The time for the person traveling near the speed of light slows down with respect to the time of those not moving.

The person traveling has no concept of this difference. What may pass as an hour or two for him may be days on Earth.

Plenty of experiments have been done to study and verify this. The US Naval Observatory send two atomic clocks on an around the world trip. One clock went east, the other west. The eastbound clock traveled faster than the westbound clock because of the Earth`s rotation. When the clocks arrived home, their times differed. The faster moving clock was behind the other clock by some measurable amount of time.

There is a formula for calculating this effect given the velocity, I just can`t remember it now. I`ll look through the old college books and see whatI find.