the energy to do so for long periods of time is the real barrier.

If these technical hurdles are ever overcome by humanity then one day it may actually be feasible to get humans out into the stars.

Thanks for reading my post. Have a nice day.