Was thinking about interstellar travel and the ability to provide artificial gravity by using a smooth acceleration and deceleration across the journey, changing from acceleration to deceleration at the halfway mark.
If we ignore relativistic effects, with smooth acceleration of 9.81 ms^-2^, you'd be going 3.1e8 ms^-1^ after the first year (3.2e7 s), if I'm not making a mathematical blunder. That's more than the speed of light at 3.0e8.
My main question, and the one that I initially came here to ask, is: if their ship continues applying the force that, under classical mechanics, was enough to accelerate them at 9.81 ms^-2^, would the people inside still experience Earth-like artificial gravity, even though their velocity as measured by an observer is now increasing at less than that rate?
A second question that I thought of while trying to figure this out myself as I wrote it up, is... My understanding is that a trip taken at the speed of light would actually feel instantaneous to the traveller, while taking distance/speed of light to a stationary observer. In the above scenario, would the final time taken, as measured by the traveller, be the same as if they were to ignore the speed that they are travelling at according to an outside observer, and instead actually assume they are undergoing continuous acceleration?
Touching on the second question, since the ship would never actually reach the speed of light, the trip would not seem instantaneous to the people on board. However, the trip would seem much shorter to the people on board than it would to external observers. The people on board the ship would experience length contraction in the direction of travel making their destination closer to themselves, while external observers would notice the people onboard the ship moving slowly, ie, experiencing time at a reduced rate. Either way, the effect is that the people on board perceive the trip to be much shorter (in terms of both distance and time) than an external observer watching their ship. In principle you can get the perceived length of the trip (both distance and time) to approach but not equal zero, though in practice this would involve killing everyone on board and destroying the ship (and maybe even the galaxy).
I agree with the other commenters that the people on board will experience a consistent acceleration of 9.81 m/s² in your described scenario. It might help, conceptually, to imagine an external observer watching someone on the ship jumping up and down at this near-light speed, taking into account the severe time dilation they'd be experiencing: The difference in perception comes because, from the external observer's point of view, the person on the ship is moving in extreme slow motion.