Experimentally, it is still an open question whether antimatter falls down or falls up. More formally we ask about the gravitational interaction coefficient of antimatter.
Our assumptions all say that it must fall down, but because of reasons that really shouldn't hold. You can construct a violation of "conservation of energy" if your equations say antimatter falls up. Yet our examples and experiences that say "conservation of mass-energy" is a thing only apply for data gathered with matter, or with a small amount of anti-matter not meaningfully interacting with the gravitational force.
We interpret General Relativity to mean that mass consists of positive-space-time. It is a presence of space time. The remainder of space-time stretches to compensate.
Suppose we were to measure the surface of the earth very carefully, and then dig a hole to the center and measure the radius. From the surface area we could calculate the predicted radius we would get from setting the area equal to 4πr^2. When we compared the predicted radius with the actual radius, we would find that the actual radius exceeded the predicted radius by the amount given in Eq. (42.3). The constant G/3c^2 is about 2.5×10^−29 cm per gram, so for each gram of material the measured radius is off by 2.5×10^−29 cm. Putting in the mass of the earth, which is about 6×10^27 grams, it turns out that the earth has 1.5 millimeters more radius than it should have for its surface area. Doing the same calculation for the sun, you find that the sun’s radius is one-half a kilometer too long. - Richard Feynman
It is farther to the center of the earth than the calculation from going around the surface would suggest. There is "more" spacetime inside the sphere than Minkowski space would suggest. The mass is correlated with positive amounts of spacetime. To give a numerical approximation: suppose a vacuum contains 100 percent units of "spacetime". A planet may contain 100.001. The sun may contain 100.05 units.
Does a substantial volume of antimatter contain a positive or negative "unit" of spacetime?
We note the historical record reflects that the initial description of TAI did not specify a gravitational field. An amendment in the late 70s adjusted for an estimate of nominal mean sea level time. See Epoch for more detail.
These calculations are done regularly by the Global Positioning Service.
The Epoch of Space-Time