Here’s a totally impractical but fun back-of-the-envelope calculation from Bob Martin.
Suppose you have a space ship that could accelerate at 1 g for as long as you like. Inside the ship you would feel the same gravity as on earth. You could travel wherever you like by accelerating at 1 g for the first half of the flight then reversing acceleration for the second half of the flight. This approach could take you to Mars in three days.
If you could accelerate at 1 g for a year you could reach the speed of light, and travel half a light year. So you could reverse your acceleration and reach a destination a light year away in two years. But this ignores relativity. Once you’re traveling at near the speed of light, time practically stops for you, so you could keep going as far as you like without taking any more time from your perspective. So you could travel anywhere in the universe in two years!
Of course there are a few problems. We have no way to sustain such acceleration. Or to build a ship that could sustain an impact with a spec of dust when traveling at relativistic speed. And the calculation ignores relativity until it throws it in at the end. Still, it’s fun to think about.
Update: Dan Piponi gives a calculation on G+ that addresses the last of the problems I mentioned above, sticking relativity on to the end of a classical calculation. He does a proper relativistic calculation from the beginning.
If you take the radius of the observable universe to be 45 billion light years, then I think you need about 12.5 g to get anywhere in it in 2 years. (Both those quantities as measured in the frame of reference of the traveler.)
If you travel at constant acceleration a for time t then the distance covered is c^2/a (cosh(a t/c) – 1) (Note that gives the usual a t^2/2 for small t.)
Relativity says that no mass can reach speed of light.
So, ignoring relativity is a major setback here!
Relativity does allow things to travel at near the speed of light, though the nearer you get the more energy it takes to get closer. Particles in an accelerator move fast enough that engineers have to consider relativity when building the machines.
Um, no, that’s not how it works. You don’t accelerate to the speed of light and then time stops, rather as you get closer to the speed of light the more energy you expend accelerating further (from the point of view of the universe) or the more the universe around you slows down (from your point of view).
If you can only accelerate at 1 g, aren’t you stuck on the earth’s surface?
Yeah, that’s not right. But it does pose an interesting question. If you could accelerate constantly and then decelerate constantly (from your own point of view), how long would it take to traverse the observable universe (again, from your point of view)? Special relativity can handle this (http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html).
Time would never stop completely, but it would start slowing down the instant you set off. Is there an upper limit on how much time you’d experience in a long journey?
Marmaduke: I believe the update to the post with Dan Piponi’s comments answers your question.
Of course the approximate 2 years answer also ignores relativity. And the actual answer depends on the size of the universe anyway. The series “Cosmos” (Carl Sagan) actually says that you can cover the known universe in 50 years (in the moving reference frame of course). But as you say there are lots of tecnical problem, of which the “dust” collisions is not the most important (at some point in the velocity near the speed of light “dust” can traverse the ship without doing damage). The main problem is the “dimension” of the trip, probably you need the whole available energy of the universe to sustain constant acceleration. Besides you will reach an arbitrary point in the far future, probably after any kind of end of the universe. A trip like this will by itself fundamentally change the universe, because of the energy densities that it will generate.
Won’t the stuff at the edge of the observable universe have expanded beyond the cosmological event horizon and have become forever unreachable by the time you get where it used to be, even if you can travel around at c?
A more interesting question (to me) than “how much acceleration, to get anywhere in 2 years?” is “how long to get anywhere, at 1g?”. If I’ve applied Dan’s formula correctly, the answer is about 24.5 years.
Of course this ignores the issue of whether you need more energy than the universe contains, etc. (I haven’t done any calculations but my feeling is that alfC overestimates the difficulties here. You’re travelling at close to c for more or less the whole trip, which means the time taken is on the order of tens of billions of years, far from “after any kind of end of the universe”. And if the acceleration turns out to be expensive on the scale alfC suggests, I bet one can reduce it by several orders of magnitude with a pretty modest increase in subjective travel time.
(I see that there’s a 2x discrepancy between my figure and the one alfC says Carl Sagan quoted. I’ll leave that to people who actually know about this stuff to resolve.)
A few thoughts
1) I agree with g that it is a more interesting question to fix the acceleration and then look at the travel distance/flight time relationship.
2) If I take the distance to be traveled to be 45 Gly, so 22.5 Gly accelerating + 22.5 Gly decelerating, I get
23.8 years + 23.8 years = 47.6 years. About the Carl Sagan number.
3) I think that the energy required to accelerate the ship actually ends up just being ma*(0.5*d), just like a non-relativistic calculation. This is about 0.5kg*c^2 of energy per kilogram of ship mass per light year of travel. If you need to do your own work to slow down, double that.
Expensive way to travel.
4) The 45 Gly number is kind of silly… you’ll never get to the edge of the observable universe because of the expansion.