I’ve just finished reading Andy Weir’s new book “Project Hail Mary”. Great story, I heartily recommend it to anyone who is into hard sci-fi. There’s minor plot point that I found hilarious: an alien civilization managed to do interstellar travel before discovering relativity (it makes perfect sense in the story). The poor bastards were very confused: the trip was shorter and took much less fuel than they had planned for!

I was aware that relativity make trips shorter from the point of view of the traveller, but fuel? Never heard about relativity saving fuel as well. I just had to check it out!

First let’s loot at the time, though, as it is also rather interesting. Let’s say you want to travel to a star $L$ light-years away, and you have an extremely powerful and efficient rocket, so the only limitation you have is the acceleration your body can take. What you do is then accelerate at a constant rate $a$ up to half-way point, where you start decelerating at the same rate $a$, so that you actually stop at your destination. How long does that take? Luckily the Usenet Physics FAQ already has all the calculations, so I won’t do them again, I’ll just copy their results.

From the point of view of the Earth the number of years1 it takes is

\[t = \sqrt{L^2+4L/a}.\] You see it is always larger than $L$, which is the familiar result that you can’t travel faster than light. More interesting is that time it takes from the point of view of the traveller, the *proper time*:

\[\tau = \frac2a \cosh^{-1}(aL/2 +1 ).\] You’re probably not so familiar with the hyperbolic cosine, so it’s worth noting that $\cosh(x) \approx e^x/2$ for large $x$, which results in

\[\tau \approx \frac2a \log(aL+2)\] for large $L$. Yep, that’s a logarithm! It’s much, much faster from the point of view of the traveller. And those poor bastards that didn’t known relativity? They’ll calculate the trip time to be

\[t_\text{Newton} = 2 \sqrt{L/a}, \]which is indeed faster than the time from the point of view of the Earth $t$, but much slower than the time from the point of view of the traveller, which is what matters here.

For concreteness, let’s look at the time it takes to reach Tau Ceti, which is roughly 12 light-years away, at a nice comfy acceleration of one $g$. We have that $t=13.8$ years, $\tau=5.2$ years, and $t_\text{Newton} = 6.8$. Ok, not so brutal, but you’d definitely notice if you’d arrive at your destination one and a half years before schedule. This is a bit misleading, though, because this distance too small for the power of the logarithm to really kick in. If instead you go to the centre of the Milky Way, roughly 30,000 light-years away, you get $\tau=20$ years, $t_\text{Newton}=341$ years, and $t = 30,002$ years. Now that’s brutal.

How about fuel, though? I don’t think we’ll ever do interstellar travel with rockets, much more plausible is a laser-on-light-sail propulsion as investigated by Breakthrough Starshot, but it’s fun to do the calculations anyway. The Newtonian calculation is an easy application of Tsiolkovsky’s equation: for each kilogram of payload, we need

\[M_\text{Newton} = \exp\left(\frac2{v_e}\sqrt{aL}\right)-1 \]kilograms of fuel. Here $v_e$ is the effective exhaust velocity, which we take to be 1, as we are using an amazing photon rocket. The relativistic rocket equation is surprisingly simple: we just replace the time in Tsiolkovsky’s equation with the proper time. The result is that we need

\[M = \exp\left(\frac2{v_e}\cosh^{-1}(aL/2 +1 )\right)-1\] kilograms of fuel for each kilogram of payload. Using the same approximation for the hyperbolic cosine as before, we get

\[ M \approx (aL+2)^{2/v_e}.\] Now this is fucking amazing. We have an exponential in the Newtonian case, but in the relativistic case it’s just a polynomial! The exponential from Tsiolkovsky gets negated by the logarithm in the proper time. Even for Tau Ceti the difference is already extremely large: $M=204$ kg, whereas $M_\text{Newton} = 1,138$ kg! And for the centre of the Milky Way? $M \approx 10^9$ kg, and $M_\text{Newton} \approx 10^{152}$ kg. Ok, both are impossible, but the Newtonian one is much more impossible!

So don’t complain about the lightspeed barrier. Relativity actually makes life much easier for would-be interstellar travellers. A Newtonian universe would be a shit place to live.

Mateus,

This is really cool! Although, I’m a bit confused:

Beginning at rest and accelerating at 9.8m/s/s, an object would reach 3*10^8 m/s before 355 days:

((3*10^8)/(9.8))/(60*60*24) = 354.308…

So (to my limited knowledge) it seems that the 20 years estimate on the proper travel time to the center of the milky way is inaccurate. Wouldn’t that traveler exceed the speed of light before beginning to decelerate?

Thanks in advance.

I’m glad you enjoyed it. You’re using the Newtonian formula for the velocity. The correct, relativistic one, is given in the Usenet FAQ I linked in the post:

\[v = \frac{at}{\sqrt{1+(at)^2}}. \] Here $v$ and $t$ are the velocity and time from the point of view of the Earth, and $c=1$ as usual.