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The velocity of a rocket, which moves withe a constant acceleration g, is equal:

v(t) = gt

but I want to keep the const acceleration inside the rocket, not in the absolute sense.

An acceleration has a dimension: L/T^2 [m/s^2];

and in a moving rocket the meter is contracted gamma times,

and a time is dilated gamma times too,

so finally the absolute acceleration must be: ##g / \gamma^3##

therefore:

##dv=\frac{g}{\gamma^3}dt##

which can be easili integrated:

##\gamma^3dv = gdt\to\, v\gamma = gt\to\, \frac{v}{\sqrt{1-v^2}}=gt##

finally:

##v(t) = \frac{gt}{\sqrt{1+(gt)^2}}##

Is this a correct speed of the relativistic rocket?