I want to make a world similar to ours, with one additional feature: Time travel. There is no difference to me if its loop-based or infinite universe based. But there is a catch: The theory of relativity has not been disproven, disallowing faster-than-light travel.
Is there any viable way to do this that's consistent with modern science (no matter how unlikely, say quantum stuff)? I'm talking both-way time travel here, both past and future.
General Relativity does not disallow time travel nor does it disallow travel to the stars in times quicker than light can get there. Both theories are "local" theories and their symmetries only apply locally. That may sound simple, but it's a fairly technical statement. Basically, the prohibition on FTL relative velocities applies only at each point in space: At a given point in space, no two objects can have a relative velocity exceeding that of light.
But in the currently accepted theory of cosmology -- which is 100%-compliant with Relativity -- there are galaxies receding from Earth at speeds greater than that of light due to the expansion of the universe.
There are several ways to time travel allowed by Relativity, none of them easy.
First, there's the Tipler Cylinder. Construct an very long, very massive cylinder and spin it so that the surface speed is near the speed of light. General Relativity says that there will be paths that can be followed near to the spinning surface which (among other things) travel backwards in time.
(The construction of infinitely long cylinders that hold together when spun up to light speed, the source of energy to spin it up, and the construction of space ships which can fly near it are left as an exercise for the reader...)
Second, there are wormholes. Create a stable wormhole pair (again, left as an exercise for the reader). Take one of the pair and put it in a relativistic spaceship (LaaEftR) and fly off to the stars and back (LaaEftR) so that that the people on the ship have experienced ten years less time passing than those on Earth. The two end of the wormhole pair are now a time machine that allows you to go through it (LaaEftR) ten years into the future or ten years into the past.
There are lots of similar constructs permitted by General Relativity: spacial tubes, the Alcubierre Drive, etc.
It's noteworthy that all of them share one characteristic: We have no idea at all how to actually build one. They invariably require material of outrageous strength, outrageous properties ("exotic matter"), or outrageous amounts of energy (usually all three). It's not that it's an engineering challenge to build them, but we can't even begin to suggest how to do it -- it's completely outside any plausible extrapolation of current technology.
Other that that, no prob!
If time travel does exist the only thing I can think of that would prevent it being used for FTL is an application of Causality Paradox such that attempts to use time travel for the express purpose of FTL would fail. There are two failure modes:
Outright failures, time travel is impossible when you're trying for FTL.
Failure in Detail, detail failure is different and more fun; you can travel back in time with the intention of defying the laws of nature with an FTL transit but you can't actually succeed. No matter what you try there is always some form of roadblock, accident, or detour that prevents you from arriving any sooner than you would travelling by less esoteric means.
I suspect there is a cool science-y sounding explanation for why that might be, maybe something involving "inviolate temporal reference points" I don't know.
Do note that this allows for accidental FTL travel in the course of temporal sight seeing which could be useful to the storyteller and potentially disastrous to the traveler.
By the way I love this question; normally it's "how do I have FTL without Time Travel?" which is easy, and a little irritating, since the Arrow of Time seemingly prevents Time Travel anyway and FTL based Time Travel models assume a shifting reference frame which as far as we can determine is impossible to create due to macro-process temporal asymmetry (entropy). This is much more challenging and interesting.
