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Antimatter is similar to ordinary matter except they have the opposite charge, i like to know why this difference in their spin would affect their motion in the curvature of our 4-dimensional spacetime? It can't be due to Earth's magnetic field or the air so what good reason to suspect antimatter would fall up since no test to date had suggest quantum particle falls differently compared to their big counterpart?

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People don't suspect antimatter falls upward: they suspect very strongly that it falls downwards in exactly the same way ordinary matter does. Indeed, it's almost inconceivable that antimatter would fall upwards if General Relativity is correct, since it would mean that somehow the geometry of spacetime would have to be different for different sorts of matter. But that's why the experiments are worth doing: if antimatter does fall upwards or behave differently under gravity then we've found an experiment where GR almost certainly must fail in a horrible way. And that would mean we need a new theory of gravity.

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To add to the above answers, in fact, you could create a perpetual motion machine if antimatter fell up as follows:

  1. create electron-positron pair.
  2. transport them to a higher altitude. This will take zero work, because the positron will want to gain speed as it goes up, while the electron will want to lose speed.
  3. annhilate them to a photon at the higher altitude. Direct beam downward
  4. collect the photon at lower altitude. It will be blueshifted by the downward trip, and thus will have more energy than the original electron-positron pair.
  5. repeat.
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    $\begingroup$ Oh, that's a lovely argument! $\endgroup$ – tfb Apr 26 '19 at 20:16
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No one suspects antimatter would fall/rise upward. Some literature does mention negative mass would result into that kind of behavior, but there is no negative mass.

Only other way to make something (matter, or antimatter) fall upward, would be - if we were able to invert the curvature of space outwards/upwards, by exploiting the very power of commonly known inward/downward curvature of space.

Only outcome of naturally available tensors is inward curvature of space. This curvature is formulated in GR. So, nature by itself is not capable of producing such upward curvature. That is why, we never observe inverted gravity in nature. But GR does not explicitly rule out inversion of this curvature inside a limited/restricted region via intelligent intervention. As this inversion would use the power of inward curvature, it would not violate GR or energy conservation.

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