Time dilation and time travel have always fascinated me since I watched Carl Sagans Cosmos as a kid but as I tried to better understand it a couple of things about the formula to calculate the amount of time dilation have always bothered me.

Simply stated the formula is:

te = ts / sqr( 1 - (vs^2 / c^2))

where te is Earth time, ts is ship time, vs is ship velocity, and c is speed of light.

If traveler's on the ship were to try to calculate the amount of time dilation in regards to the Earth the traveler's would have to divide their time on board by a square root that involves their velocity. When you figure the square root of anything you intuitively get a positive result but forget that there is always a corresponding negative result. Employing this logic a traveller on the ship can either determine that a longer amount of time passed on Earth than on the ship or the traveler can figure that the ship moved backwards in time in relation to the Earth.

Is this a paradox?

My second concern regarding this formula involved the apparent velocity of the ship as experienced on board the ship and observed by the Earth. If a traveler on the ship and an observer on the Earth both use this formula to calculate the ship velocity by observing each others clocks then fastest velocity the ship can achieve is something in the range of 70.71% the speed of light. I figured this out when I was a teenager programming an old Texas Instruments computer. I ran a simple program that stepped up the ship velocity and compared the apparent ship time and apparent Earth time and when the velocity passed the 71% mark an observer on the Earth would see the ship time passing so slowly that it would appear frozen. Likewise, a traveller on the ship would observe the time on the Earth passing so fast that they would think the Earth aged, died, and turned to dust.

My second concern may seem a bit counterintuitive but if you consider that the denominator in the formula starts to approach zero then you can see that the apparent Earth time begins to increase exponentially.

Is this another paradox?

I understand that Einsteins Special Theory of Relativity supports the idea of time dilation and I do not know what advances have been made in this field recently but considering these two problems I believe that there is much that we do not understand about objects approaching the speed of light.