This is a subject I spend a LOT of time studying - kind of an obsession of mine.
- Travelling faster than light does not make you go backwards in time. The entire idea comes from Lorentzian transforms - time and length shrinks, mass grows.
t’ = t * sqrt(1 - v^2/c^2)
As your velocity (v) approaches the speed of light (c), v squared approaches c squared and the ratio approaches 1. One minus one is zero. Square root (sqrt) of zero is zero and local time times zero is zero relative time.
Simple really - until you go FASTER than the speed of light. Then things get… mm… imaginary.
Then the v squared over c squared part is MORE than one, and one minus it becomes a negative number (which is why people keep thinking this means ‘time goes backwards’) except they forget the ‘square root’ part… and there is no square root of a negative number. Which means the equation doesn’t say ‘going faster than the speed of light means going backwards in time’ - it means ‘you can’t do that’.
Consider length - which is governed by the same equation. This interpretation of Lorentz’ equations would imply that if you’re going faster than the speed of light, you’d have a negative length, which is literally meaningless.
- The Thorne time machine doesn’t work. The original idea for this came out of a phone call from Carl Sagan to Kip Thorne asking for help in the writing of the book Contact. Thorne had grad student work out the math and to their surprise, they realised that you could, in theory, make a time machine. Cecil’s description of it was more or less right - but, ended a little to soon. Sometime later, they ran the numbers again and realised there’s a problem: the amount of radiation created in the wormhole being held open would basically be so huge that anything, even an atom, trying to pass through would be wiped out.
There’s also a bigger issue - moving one end of the wormhole doesn’t actually make it stick in time - it slows the clocks. It’s a subtle but important distinction. Imagine you have a spaceship and it’s travelling to Alpha Centauri (around 4.2 light years away) and you’re doing .5c (half the speed of light). Plugging that into the equation above and you get: .866 which means time is slowed to 86.6% of the time experienced by someone ‘at rest’. Since you’re going 1/2 the speed of light - it takes you 8.4 years to get to Alpha Centauri and indeed - that’s exactly how long it will take… but by your clock, only 7.2 years will have passed. They open up your ship, look inside and to them, you’re in the ‘past’ - except you’re not. You’re there now - with clocks that are wrong.
If you built a Thorne time machine, you wouldn’t go back into the past, you’d just see the local clocks get further and further off as you travel down the wormhole. The time at the other end would be the time you started plus transit time.
- Baryon numbers. This one is going to sound weird - but when the universe was created in the big bang, a certain number of particles (quarks) were created. Most of them were annihilated into photons in the first second - but, for reasons we don’t entirely understand yet, a little bit survived - which is us. But from that moment on, you cannot create new matter out of energy without maintaining the baryon number - it’s like bookkeeping for the universe.
If you went into the past, you’d change the baryon number here and in the past. Which isn’t permitted.
- Temporal asymmetry. For a long time, physicists were sure time was symmetric - that anything that happens forward in time can be run backwards. Except we now know that’s wrong. B and K mesons have decay modes which are NOT symmetric - they can only work one way - forwards. This strongly suggests that the universe has a ‘time handedness’.
Interestingly, we also used to believe that the universe couldn’t tell the difference between matter and antimatter until we discovered that weak interactions can tell them apart. Neutrinos also have a ‘handedness’ which break symmetry.
- Supertime. If you can travel in time - you have to travel IN some additional time axis. The action of movement implies a temporal axis. Movement is distance over time… and in this case, the distance IS time, so you need time over time… they can’t be the same time axis, so we’d need another time axis.
There’s no evidence such a thing exists.
- Information theory. Travelling in time - especially backwards in time - violates information theory and causality. It creates a situation where the creation of information is the information itself (consider: invent something - then go back into the past before you invent it and tell yourself how to invent it… where did the information come from?)
I can actually give you a ton more reasons time travel is impossible, but I’ll give you one last thing to think about. It’s not an argument for or against time travel, rather a caution.
To travel into the past or into the future - there’s an assumption that there’s somewhere to travel to. If the past exists in a concrete sense then it’s either immutable and fixed because it’s happened, or it’s variable. If it’s variable, then quantum mechanics would cause it to change continuously - yet reality seems pretty stable, which suggests it’s immutable… but if it is - then the same has to be true for the future (because for any point in the future - everything before it is the past and has to comply with the previous statement).
Now the scary part. If it’s immutable - then we have no free will. Everything we do is predetermined… even me writing this post. I’m not doing it because I choose to - I do it because this is what is supposed to happen right now. Even me thinking I have a choice as to what’s supposed to happen isn’t my decision - it’s what’s written into the universe at this point in the space time continuum.
So… be careful what you wish for. 
