Very Fast Speeds
If you recall, special relativity had an expression for the relationship between proper time, Tau, and the coordinate time, t. Let us now examine that relation.
As you can already see the time interval that the a stationary observer
measures for the moving observer is dependent on the relative velocity, v. Imagine that
two stationary observers are on Earth, lets name them Jennifer and Jan. Then imagine a
moving observer, Bob, is aboard a space ship flying away from Earth. Any time period on
Bob's ship will be measured by Jennifer and Jan to be longer than Bob will measure it.
This effect grows larger as Bob's velocity increases. This is known as time dilation. The
dilation grows until a critical velocity is reached, the velocity of light. At the speed
of light any event that occurs on Bob's ship will appear to take forever to Jennifer and
Jan. If Bob's velocity exceeds c the time dilation factor becomes imaginary and Bob would
actually be moving backward in time. However, there are other constraints that limit Bob
from reaching c. As v increases mass also increases by the relation 
. So Bob's mass would become infinite
at c. Also it would take an infinite amount of energy to accelerate Bob to c. So Bob, and
any other particle with mass, may get arbitrarily close to c but may never reach or exceed
it. There is one loophole in the equations that does allow things to be traveling faster
than c. If such a particle existed it would have to have been created moving faster than c
and it could never travel at c or slower. Such theoretical particles are called tachyons
and would travel backward in time. But Bob is not a tachyon so there's no hope there.
But Bob has indeed become a time traveler. Suppose that Bob reached a speed of .99c at some point and maintained it for 5 years and then decelerated, which is not so easy to do since Bob has a gigantic momentum. Bob then decides that he is just way too far from home so he heads back to Earth to visit Jennifer and Jan. Again he reaches .99c and maintains it for 5 years. 10 long years have passed for Bob on that space ship but he looks forward to seeing his friends Jennifer and Jan. When he sees them he finds that they are very old women now. 70 years have passed on Earth since Bob's departure. Bob has made a trip to the future. Can Bob get back to the time he left? We'll find out in good time.