WARNING: This is a highly speculative writeup. If you insist on arguing from physics as known in the early 21st Century, you must conclude that FTL and time travel are pure nonsense. They may well be -- or not. Please, expand your mind beyond the known.
There are all kinds of objections to Einstein's Speed Limit: c, lightspeed, roughly 300,000 km/s. Most of them simply bow to Einstein -- to the General Theory of Relativity. Even the Special Theory seems to forbid FTL travel. An object with non-zero rest mass moving at lightspeed should have (as perceived by a relatively stationary observer) infinite mass and zero length in the direction of motion. A clock moving at lightspeed should appear to stop. This is clearly impossible. We don't want to have objects in our universe that outweigh everything else, nor do we wish them to be two-dimensional and eternal.
But what if Einstein is not the last word on physics?
Newton's theories sounded very good and stood for hundreds of years; nobody could find a better way to describe motion. But then Einstein extended Newton. He didn't actually say Newton was wrong, but he presented a much broader picture of the universe. Newton's theories are contained in Einstein's; they are shown as a useful approximation in the special case of objects moving at very slow speeds -- say, 1% of lightspeed or less.
What if somebody surpasses Einstein with an even broader theory? FTL travel -- and time travel, which is exactly the same thing -- may indeed be possible, at least in theory.
The problem of correcting for relative motion of celestial bodies is trivial. After all, when I drive from Chicago to New York, I don't want to go to where New York was when I left Chicago. But I manage to get there anyway.
The practical objection to time travel is the magnitude of c. If it becomes possible to treat time as we treat a spatial dimension, then we find it will take the same amount of energy to move one second into the past as it does to move 300,000 km in the direction of, say, Mars. (Quibble acknowledged.) A journey back to the 13th Century, comprising over 700 years, or over 22,000,000,000 seconds, is equivalent to a spatial journey of 6,000,000,000,000,000 km. You'll need a big gas tank on your time machine, and a plentiful supply of reading matter.
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Some people just know too much to enjoy the above writeup. I'll try to make it simpler. Picture the guy in his wife-beater, beering it up in front of the playoffs. At halftime, there's an ad from GM for time-machine-equipped Chevy Blazers. What Joe does not understand is that in order to drive his new ride *one* second into the past, he will need to drive over 186,000 miles. He will *not* be able to go back even as far as *yesterday* to call his bookie and bet the game. And his grandfather is safe.