If we can still see the galaxy receding from us, even heavily redshifted, it is still only traveling Slower Than Light.
Still incorrect: in all reference frames, light always
moves at 299,782 km/s, regardless of the velocity of the object that emits that light, regardless of the observer's relative motion.
That's Einstein's Postulate, the fundamental assumption of special relativity. You need to get your head around that for anything else to make sense.
You know that is silly. Your first interpretation, a FTL ship will arrive before it's emitted photons is correct. Your regression into this silly idea that somehow photons traveling at light speed will outrun a ship traveling faster-than-light is just bizarre.
It bears repeating that the photons arrive first only in the traveler
's reference frame; they arrive second in the observer
's reference frame. This theoretically allows for a causality violation, which is why FTL travel is sometimes assumed to be a form of time travel.
And I agree, like a lot of things in special relativity it seems very
Wrong. FTL means traveling faster than light (in a vacuum). Special Relativity postulates an invariant light speed (in a vacuum)...
... in all reference frames. That means that no matter how much you accelerate, you cannot and will not ever encounter a photon that is not
moving away from you at the speed of light.
You forget, in the ship that is traveling Faster-Than-Light's frame, the speed of light will still be SLOWER than the ship.
Think about that for a moment. You just said above that the speed of light is the same in all reference frames, then no amount of forward acceleration can change the speed of light in your frame: If you're at .5C, light still moves forward at C. If you're at .9C, light still moves forward at C. If you're at 2C, light still moves forward at C.
This is because, as you said, the speed of light is a constant in all frames, and that means that TIME is variable. At FTL velocities your clock implicitly moves backwards and therefore your photons appear to be moving away from you while in other frames (whose clocks are moving forwards) those photons are moving towards you.
Velocity is a function of distance over time; that is, the distance to an object at T=0 plus the distance to that object at T=1, T=2, T=3, etc. it is not something you can measure against, say, the vacuum of space or the center of the galaxy and say "I am moving at X velocity." That would be a universal/all-encompassing reference frame, something that SR explicitly disavows.
I know that's confusing, and I know you're struggling with it. One thing that might help is to think of the dichotomy between, say, moral relativism and absolutism (the moral relativist claims that acceptable moral/ethical standards vary from culture to culture and that each culture's opinion of what is acceptable are equally valid). Relativistic physics is essentially the mathematical paradigm of PHYSICAL relativism: there is no such thing as objective truth, and there is no ONE answer for any particular question about velocity of time. It depends entirely on who's asking.
And that is why SR has lots of STL examples but rarely do you read about FTL examples. I recommend you look up SR and Tachyons or FTL particles.
What do you think I've been doing for the past five days?