Theoretical Possibility of "Opposite" Gravitational Mass? This is an unusual idea that I have been entertaining for some time, and I can't find anything about it online.
However, it is so simple that someone must have conceived it before.
First, I will elaborate my idea, then I will ask if it possible within the framework of General Relativity.
There are two types of electric charge, positive and negative. Like charges repel, unlike charges attract.
Could there be two types of gravitational mass? Let's call them Д - mass and Ξ - mass.
They could a follow a similar but opposite rule to electric charges: like masses attract, and unlike masses repel.
We assume that both Д and Ξ masses have the same inertial masses.
Д - mass is the type of mass that we're all made out of, our bodies, our planets, our solar system. Ξ - mass would be the"opposite" type of mass.
Like masses attract, so we see that every bit of Д - mass gravitationally attracts every other bit of Д - mass. Using Newton's laws, we can obtain Galileo's Law of Falling Bodies, which is the basis of Einstein's equivalence principle.
Inertial masses remain the same. Falling objects on a planet made out of Ξ - mass would be kinematically indistinguishable from one made out of Д - mass.
Let's say one day a meteorite crashes onto Earth. It is a relatively ordinary meteorite, except that embedded within it are chunks of very pure Ξ - mass. When such a chunk is pried out, it falls up! It would fall towards the sky and keep going.
If we measure the acceleration of the up-falling chunks, we would see that it is also 9.81 ms-2.
If we combine two equal Д and Ξ masses, we can produce a gravitationally "neutral" mass, one that can float weightlessly. However, it will still have inertial mass.
We have not observed any neutral or Ξ masses. This is similar to the issue of baryon asymmetry. Due to like masses attracting and unlike masses repelling, this could result in increasing separation between the two types, and any Ξ masses in our universe might be really, really, really far away.
My final question is whether the existence of this "opposite" Ξ - mass is possible within the framework of General Relativity. Does the resulting repulsion and "falling up" violate the equivalence principle?
 A: This is a relevant article 

Does antimatter behave the same way as matter under the influence of gravity, or does antimatter even fall up? One of the keys lies in free-fall experiments to measure the acceleration of gravity, g. These are notoriously difficult with charged particles, but first measurements on neutral antimatter have recently been made and there are plans for dedicated experiments on antihydrogen at CERN with AEgIS and GBAR. There are also schemes to study gravitational effects in positronium and muonium, which are neutral atom-like systems consisting of a particle – an electron or a muon – and its antiparticle.

See the article and also the wikipedia entry.
A: Under some assumptions, it is possible to prove that mass in general relativity is positive, and that gravity is always attractive. 
However, GR does not couple strictly to mass, anyway. Photons are massless, but it is well known that light can be deflected gravitationally. This is because photons have energy and momentum despite being massless. So a gravitationally neutral body can't exist -- it would have to have no mass, energy or momentum, i.e. not "be there." 
A: This kind of a universe will put the Equivalence Principle in question. Because for the same inertial mass, there are two possibilities of gravitational mass. So one has to consider that the inertial mass should also come in two varieties to ensure the equivalence principle.
But now one has another trouble. If a particle has negative rest mass then its rest mass energy is negative. But in physics this is just not allowed. Because if the rest mass energy of any particle is negative then there is no reason why there would not appear moving particles with negative rest mass (thus, zero net energy) out of just empty space.
So I see no way this can be a valid case within the accepted basic principles of gravity Physics.
A: A type of particle that "falls up" in a vacuum at the Earth's surface would be an immediate, very clear-cut violation of general relativity.
In Newtonian gravity, there's a gravitational field that's a vector field, that points in the "down" direction.  But in general relativity, there is no such vector field.  In general relativity, there is no gravitational field, other than the metric tensor which measures spacetime.  And within a small enough region of spacetime, the metric tensor is indistinguishable from the Minkowski metric.  No matter what expression you form out of just the Minkowski metric, there's no way to form a "special" spatial direction that's physically distinguishable from any other spatial direction.  I.e., no matter how a particle might couple to the metric somehow, the particle can't fall up, because there's no way for the particle to "know" from the metric which direction "up" is.
Alternative models of gravity have been proposed, some of which include a vector field, in which case an "up" direction could be locally distinguishable.  So doing an experiment to see if antiparticles "fall up" would be one way of testing the validity of general relativity.  But the null hypothesis in such an experiment would be that general relativity is correct, and antiparticles will fall at constant velocity in any inertial frame of reference, just like a non-anti-particle.
