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Q: If time slows down when you travel at high speeds, then couldn’t you travel across the galaxy within your lifetime by just accelerating continuously?

Physicist: Yup! But sadly, this will never happen.

This is a good news / really bad news situation. On the one hand, it is true (for all intents and purposes) that if you travel fast enough, time will slow down and you’ll get to your destination is surprisingly little time. The far side of the galaxy is about 100,000 lightyears away, so it will always take at least 100,000 years to get there. However, the on-board clocks run slower (from the perspective of anyone “sitting still” in the galaxy) so the ship and everything on it may experience far less than 100,000 years.

First, when you read about traveling to far-off stars you’ll often hear about “constant acceleration drives”, which are rockets capable of accelerating at a comfortable 1g for years at a time (“1g” means that people on the rocket would feel an acceleration equivalent to the force of Earth’s gravity). However! Leaving a rocket on until it’s moving near the speed of light is totally infeasible. A rocket capable of 1g of acceleration for years is a rocket that can hover just above the ground for years. While this is definitely possible for a few seconds or minutes (“retro rockets“), you’ll never see people building bridges on rockets, or hanging out and having a picnic for an afternoon or three on a hovering rocket. Spacecraft in general coast ballistically except for the very beginning and very end of their trip (excluding small corrections). For example, the shuttle (before the program was shut down) could spend weeks coasting along in orbit, but the main rockets only fire for the first 8 minutes or so. And those 8 minutes are why the shuttle weighs more than 20 times as much on the launch pad than when it lands.

The big exception is ion drives, but a fart produces more thrust than an ion drive (seriously) so… meh.

In order to move faster, a rocket needs to carry more fuel, so it’s heavier, so it needs more fuel, etc. The math isn’t difficult, but it is disheartening. Even with antimatter fuel (the best possible source by weight) and a photon drive (exhaust velocity doesn’t get better than light speed), your ship would need to be 13 parts fuel to one part everything else, in order to get to 99% of light speed.

That said, if somehow you could accelerate at a comfortable 1g forever, you could cross our galaxy (accelerating halfway, then decelerating halfway) in a mere 20-25 years of on-board time. According to every one else in the galaxy, you’d have been cruising at nearly light speed for the full 100,000 years. By the way, this trip (across the Milky Way, accelerate halfway, decelerate halfway, anti-matter fuel, photon drives) would require a fuel-to-ship ratio of about 10,500,000,000 : 1. Won’t happen.

The speed of light is still a fundamental limit, so if you were on the ship you’ll still never see stars whipping by faster than the speed of light (which you might expect would be necessary to cross 100,000 light years in only 25 years). But relativity is a slick science; length contraction and time dilation are two sides of the same coin. While everyone else in the galaxy explains the remarkably short travel time in terms of the people on the ship moving slower through time, the people on the ship attribute it to the distance being shorter. The stars pass by slower than light speed, but they’re closer together (in the direction of travel). “Which explanation is right?” isn’t a useful question; if every party does their math right, they’ll come to the same conclusions.

Answer Gravy: Figuring out how severe relativistic effects are often comes down to calculating , which is the factor which describes how many times slower time passes and how many times shorter distances contract (for outside observers only, since you will always see yourself as stationary). Photon ships make the calculation surprisingly simple. Here’s a back-of-the-envelope trick:

If your fuel is antimatter and matter, then the energy released is E=Mc2 (it’s actually useful sometimes!). If the exhaust is light, then the momentum it carries is P=E/c. Finally, the energy of a moving object is γMc2 and the momentum is γMv. It’s not obvious, but for values of v much smaller than c, this is very nearly the same as Newton’s equations.

For a fuel mass of f, a rocket mass of m, and a beam of exhaust light with energy E, lining up the energy and momentum before and after yields:

So, when v ≈ c (when the ship is traveling near light speed), . That means that if, for example, you want to travel so fast that your trip is ten times slower than it “should” be, then you need to have around 20 times more fuel than ship. Even worse, if you want to stop when you get where you’re going, you’ll need to square that ratio (the fuel needed to stop is included as part of the ship’s mass when speeding up).

More tricky to derive and/or use is the math behind constant acceleration. If a ship is accelerating at a rate “a”, the on-board clock reads “τ”, and the position and time of the ship according to everyone who’s “stationary” are “x” and “t”, then

this is lined up so that x(0) = t(0) = 0 (which means that everyone’s clocks are synced when the engines are first turned on).

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