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Comment by mdturnerphys

This is something I hadn't thought through before and made my morning bus ride substantially more interesting :)

I'm moving this disclaimer to the beginning, as I don't want anyone to get too excited: We haven't observed differences between the two yet, at a level of a part in 10^13. The following outcomes would only be possible with a different sort of matter than what we're aware of.

Energy is still conserved, so there aren't any perpetual motion machines that result. Such materials would fall differently--both in terms of acceleration and direction. (The later is because objects are also subject to inertial "forces", e.g. centrifugal force, Coriolis force, due to us living on a rotating reference frame called the Earth.). Orbital radii for given orbital periods could be different, so you could have geostationary satellites at different distances from the Earth. You could also make objects that are harder to push than to lift, or vice versa, but I can't think of a practical application for this other than demonstrating the effect.

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