The construction of an anti-gravity device requires material which can stand an enormous negative pressure and be so light that the positive gravity of its mass does not win the anti-gravity.
Positive gravity causes light to travel slower from the point of view of an external observer. Space appears optically dense. Gravitational lensing is a result of this effect.
Conversely, negative gravity would make light travel faster than light from the point of view of an external observer. Then we would have all the paradoxes of causal loops.
We were able to derive a minimum mass for a force field in an earlier blog post. The minimum mass was derived from momentum conservation.
Now we may have found a stronger limit: the mass of a force field must be high enough to offset the negative gravity produced by pressure on the material. That is, it must not make faster-than-light travel possible.
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