[Sir Parsifal]

Flat-earthers negate the existence of gravity. If it existed, the Earth would take the shape of (almost) a ball, that's how gravity works.
To counter the problem of the Earth's pull happening, somehow, without gravity, flat-earthers devised the concept of "Universal Acceleration."

There is a problem with Universal Acceleration though. And the problem isn't that "the Earth is just going to accelerate to the speed of light!"
Since flat-earthers seem to respect the theories of relativity (the special theory of relativity is, after all, mentioned on the FAQ page as a solution to the above problem), I will point out what that other, actual problem with Universal Acceleration is (and it is the most important problem with it), based on the grounds of the special theory of relativity as well (funny, isn't it? you live by the sword and you die by the sword ).

So here's the quick refutation of Universal Acceleration, and with it, of the entirety of the so-called Flat Earth Theory:
Earth cannot accelerate in a uniform matter ad infinitum, because with increasing velocity, the relativistic mass of the object increases as well. This is Physics 101. This would have crucial implications on the outside world, including any kind of measurements, to such proportions that these changes would be well visible to the naked eye. In fact, the Earth's acceleration would have to keep slowing down (which is not happening) to compensate for the potential increase in relativistic mass. Obviously, none of the above is happening at all, thus Universal Acceleration does not exist and is hereby proven to be false. And thus the Flat Earth Theory as a whole is proven to be false.

QED

[Adrenoch]

UA doesn't work for a number of reasons, such as the same mass weighing differently at the poles versus equator, or on a mountaintop versus a valley - but relativity isn't one of the reasons.

To an observer on a moving object, like a flat Earth, you can always accelerate. Firing a rocket engine on the "bottom" of the FE will accelerate you (in your frame) just the same regardless of how long you've been accelerating in the past. The acceleration will feel exactly the same.

You're right that your mass would increase, and your time would dilate, both relative to other non-accelerated objects, but your mass wouldn't seem to change to you.

An easy way to picture it is to imagine you and your rocket are the only things in the universe. With the engine off, there's no way to tell your speed. You could be travelling 99% c relative to your starting speed, or you could be stationary. If you light your rocket, you're going to feel the exact same acceleration regardless of how fast you are going relative to your starting speed.

Now, if we're talking about UA on a flat Earth that doesn't have a dome, and you can measure yourself against all the stars, then you'd still feel the same acceleration, but you'd readily notice the difference in how other bodies appear.

But either way, the acceleration you feel is no different regardless of your speed.