spacetime - How to travel between two stationary worlds in the least amount of time? (time dilation)

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?

Answer

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.