Q about relativistic mass increase, starship

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A warning is often given that if one wants to have a spaceship traveling at relativistic speeds (usually to get to another star), the ship's mass increases, making acceleration more difficult. But if it where an antimatter propelled ship, with mass being turned into energy, then wouldn't the mass increase be offset by the increased energy released?

In other words, say you have a best case scenario: it's a positron-electron reaction, producing all gamma rays, which by then we know how to redirect and use, almost 100%, for forward propulsion. So all propellant mass is being turned into energy, almost all of which is being spent on forward propulsion. Isn't the propellant mass increasing, along with the whole ship's mass, and so the ratio of ship's mass to energy being released in the "combustion chamber" always the same?
 
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No. The easiest way to see this is to not even use the concept of "relativistic mass". For detailed calculations, see for instance

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
 
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