Falling test electric dipole in the Schwarzschild geometry

The infall of a dipole and anti-dipole towards a Schwarzschild Black Hole (SBH), within the context of spherically symmetric spacetime, serves as a model for understanding the gravitational behavior of matter and antimatter. SBH is chosen as a prototype to represent the gravity of our Earth or any spherically symmetric heavenly object. Our findings indicate that in the geometrical units, the acceleration of dipoles and anti-dipoles can be either attracted or repelled, depending on the mass ($m$) of the gravitating center, dipole moment ($P$), and distance ($r$). In the physical units, however, the difference is far from being detectable, at least with the present technology. Therefore, the assumption that they always fall down is valid only by virtue of the huge mass of the Earth. Consequently, the detectability of the dipole correction is highly suppressed for massive central bodies such as the Earth, whereas satellite-based experiments in low-gravity environments may offer the sensitivity required to observe the effect. These results may shed light on the question of what happened to antimatter believed to have formed in the stages of the Big Bang. By providing new insights on the universe's imbalance between matter and antimatter, they may be able to clarify why matter eventually took over and gave rise to planets, stars, and galaxies, as well as our own existence.