Why is motion in spacetime impossible? It has to do with the definitions of space and time and the equation of velocity v = dx/dt. What the equation is saying is that, if an object moves over any distance x, there is an elapsed time t. Since time is defined in physics as a parameter for denoting change (evolution), changing position from one point in a time dimension (time axis) to another is self-referential. Why? Because the equation for velocity along the time axis would have to be v = dt/dt which is meaningless, of course. It is logically impossible for the t coordinate of an object to change in spacetime. Et voil´! It's that simple. No time travel, no motion in spacetime, no spacetime and no time dimension.
Notorious Spacetime Crackpots [1]
I've always been unconfortable with the notion of time being considered the fourth dimension. You just don't have the freedom of motion as with the other three spatial dimentions and that wierd things start happening if you reverse time or otherwise can affect the past. The fourth dimention as another spatial dimention I can live with.
But I'm not about to go calling Richard Feynman a crackpot though!