How long would it take to reach the edge of the reachable universe?

Question

How long would it take to reach the current edge of the reachable portion of the universe, with the following bounds in mind:

• We figure out a way build a space ship to provide a constant 1 G acceleration for an extended period of time.
• We accelerate until we are halfway there, then decelerate past this point (so we stop when we get there.
• Our target is about 15 billion lightyears away. As per http://curious.astro.cornell.edu/about-us/104-the-universe/cosmology-and-the-big-bang/expansion-of-the-universe/616-is-the-universe-expanding-faster-than-the-speed-of-light-intermediate galaxies past 4,740 megaparsecs are no longer reachable. Using this site http://www.convertunits.com/from/megaparsecs/to/light+years to convert megaparsecs to lightyears means this "unreachable" point is currently 15,459,812,396.5002 lightyears away. Let's simplify to 15 billion lightyears, which is still supposed to be reachable.
• We somehow work out all the details of life support, collision with particles at relativistic speeds, etc...

I would like to know the time the traveler experiences, and the time that everyone else (e.g. those on earth) experience.

Answer 1

Jonathan's answer is essentially correct, but as Rob Jeffries comments, he doesn't take into account that the Universe is expanding during the journey.

The edge of the observable Universe is 47 billion lightyears (Gly) away. Even if you are a lightbeam, you cannot reach that point. The farthest you can go if departing today is roughly 5 Gpc, or 17 Gly, but this journey would of course take infinitly long (or else it wouldn't be "the farthest you can go"). This distance is probably what the linked article is referring to (I didn't read the article; it's very, very long).

So, in order for the answer to be any fun, you have to freeze the Universe, using magic, which is what Jonathan's calculator is doing. Here I'll just provide the analytical solution: In that case, the proper time $\tau$ (i.e. the time experienced by the traveler) to reach a distance $x$ when traveling at a constant acceleration $a$ is $$ \tau = \frac{c}{a} \cosh^{-1} \left( \frac{ax}{c^2} +1 \right), $$ where $c$ is the speed of light. If you wish to decelerate after having reached halfway, you just divide $x$ by $2$ and multiply the result by $2$.

If you plug in the $x=15\,\mathrm{Gly}$ you request, you get roughly 45 years. To get to the edge of the Universe at 47 Gly actually only takes a few years more. The reason for this is simply that traveling at 1G gets you to (almost) the speed of light in only a couple of years, and hence you experience (almost) no time, no matter how far you go.

The time experienced for the Earthlings for the traveler at constant acceleration is given by $$ t(\tau) = \frac{c}{a} \sinh \left( \frac{a \tau}{c} \right), $$ which works out to 15 Gyr for the 15 Gly, and 47 Gyr for the observable Universe. The reason is simply that the traveler, from the point of view of the Earthlings, extremely fast reaches a speed which is almost the speed of light.

Answer 2

I welcome everyone else's input on both this answer and possible different solutions! This is what I came up with:

First off, providing a 1G thrust the whole way (switching to decelerating half way there) would be a great way to provide "artificial gravity". I found a site that actually has scripts to help with such calculations! Assuming the site is correct, here is what we come up with: http://www.cthreepo.com/lab/math1/ Using the Long Relativistic Journeys calculation:

• Enter acceleration: 1G
• Enter light years to travel: 15000000000
• Time experienced by observer: Infinity (incorrect, and the site points out that due to a limitation in the java script, such large numbers do not calculate properly)
• Time experienced on Earth: 15010384285.855611

However, I noticed something interesting that might still get us there:

• For a trip of 15 light years, it shows the observer experiences about 5.54 years, and the earth time experienced is about 16.84 years.

• For a trip of 150 light years, it shows 9.80 years for the observer, and 152.03 years for earth

• For a trip of 1500 light years, it shows 14.24 years for the observer, and 1502.97 years for earth

• For a trip of 15000 light years, it shows 18.70 years for the observer, and 15012.32 years for earth

I am noticing a pattern here... The time for the observer increases by about 4.5 years every time the distance is multiplied by 10. The time experienced by earth grows closer to the number of light years as distance is increased too. Now, let's see where the calculation breaks down, and see if this pattern is still happening before that point.

• It breaks down at 1.5 billion light years, so lets take a few steps back and see if we still see the pattern.

• For a trip of 1,500,000 light years, it shows 27.63 years for the observer, and 1501040.36 years for earth. Note that commas were added to easier see what number we are talking about, when entering numbers into the calculation, don't use commas.

• For a trip of 15,000,000 light years, it shows 32.10 years for the observer, and 15010386.22 years for earth.

• For a trip of 150,000,000 light years, it shows 35.61 years for the observer, and 150103844.77 years for earth.

Yes, the pattern still largely holds. It seems be about 3.5 - 4.5 years for each additional factor of 10 for distance (and seems to be getting less time "near the breaking point"). Let's assume based on this that each factor of 10 increase produces an approximate 4 year increase on the observer's time, and that the time experienced on earth is very close to the number of light years travelled.

• For a trip of 1.5 billion light years, this would mean 39.61 years experienced for the observer, and slightly over 1.5 billion years experienced on earth.

• Now for the big one and our answer, a 15 billion light year trip should take about 43.61 years for the observer, while slightly over 15 billion years would be experienced on earth!

This has some astonishing implications:

• With such a ship, we could theoretically travel to the end of the reachable universe within a human lifetime (though 15 billion years would have passed on earth)!
• We could also theoretically reach any point within this area in less time (so we could travel practically anywhere, to a neighboring galaxy, to a galaxy off in the distance, to the center of our galaxy, etc...) within a human lifetime (in less than 44 years)!
• We could time travel into the distant future by going out somewhere very distant at relativistic speeds, then coming back to earth! Note: going back in time still hasn't been figured out yet as far as we know. For large distances, the time travelled would be slightly more than double the distance in light years travelled (accounting for the return trip).