Because I see that you don't know much (if anything) about the actual time travel possibilities that Physics allows, you should start with the predictions made by General Relativity that have been shown to be correct. These predictions were made over a hundred years ago.
First, time travel due to time dilation.
Time dilation occurs anytime you move with respect to the inertial frame you were at rest in. Obviously, the effect is minuscule for ordinary velocities, but increases exponentially when your velocity approaches about half of light speed.
This prediction was confirmed over half a century ago by measuring the half-life of particles in an accelerator. The half lives of the particles (as they gained velocity) increased exactly as predicted by time dilation.
General Relativity also predicted the same effect due to gravity. This prediction was also proven to be correct, to the finest detail - the furthest decimal that they can measure to (significant figure.)
Gravitational waves, which were only first observed a few years ago, is another century-old prediction that has proven correct, but there's not much time-travel action in them (as far as I know.)
One prediction from General Relativity that hasn't been shown to be true is called an "Einstein-Rosen bridge." You may recall Jane Foster talking about them in the Thor movies. We usually call them wormholes. Here's a pdf about them:
http://u2.lege.net/cetinbal/pdfdosya/EinsteinRosenBridge.pdf
This should be more than enough for you to see that there are ways of time travel forward
and backward that are allowed by the current paradigm in Physics.
PS:
Regarding a time-like loop, didn't I say that you'd need dark energy? Maybe it's negative energy. Dark energy, negative energy, whatever. LOL
You're not asking me to google these things for you, are you?
Harte