How far could I travel in a lifetime if I had a constant acceleration of 1 g?
Short answer: Farther than any distance you could possibly imagine.
Long answer: This is a really interesting take on acceleration. After all, you’ll spend most of your life experiencing a g-force exactly equal to 1, just by virtue of being on the surface of the earth in earth’s gravity. The difference is that there is an upward force exerted on you by the ground which exactly cancels gravity, so you don’t accelerate.
But what if you were accelerating? What if you spent your entire life in a space ship that was accelerating away from the earth, starting from the moment you were born? Never mind the feasibility of such a spaceship. That’s a problem for the engineers, and we’re theorists. Imagine you were born, your parents dropped you in a spaceship, and shot you into space.
It turns out that if you were accelerating like you were dropped in earth’s surface gravity, but without ground to stop you, you would be going the speed of light in a little less than a year according to Newtonian mechanics. Of course, that’s not actually possible; the speed of light is a hard limit for massive objects. In your reference frame you’re accelerating uniformly, but in the earth’s reference frame you’ll only asymptotically approach the speed of light (read more about that here). Anyway, the whole point of that was to convince you that we need special relativity for this problem.
It’s a common misconception that special relativity can’t handle accelerations, and that accelerations fall under the purview of general relativity. In reality, special relativity can handle accelerations but the math is much more complicated, and we have to be very careful about whose time and whose distance we’re considering. After all, special relativity tells us that times will dilate and lengths will contract in different frames. Because of the frame dependence all the distances and times below are going to be given in the earth’s frame, except for the age of the passenger on the space ship.
So let’s get moving.
After one year on the ship, you will have gone 0.56 light years. That gets you about halfway to the edge of the Oort Cloud. The solar system is far behind you. The sun is still probably the brightest star in your sky, but it’s starting to look more like just another star rather than the star.
In two years your time, you’ve traveled 2.91 light years. That’s crazy, and it might make you think that you’re traveling faster than the speed of light, but this trick is due to time dilation and length contraction of your frame. At this point, you’re closer to the Centauri system than to the sun and earth, and although you’re only two years old on your spaceship nearly three years have passed on earth.
By your fifth birthday you’ve traveled 83 light years from earth. You’re now farther away from the earth than several hundred stars. About 84 years have passed on earth, so your parents are probably long dead along with any identical twin you may have left behind. By your tenth birthday you’ve traveled 14,700 light years. That’s almost a third of the radius of the galaxy. If you chose to travel up and out of the galactic disk, then the galaxy offers a spectacular view underneath you, like no human has seen before.
In twenty years you’ve traveled 440 million light years. The local super cluster is receding in your rear view mirror, and the next few years of your life start to get weird. Some time around your 23rd birthday, in the earth’s reference frame, the sun will swell into a red supergiant consuming the earth, but you’re too far away to see it. The sadness is short lived, as the sun will have contracted to a white dwarf by your 24th birthday. By now you’ve traveled 77 billion light years from earth — welcome to the other side of the observable universe. At this point, I’m out of helpful highway markers to tell you how far you’ve gone. There’s simply no sense of scale for comparison that’s larger than this, and if you noticed how fast the numbers blew up in the past three paragraphs you can imagine how much bigger these numbers will get if you keep on the gas for the rest of your life.
So I guess we’ll have to make do with some really big numbers.
If you’re an average man in the US, you can expect to live 76 years. If you spent your entire life on this journey, you can travel a distance 1022 times the current size of the observable universe. If you’re an average woman, you live 5 years longer, which lets you cross the universe about 1025 times. But what is the universe like when you die? It is billions of times older than its present age and the stars have all burnt out, leaving only dwarfs, neutron stars, and black holes, as the universe grows dark and cold for all eternity.
The moral? If you’re brave enough, no distance is too far.
asked by /u/tyy365
image credit: Wikimedia Commons
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