Let's say you're traveling at 99.5% the speed of light. That's 299792458 meters per second, or 185351 miles per second. You're heading straight towards Proxima Centauri, the nearest star besides the Sun. Proxima Centauri is about 4 light years away, meaning that it takes a beam of light four years to get from here to there. 4 light years is about 2.3 x 1013 miles.
Large numbers, eh?
It might seem as if you would get to Proxima Centauri in about 4 years. After all, you're practically like a beam of light yourself now. But things are not so simple when we get close to the speed of light. To continue with our calculations, we will need something known as the Lorentz factor. The Lorentz factor is just some number, symbolized with (the letter gamma).
= 1/sqrt( 1 - (v/c)2)v is the velocity, and c is the speed of light. In our example, is approximately equal to 10.
One of the things that happens when you travel near the speed of light, is that all distances in the direction of travel get shortened by a factor of . The Earth will seem like a squashed spheroid to you. It will also seem as if Proxima Centauri is not 4 lightyears away, but 0.4 lightyears away. So you will get there in 0.4 years without ever breaking the speed limit set by Relativity!
I would be on Earth, watching you. I would see you arrive at Proxima Centauri 8 years later. From my perspective, it took you 4 years to get there, and 4 more years for your signal to return to me. Let's say that you decided to return home, and sent a message to me saying so. It would take the message 4 years to get to me, and you will appear about one week later, since you travel at nearly the same speed as your own message.
By the time you returned, about 8 years would have passed on Earth. From your perspective, the entire trip only took 0.8 years. From your perspective, Earth has skipped ahead in time by about 7.2 years. From my perspective, it seems like you hardly aged over that long trip.
Let's say that you're about 100 kg (220 lb). Once you start moving near the speed of light, your mass increases by a factor of , so you will appear to be 1000 kg. Since your mass increased by 900 kg, guess how much energy it required to mobilize you? That's right, E=mc2. In your case, this will be 8.1 x 1019 Joules, or about twenty thousand megatons. And that's just you by yourself. If you're in some kind a space ship, the space ship might carry about a million more megatons of energy. You would probably make quite an explosion if you hit one of Proxima Centauri's planets, if it has any. Not that we were planning to do anything like that.
You and your spaceship will appear to be flattened by a factor of . You will seem as flat as a pancake. You will probably look quite absurd, a massive, flying, pancaked person. What's more, your colors would be all shifted. When you're flying away, you will be redshifted by about a factor of 20. Basically, I will only see what little ultraviolet light you emit. But when you're flying towards me, you will be blueshifted by a factor of 20. Much of that infrared light which you normally radiate will now be blueshifted into visible light and x-rays. I'm not sure what color you would appear, but I estimate about the color of a star of 6,000 degrees, like the sun--a yellowish white.
You, while on your journey, will see something even more exciting. Everything behind you will be redshifted by a factor of 20. You would see the extreme ultraviolet range. In the ultraviolet range, most stars will seem much dimmer, except for the hotter ones, including white dwarfs. If you look in front of you, light is blueshifted by a factor of 20, so you will see part of the middle infrared spectrum. You can see giant dust clouds and nebula in this range. Some of these dust clouds are where stars will form. And as you swing your head around, looking from front to back, you will see all the ranges between middle infrared and extreme ultraviolet. I'm sure it would be quite an experience.