What happen to rocket travelling @ 0.8c

[Manula]

We all know that any biological or an other chemical activity occurring inside a rocket traveling at 0.8c appear to be slower to an external observer.
Then the rate of combustion of fuel in the rocket should also be reduced according to that.
Then what prevents the rocket from slowing down according to that...

 

[Nugatory]

We all know that any biological or an other chemical activity occurring inside a rocket traveling at 0.8c appear to be slower to an external observer.
Then the rate of combustion of fuel in the rocket should also be reduced according to that.
Then what prevents the rocket from slowing down according to that...

If the rocket is traveling at a constant speed of .8c (or any other constant speed) there's no combustion of fuel going on at all - the motors are switched off and the rocket is coasting at a constant speed (no friction in outer space, remember) without slowing down.

Fire up the motors, and the rocket will start accelerating and increasing its speed. Now the math becomes a bit more complicated (so I would advise you to get comfortable with the constant-speed math first) but at a hand-waving level... Yes, effects such as the rate of combustion being reduced according to an external observer will conspire so that the rocket's speed will not increase without limit - indeed, the rocket won't be able to quite reach the speed of light relative to the external observer, no matter how long and hard the motors fire.

 

[A.T.]

Yes, effects such as the rate of combustion being reduced according to an external observer will conspire so that the rocket's speed will not increase without limit

Isn't the force from the rocket engine frame invariant? I would think that the slower burn rate is exactly compensated by the increase of the fuel's inertia, so the thrust stays the same. But the increase in inertia also prevents that constant force from accelerating the rocket to c.