Crossing the Milky Way

I was pondering this thought: how long would it take to cross the Milky Way (our home galaxy) in a spaceship under a constant acceleration of one standard earth gravity (1 g or 9.8 meters per second per second). Imagine we start on the edge of the galaxy and accelerate to some final velocity at the opposite edge of the galaxy.

There is no known practical way to produce a 1 g acceleration for the long time required to make the trip, so this is a thought experiment or (from German) a Gedankenexperiment.

An initial velocity of zero makes the formulas we need very simple:

d = 1/2 * a * t²   where d is distance, a is acceleration, and t is time,

and

v = a * t   where v is velocity, a is acceleration, and t is time.

The constants we need are the speed of light (3.0e+8 m/s) and the diameter of the Milky Way galaxy (100,000 LY).

I’ll skip the math and get straight to the answers.

Accelerating at one standard earth gravity, the trip would last four and a half centuries. How fast would our spaceship be traveling after undergoing four and a half centuries of acceleration at one standard earth gravity?

The answer is: 1.362e+11 meters per second.

How fast is 1.362e+11 meters per second? It’s over 450 times the speed of light! Oops! Nothing can travel faster than the speed of light! Obviously, we need to account for the effect that traveling at relativistic speeds will have on the mass of the ship. The closer the ship’s speed gets to the speed of light, the more massive the ship becomes.

The mass of an object at relativistic speeds is computed as
m = m₀/((1 - v²/c²))^1/2 where m₀ is rest mass, v is velocity, and c is the speed of light.

Assume the ship’s mass is 1000 kg. As its speed increases, so does its mass. In the table below are some sample velocities with their associated relativistic mass.

Percent of light-speed	Mass
0	1000
10	1005
50	1155
90	2294
95	3203
99	7089
99.9	22,370
99.99	70,710
99.999	223,600

What we need is not constant acceleration but constant propulsive force. For our thought experiment, that force will produce an acceleration of one earth gravity at the start of the trip, but acceleration will decrease as the ship reaches relativistic speeds due to the increased relativistic mass.

Let’s re-compute. Our spaceship is going to cross our galaxy, starting with zero velocity, and with a constant propulsive force that produces a 1g initial acceleration. How long will the trip take? How fast will the spaceship be traveling when it reaches the opposite edge of the galaxy?

In order to take into account the constantly changing mass at relativistic speeds, the computation requires integral calculus. There is one more complication. There are two frames of reference: the stationary frame (centered on the galaxy) and the acceleration frame (centered on the spaceship). As the ship reaches relativistic speed, it not only becomes more massive but the passage of time slows aboard the ship. This time dilation is described by the Lorentz transformation.

With a constant force producing a 1g initial acceleration, the trip will appear to earthbound observers to take 100,000 years, but to an observer on the ship the trip will appear to take 11.8 years. Our final velocity will be 0.99999999995 of light-speed. The ship’s relativistic mass will be five orders of magnitude (100,000 times) larger than its rest mass.

Despite the optimists who create science fiction space adventures, it appears that galactic travel is not in the cards for humans. Even if we invent a way to travel significant distances in our galaxy, everyone back on Earth will be long dead by the time we reach our destination and return. (We have to accelerate outbound and also decelerate outbound, and then accelerate on the return trip, and decelerate as well.) Earth could send out galactic explorers but we wouldn’t hear from them for millions of years. And if it takes that long to hear from them, would Earth send them out in the first place? In a million years, humans will have evolved into something not human, or else we will have followed the dinosaurs into oblivion.