I'll start with this quote.

"In addition to time dilation and length contraction, another unusual effect of near-light speed is stated in Einstein's second paper, titled "Does the Inertia of a Moving Body Depend on its Energy Content?" In this paper, Einstein gave the ubiquitous formula E=mc2. The formula shows the equivalence of mass and energy and illustrates the fact that an increase in mass (m) is accompanied by an increase in energy (E) by mc2. The formula also shows an increase in energy (E) results in an increase in mass (m) by E/c2. In other words, as an object accelerates by gaining energy, it gains mass. As the object approaches the speed of light, its mass approaches infinity. An infinite amount of energy is required to accelerate an object to the speed of light, so the speed of light acts as a speed limit for matter."

I don't doubt the math behind it but the interpretation stated. "The formula also shows an increase in energy (E) results in an increase in mass (m) by E/c2."

I mean, I assume that we were talking invariant mass here. From which frame would it be noticeable? The moving frame? That a accelerating frame gains momentum and energy may be true, but only relative its possible final interaction (collision for example), as it seems to me? Assume that the mass actually increase. What then is the difference between a constant acceleration at one G relative a non-constant acceleration taken under a shorter time interval? Assuming that both will reach the exact same velocity becoming 'at rest' with each other at a later point? Reading the quote it suddenly feels as if I might be able to express energy in 'time'. And that seems hard to do?

As long as we're talking a acceleration it makes a sort of sense, but what happens with that invariant mass if he stops the engines? No acceleration any more. Where did it go? Did the energy disappear too? Thinking of the litmus test, namely that possible interaction/collision, the energy must be there as well as the momentum. But if we assume a invariant mass it seems to me that it should affect all particles making up that ship, as well as the crew naturally. That is, if I assume that it is noticeable, and it should be if it is invariant mass.

Or am I reading the quote wrong here?

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As for uniform motion, assume a planet in deep space far away from any gravitational forces. Will it geodesics change as it gets another uniform speed relative some common reference? I don't think so, do you?

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To me it seems more of a mix up with the effects of acceleration induced gravity, than the invariant mass defined in the object(s) themselves?