The reason that there are time paradoxs with special relativity is that Einstein got it part wrong. Which part? In his parable of the train with the light beam bouncing up and down he mistakenly assumed that inside the train was a separate part of space-time to the outside of the train. In fact it is not separate and the observer in the train will see what the outside observer would see.. ie, the light beam zig zagging between the mirrors in the direction the train is coming from, the zig zags becoming greater as the speed increases. So, no time dilation due to speed. However, due to the tremendous input of energy to accelerate the train to near light speed the mass of the train would increase tremendously to where there would be time dilation due to gravity... a la general relativity. There are no paradoxs involved with general relativity.

So, we do not observe time dilation outside of strong gravitational fields?

 

Time dilation effects have been observed in various contexts and agree with either special relativity or general relativity if gravitational effects are significant.

 

There are some lovely recent results in

 

Optical Clocks and Relativity

C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland

Science 24 September 2010:

Vol. 329. no. 5999, pp. 1630 - 1633

 

A summary can be found here.

 

Anyway, there are no paradoxes in special relativity. Just a human attachment to 3 space and 1 time way of thinking rather than a 4-d space-time way of thinking.