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Let's imagine there are two, isolated, stationary worlds in space (called A and B), very far apart from each other. I live on World A, and some aliens live on World B.

I want to learn about the aliens on World B by talking to them in person. My lifespan is a quadrillion years, so I'm not worried about dying while traveling to them. However, I would like to see the alien civilization as close to its infancy as possible. In other words, I would rather see alien cavemen than alien astronauts.

If I travel too slowly, I give their civilization too much time to develop into astronauts—no good.

If I travel fast enough (close to the speed of light), time passes faster for World B than for me and my spaceship, due to time dilation (correct me if I'm wrong). Thus, I'm worried that if I travel too fast, time might pass so quickly for World B that they develop into astronauts before I arrive.

Am I right to worry about this? If so, what's the optimal speed to ensure that I arrive earliest in their civilization's development? If my reasoning is wrong and traveling faster is always better, then why?

There's no problem of traveling "too fast". It's true that the faster the space ship goes, the faster time will pass for the inhabitants of World B relative to how fast time passes for the spaceship traveler - but that's unimportant and not relevant to your question above. What your question is concerned with is the absolute amount of time that time passes on World B (regardless of how much time passes on the space ship) by the time the space ship reaches it, and the way to minimize that total amount of time that passes on World B is for the space ship to travel to it as fast as possible (i.e., as close to light speed as possible).

Suppose that A and B are at rest relative to each other (which you have) and in their mutual rest frame are separated by 100 light years. That means that no signal can travel from A to B (or vice-versa) in less than 100 years. Signals include optical or radio signals, which travel at the speed of light, and also material projectiles like spacecraft, which are slower.

So, if you leave in your spacecraft when you receive, at A, a signal that says "what to expect on planet B now that it's the year 2019," the earliest you can arrive at B is their year 2219. The message you got was old, and it takes time for you to arrive.

Time dilation has the effect of compressing the time in your trip. On your way from A to B, you'll receive 200 years worth of their news broadcasts: the 100 years' worth that were already in transit to you when you left, and the (at least) 100 years' worth that are emitted while you are en route. But if you travel with a relativistic factor $gamma=(1-v^2/c^2)^-1/2=100$, you'll only have about a year to study all of that news.

The time on the alien world at spacepoint B is exactly $t = fracB-Av$, when you travel with velocity v. The effects of special relativity come into account, when you look from one inertial system to another. This means that for the traveller the time moves on slower than for a person on earth, the difference is exactly the famous gamma factor $gamma = frac1sqrt1-v^2$ (note that I use natural units here with c = 1).

For a better visualization, you can imagine that a good friend of yours on planet A measures the movement of your spaceship. The time for him and world B moves on linearly. Only when he wants to observe you personally in the spaceship, he will see the effects of special relativity. So, as close to c as possible is the best choice. One other effect is that you dont age as much as resting people on the planets when moving that fast.

I hope that helped a bit.