Since Newtonian gravity is analogous to electrostatics shouldn't there be something called negative mass? Also, a moving charge generates electric field, but why doesn't a moving mass generate some other field?
Answer
General Relativity is a mathematical model that relates the curvature of spacetime to an object called the stress-energy tensor. In many cases the stress-energy tensor is dominated by mass and you can simply consider the curvature as being related to the mass. However this isn't always true as I'll mention below.
Anyhow, we can put any numbers we want into the stress-energy tensor and then calculate the curvature. If we put in a positive mass we get (in the Newtonian limit) the usual law of gravitation, but we could put in a negative mass and we'd get a repulsion just as you do in electrostatics. Matter with a negative mass is usually referred to as exotic matter and is a favourite trick for building weird objects like the Alcubierre faster than light drive or wormholes.
However just because we an put exotic matter into Einstein's equation doesn't mean it's physically reasonable to do so. No-one has ever observed exotic matter, no-one has ever come up with a convincing theoretical reason for it to exist. So while we can't prove exotic matter doesn't exist few of us think it does - though we'd all love to be able to build a faster than light drive!
Even though we've never observed exotic matter, we have (we think) observed dark energy. This isn't matter, and doesn't have a negative mass, but it does cause a gravitational repulsion.
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