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One way of creating artificial gravity is acceleration- gravity by inertial mass.

So in theory, if we would find a way to travel interstellar space by accelerating, reaching near light speed, would there be "gravity" on that spaceship because of its constant acceleration?

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Yes. There would be, because constant acceleration is equivalent to 'gravity.' Being at a velocity close to the speed of light changes nothing. Occupants of spaceships travelling close to lightspeed will experience the same acceleration.

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    $\begingroup$ Note though that in relativity proper acceleration is different from the coordinate acceleration measured in a fixed inertial reference frame (the proper acceleration at any moment is equal to the coordinate acceleration in an inertial frame where the ship has an instantaneous velocity of 0 at that moment), and it's the proper acceleration that would determine the artificial gravity onboard. A ship with constant proper acceleration will have decreasing coordinate acceleration in some frame as it approaches the speed of light in that frame. $\endgroup$ – Hypnosifl Sep 28 at 15:15
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    $\begingroup$ @Hypnosifl Thanks for distinguishing between coordinate and proper acceleration. The only acceleration I was concerned with my answer was the acceleration inside the spacecraft itself, which is its proper acceleration. It's nice to be reminded of the niceties of relativity. $\endgroup$ – a4android Sep 29 at 3:59
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What you describe is what astronauts experience in their missions: whenever they fire their rockets, they experience the inertia and feel the "weight", which they don't when they are in free fall.

The problem with this concept for giving constant apparent gravity is that you need to constantly fire the rockets, and that requires fuel, which requires further fuel to itself be accelerated. This notorious effect is know as the tyranny of the rocket equation.

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