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I thought I would indicate how I see this problem. This question involves different ways of defining the payload and total mass of the rocket. The four momentum of a body in flat spacetime, such as a rocket, is $$P~=~(E,~{\bf p}).$$ The four-momentum has the spatial momentum and the energy. The energy the rocket before it starts to accelerates has the ...

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I am adding this answer to respond to your last followup comment on L.Levrel's answer. Point One: Alice and Bob have to agree on how many times Alice's clock ticks per rotation, and they have to agree on how many times Bob's clock ticks per rotation. Therefore, if Alice says Bob's clock is slow, Bob must say Alice's clock is fast, and vice versa. Point ...

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