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?
Relativity says yes. There’s no absolute speed, only relative speed; within the local reference frame of the ship, everything will continue to work normally, including the force experienced due to acceleration.
The ship is not actually going to reach the speed of light (as seen by an outside observer) though. The faster the ship goes, the more its (observed) mass increases, and the 9.8m/s² acceleration will have less and less of an effect. But to the people inside the ship, it appears as though they can accelerate indefinitely, going faster and faster at their steady rate of acceleration. Due to relativistic effects, it’ll never look like they are passing any objects outside the ship at more than the speed of light; instead it will appear as though the distance they have to travel is compressed, so they don’t have to travel as far.
You can think about it this way. In relativity:
A minor nit pick. It’s worth noting that increasing mass is an inaccurate view. It works in the simple examples, but can cause confusion down the line.
Instead, an additional term is introduced. This term, while it could be combined with the mass, is actually a vector, not a scalar. It has both value and direction, not just value. This turns your relativistic mass into a vector. Your mass changes, depending on the direction of the force acting on it! Keeping it as a separate vector can improve both calculations and comprehension, since comparable terms appear elsewhere (namely with time dilation and length contraction).