@Longtitube
As I understand it, this one all comes down to energy, and this is the fundamental issue at the heart of the UA problem. Apologies - my previous answer was probably misleading. It is impossible to accelerate something at 1g forever as far as an external observer is concerned. But from the perspective of the 'something', it can feel 1g forever...the issue is where is the force coming from?
My understanding:
If you got in a spacecraft and had some magic propulsion source that was capable of accelerating you at 1g for as long as you wanted then, just like in our example here, it would take around a year to accelerate up to c. If you were travelling to some distant solar system and your propulsion system could keep providing the thrust, you could indeed keep on accelerating, and it would feel like 1g.
But speed, or more accurately velocity, is always relative. So if you were taking earth as your reference point, then your apparent velocity would never reach c. As far as the crew on the spacecraft are concerned, they would feel the 1g acceleration, and would actually make the journey in a time that felt, to them, as if they were exceeding the speed of light. However, to external observers, it would take far, far longer to reach the destination - a sub-c effective speed, with the difference caused by your ever-increasing mass - the constant thrust required to achieve the 1g would have less and less apparent effect to an
external observer because the 'm' in f=ma is increasing all the time. Wikipedia tackles it here:
https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration, although there's plenty of examples if you google 'space travel at constant acceleration' - it's quite a well-known proposal, common in sci-fi, as the 1g aspect of it would provide a survivable living environment for the crew. The problem is that you aren't 'travelling' faster than c - it's taking hundreds of thousands of years to get anywhere, even though the crew don't age (as much).
This is why energy becomes so important here, and why I keep asking the question that is, unsurprisingly, not being answered. Where or what is the cause of the force that is propelling earth? The problem is that, unlike centripetal acceleration such as in orbits, linear acceleration is adding kinetic energy to an object, and that energy has to come from somewhere. So the question then is where? What is causing the earth (and the dome over it, and the sun, moon and stars, whatever they may be) to accelerate? It can't come from anything external - that's really important, because the apparent mass of the earth etc would be increasing, and you couldn't accelerate past c from the frame of reference of whatever the external source was. So it has to be endogenous to the earth's accelerating system, and essentially everlasting, and somehow capable of accelerating the moon, sun and stars with it, all of which are rotating as well, due to some other magical unexplained force.
Of course all of this required magic disappears if you just accept what is entirely obvious from hundreds of years of painstaking observation, measurement and calculation, which is that we are on a spherical(ish) globe with a single orbiting moon, part of a solar system of planets with associated moons orbiting the sun, which in turn is part of a much larger galaxy. There is nothing unexplained or mysterious about this system at all - all the calculations work out. We know the distances, speeds, and masses of the various planets and moons. We can calculate planetary positions, eclipses and tides with excellent precision, and they all match perfectly to our understanding of gravity and other physical calculations. Even if, for whatever bizarre reason, you choose to believe that every picture and video of earth taken from space is somehow faked, and that every space rocket, every satellite, every intercontinental ballistic missile even...all of that, by every country who has ever engaged in a space programme...even if you think all of that was just some elaborate theatre involving millions of people and trillions of pounds...the heliocentric globe earth model still provides faultless explanation for every observed motion and force that we can detect ourselves.