Scientists announce a physical warp drive is possible (esquire.com)

"Preferred" foliations are perfectly fine in relativity when picked out by the matter. Working relativists do it all the time. One just has to be careful about preserving Lorentz-covariance for observers whose time axis is badly misaligned from the "preferred" one, because the (invariant sector of) physics must be identical for them. In particular a good choice of time axis along which to split spacetime values into time-ordered spatial values is a thermodynamic arrow of time that applies to the vast majority of the matter subject to dynamical equations.

Example preferred foliations are the cosmological scale factor, and terrestrial atomic time like UTC; they are picked out by the symmetries of the matter (large scale homogeneity and isotropy, near-sphericity and near-rotationlessness, respectively). "Lab frames" for example, are often UTC. That relativity provides mechanism to transform tensor-components as one switches systems of coordinates (or frames) is a great freedom.

The point is that in e.g. the cosmological foliation, an FTL (but not instantaneous) traveller will only be at one location on a cosmlogical values surface.

Geroch's point is that one should just accept this picture and calculate the evolution of field-values in the presence of objects with causal cones of different widths. There is no causality violation with very gentle assumptions about the spacetime. There are likely to be weird observations though, peculiar to the observers. But we can already have weird stroboscopic effects and so forth for merely very fast (not even relativistic!) travellers thanks to thinks like the persistence of vision or analogues in cameras and detectors.

However, an instantaneous traveller (>>>>> FTL, literally infinite speed and causal cones that are infinite in width) may mathematically appear up to infinite times on a single spacelike hypersurface. This does make a mess unless one can extract from the FTL traveller's configuration an equation of motion with enough constraints to make predictions. (For instance, if you have too many copies in the same "slice" of spacetime, does the entire spacetime collapse into a black hole? Can the copies be inside each other? Or only the bosons? And so on.)

Finally, a backwards time traveller need not even do FTL. Again, it will appear twice (or more) on well-chosen values surfaces, posing no problems for causality analysis. Here we return to the question of the thermodynamic arrow-of-time.

Let's consider a Rocket(with some amount of fuel) accelerating (gently) to the right, leaving an exhaust, that an intertial observer ticking at t watches :

  t_0 R(10)             
  t_1 eR(9)             
  t_2 eeR(8)
  t_3 eeeR(7)
  t_4 eeeeR(6)

But the dynamical equations in this arrangement of matter is perfectly reversible. Let's complicate this a bit:

  t_0 R(10)        eeeeeeeeeR(1)
  t_1 eR(9)        eeeeeeeeR(2)
  t_2 eeR(8)       eeeeeeeR(3)
  t_3 eeeR(7)      eeeeeeR(4)
  t_4 eeeeR(6)     eeeeeR(5)

R, moving gently through space, sees the thermodynamically-opposite copy out the window in both directions. R believes there is a weird causality violation, because one copy of R sees a copy of itself with less fuel (and wristwatches showing times in the future, but ticking backwards).

However, t just sees a time traveller, whe...

In TNG, they don't practically behave much differently from the warp core, but some of the video game spin-offs had fun with the idea, having destroyed Romulan warbirds collapse into their own engine core, which then evaporated because it was unstable outside of the confines of the gravitational field.

Yes, this is gobbledygook. But no moreso than regular "subspace" or anything else in Trek.

For power (ed: including their standard warp drive), the Discovery still has a plain old M/AM reactor. And yes, I did just use the phrase "plain old M/AM reactor" hah

Let's say your spaceship can go to Jupiter in one second (Edit: as measured from Earth), which would require FTL speed. Then you turn around and come back. You've still arrived two seconds after you left.

At Jupiter, you could look through a super-telescope and see yourself still on Earth, but you still aren't really seeing yourself, because you are not still on Earth. You are just seeing the photons you passed during the trip, catching up to you.

The explanation for this is complicated, however, so I would suggest you look at the links provided earlier in these comments for a more comprehensive explanation.

These are the sort of contradictions that break our understand of causality once you introduce FTL.

The main thing in the situation you're describing is that it's one way. Make it two way, imagine getting a message saying "don't send that message" from someone who has already seen the message you're about to send.

You can imagine this and that's part of the problem understanding the issue I think. But if this combination of events was possible very many things that we know to be true about spacetime would have to not be true.