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Let's say we'd transform earth into a giant space ship and had some planetary drive that can accelerate it without burning any kind of fuel (so the amount of matter on earth stays the same - we'd collect the energy for the acceleration from the outside environment as we go).

If I understand correctly, as we approach the speed of light (let's say relative to the CMB rest frame) earth will bend space-time more and more since that added kinetic energy also contributes to the bending (sometimes referred to as becoming heavier as one gets closer to $c$).

What I wonder is - once we stop accelerating further, would we locally on earth notice any difference in terms of earth's gravitational pull acting on us? Will we "get heavier" in our local earth frame of reference?

matthias_buehlmann
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The mass doesn't increase when you speed up. The relativistic mass is a concept long abandoned.

This means that whether the earth is moving (constant speed), or stationary, it would create the same curvature because all inertial points of view are valid. It's true that moving earth has more kinetic energy, it has more momentum which creates a repulsive curvature. More on here

Returning to your question, an accelerating earth is noninertial, similar to a car accelerating. This means that, relative to the earth, you would experience a backward pull (of course not a real force, a fictitious force). But an extra curvature caused by the earth, then no.

Habouz
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