Time, matter, energy, and a lot of speculation.
Someone probably thought of this before, but even so, here is a bit of speculation probably loaded with wrong assumptions and horrible oversimplifications.
One of the odd effects of relativity is time dilation. As things speed up, they age more slowly as seen by an outside observer. At the limit, the unreachable speed of light, the aging should stop. Above the limit, the age should reverse. This is putting it a bit simply, but a clock moving faster runs slower as seen by others. This has a neat graphic explanation showing that if light speed is universally constant, time must do things in a manner people are not used to. And one of the results is that time travel (at least in the reverse direction) is impossible for us, at least without invoking folded spacetime or something even more exotic than the already mindblowing relativity.
But time travel is possible. We're all travelling in time right now, going forward at the standard rate of 1 second per second. That is we go forward one second in general (the first second) for second (the per second one) elapsed experience.
Ignoring how it works, lets suppose a time travel vehicle. To try to go back in time, the thing must start 'at rest' as far as we are concerned: one second per second. Then it has to slow down. Dropping through 0.9 seconds per second, and 0.8 and through 0.5 and so on we get to, say 0.1 second per second. Looking outside the vehicle one would see the world going by at 10 times normal speed and the world looking back would, for a fraction, see the occupant moving at one tenth normal speed. At 0.01 seconds per second the outside world is sped up one hundred times and the inside world is slowed to a hundredth normal speed, as seen from the outside.
But 0.1 or even 0.01 second per second isn't going backwards, it's just going forward slowly. What happens at zero seconds per second? Looking out, the world must move infinitely fast. *Poof* Welcome to the end of the world. Great. Now where, nevermind when, are you? You've run out of universe. Or something else strange happens - an infinite time gone by instantly. It's still a navigational nightmare of some kind.
And the world looking back sees... nothing. The vehicle has disappeared, stuck in time, not going infinitesimally slowly, but infinitely slow - stopped. The matter that was there isn't there anymore. But that can't happen. Matter is conserved. The only time matter disappears, a great lot of energy has to be in its place. But this isn't just at the time, so to speak, that the vehicle reaches 0 seconds per second. Even when slowing down, it's gone missing from where it should be.
Imagine a line of cars all travelling in the same direction at the same speed. If one slows down, even a little, it will leave a gap. It does not have to stop for the gap to exist. Slowing down is all it takes. But now the road is time.
The time vehicle did not turn into energy and cause a great explosion. The matter was not converted, it was left behind. But the matter-energy relation is still there. (Okkay, so I did just invoke relativity).) If the matter is missing, energy must be there in its place. Aha, the gotchya. The energy isn't coming out of the time vehicle, it's being pumped into that space to fill in the gap. This can be done for a little while, and things can be slowed down, for a little while. But to stop something would require that energy be pumped into that space for the rest of time. That's a lot of energy. That's too much. If there is no end of the universe as such, then it has be supplied in an amount to last forever. This is effectively an infinite requirement. If the universe doesn't last forever, then the amount is not infinite, merely huge, and for most purposes huge is as good (or as bad) as infinite.
Relativity explains it by saying that object grows more massive and thus takes more energy to move. Thus a particle accelerator can get things going very, very fast, but not quite reach light-speed as the particles get heavier and heavier the faster they get.
Either explanation gives the same result, that time travel in reverse is effectively impossible. The energy requirements just to stop, nevermind go backwards, are too high for it be workable.
Making the crazy assumption that somehow the chrononauts have managed to shift their vehicle into reverse, then what? Then the energy replacement has to be sent after them. As if an infinite energy requirement wasn't bad enough, now yet more must be used to send the energy itself backwards in time, rather than just expended as a current and future matter replacement.
Ah, but if the vehicle returns to the current time and stops, so to speak, again at 1 second per second, then the energy doesn't have to be supplied anymore. It looks tempting. Make the trip, come back, get away with it. But there are a couple problems. One is the passing through the stop when the world goes by infinitely fast so the end, if any, is reached instantly. The other is... the net energy might end up being zero (After all, if it takes energy to slow matter in time, shouldn't speeding matter in time (the return trip) generate a similar amount of energy? Not infinite, as it's not being removed from the past, merely sped up in the future.) but how do you get the universe to take an IOU?
There is something interesting right there. Sending matter forward in time at a rate faster than the usual one second per second should yield energy. What's happening? The matter is displacing itself, so to speak, or its energy equivalent in the future, that is the future as we see it. Assuming inertia, the mass will continue at the new rate unless acted upon. Having no idea what could act upon it in time, it might just coast at the new rate, or maybe feed of off its own displaced energy. (Yes it is getting rather wild here, isn't it?) And if not really accelerating, would the energy we see only show up at the time of initial acceleration? A sort of "chronic boom"?
Whatever happens, we only get the energy at the instant of departure. The matter is moving away from us in time and digging or plowing its trench with it - forward in time, away from us. This would be another reason going backward won't work: we'd have to keep dumping energy into the void to fill in for the matter, but the hole to pour it into has moved! Or rather, it hasn't moved. It's still at the same point in time and the world has moved on.
Is that how nuclear reactions work? Matter isn't really destroyed but sped up in time and the energy release is a payment for it? An instantaneous payment, as far as we can see, even if might be continuous as it travels on?
If nuclear energy really is energy from matter made time-fast, is the time-fast matter part of the 'dark matter' or 'dark energy' that has been so elusive? And where does it eventually get to, anyway? If there is an end to the universe, does it hit it and stop? Disappear.. into what? Bounce back time-reversed, doing even stranger things? And is there a more friendly way to speed up matter in time to get a "cleaner" energy supply? And as all the stars are nuclear reactors, what does that do for, or to, cosmology?
Someone likely has already shown all of this to be bunk. Or else is working on a grant about it. Or perhaps both. If nothing else, it might make for some interesting science fiction. Just what are all the effects of someone being knocked into the middle of next week?