Is that what relativity predicts? Of course not - special relativity predicts the opposite. A less dishonest Einsteinian would say:
"I can send you off on a mission traveling near the speed of light, and time would SPEED UP immensely for you."
Here are two less dishonest Einsteinians:
David Morin, Introduction to Classical Mechanics With Problems and Solutions, Chapter 11, p. 14: "Twin A stays on the earth, while twin B flies quickly to a distant star and back. [...] For the entire outward and return parts of the trip, B does observe A's clock running slow..." http://www.people.fas.harvard.edu/~djmorin/chap11.pdf
"The situation is that a man sets off in a rocket travelling at high speed away from Earth, whilst his twin brother stays on Earth. [...] ...the twin in the spaceship considers himself to be the stationary twin, and therefore as he looks back towards Earth he sees his brother ageing more slowly than himself." http://topquark.hubpages.com/hub/Twin-Paradox
It is particularly sadistic to brainwash the gullible world by teaching statements that contradict relativity:
Neil deGrasse Tyson: "One of the towering great achievements of the human mind in our understanding of the universe is Einstein's theories of relativity. It makes only two assumptions: That the speed of light in a vacuum is constant no matter who is doing the measurement and no matter in what direction you are moving or how fast. You always get the same measurement for the speed of light. That's Assumption 1 which by the way the experiment has shown to be true. Assumption 2: The laws of physics are the same for all observers in uniform motion relative to one another. Those are the only two tenets that you have to buy into. Given those two tenets, extraordinary spooky phenomena derive from them. For example: As you travel faster, time ticks more slowly for you than it does for other people who are not. The traveling twin when he comes back is only 5 years older, we have aged 110 years." https://www.youtube.com/watch?v=n2s1-RHuljo
Note the statement
"As you travel faster, time ticks more slowly for you".
This is the same pernicious lie discussed above. Actually special relativity predicts the opposite:
As you travel faster, time ticks FASTER for you.
The special relativity's prediction is wrong of course but at least it VALIDLY follows from Einstein's 1905 postulates. Neil deGrasse Tyson's statement is non sequitur - its presence makes any rational analysis of the twin paradox virtually impossible.