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1

Relativistic mass will not increase the gravitational pull, the gravitational force depends on the rest mass of an object.

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You need to accelerate and slow down the flywheel. In order for the system to be self contained, the energy has to come from within your device, The total energy of the device does not change with the accelerating flywheel since the energy has to come from something else in the device.

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A higher speed does not equal a higher mass. This is something commonly taught to beginning students of relativity as an explanation of relativistic momentum because it is easy to understand (but does lead to misconceptions). Also, momentum must be transferred to slow down the flywheel.

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If your goal is to consider the Alcubierre metric then you'll notice the energy density is sometimes negative. The classical vacuum has an energy density that is zero. So this tells you that you either need quantum theory (which is wishful thinking since if you have quantum gravity you need a different theory than GR and so there might not be Alcubierre ...

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When you talk about the energy density in the vacuum I assume you mean the energy of the gravitational field. Obviously the stress-energy tensor is zero because otherwise it wouldn't be a vacuum. In that case the object you need is the stress-energy-momentum pseudotensor. This is constructed entirely from the metric tensor (which is sort of your function ...

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