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?