An article has recently been published in Nature that shows anti-matter is also pulled toward the mass in gravity. In FS, the direction of time is incorporated in SR and specifies whether the object recedes or approaches the origin. The same idea can be extended to gravity. It means the direction of time should be a basic input to the theory of gravity. When time moves forward, that is the default direction of time for matter and the anti-default direction of time for anti-matter. In the forward direction of time matter is gravitated toward massive objects. in the anti-default direction of time for anti-matter, it again gravitates toward the massive object because, in the default direction of time for anti-matter, space expands. To clarify if the direction of time is a basic input of the gravitational theory and if it is fixed for matter and anti-matter in an experiment, then both of them will behave similarly.
UPDATE 1:
However, the universe expands in the default direction of time for matter. This is still unclear but there might be two distinct types of inversions (similar to additive and multiplicative inversions in R). One inversion is based on CPT (and turns matter into anti-matter). In the other inversion, the direction of time is not changed. It turns gravity into anti-gravity. This inversion is not based on CPT and is a totally different type of inversion. The new inversion might have something to do with the inversion of the black hole. The origin is fixed under CPT (similar to zero in additive inverse), but the mult inverse turns zero into inf. The same appears to be true here. The origin (singularity) should be inverted to achieve anti-gravity.
UPDATE 2:
Anti-matter is the inverse of matter. They turn into pure E in annihilation. There should be two situations (in gravity/expansion) that are inverse of each other. For example, the Big Bang is the inverse of the Big Rip. Similarly, at any point in time, there is an inverse situation in the future. The combination of these two situations is always fixed. The identity element of this operation (i.e. the fixed value; for example, 1 in mult inverse) might be a breakthrough in the universe but is unclear at this point what it is.
UPDATE 3:
As a side note, 0*inf equals any real number. It means the mult inversion of zero/inf is fixed no matter which value is set to be the identity element. That is a type of relativity.