Let's say that we have a spaceship. From the beginning, the ship is stationary (relative to me) and not affected by outer forces (like Earth's or Sun's gravity). Then the ship starts to accelerate at $10\space m/s^2$, that should be comfortable for the astronaut onboard (artificial gravity near $g$). Normally this is an easy problem: after 1 s the speed is 10 ms/s, after 2 s the speed is $20\space m/s$, etc. But the perceived acceleration is constant for the astronaut, meaning that we need to account for time dilation (otherwise, after a year the ships speed would be higher than $c$).
How do I calculate the acceleration of the ship after $x$ seconds? (the ships acceleration from my point of view, not the ships) As I understand the acceleration constantly increases the velocity, which increases the time dilation, which decreases the acceleration.