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We show Zeeman-like splitting in the energy of spinors propagating in abackground gravitational field, analogous to the spinors in an electromagneticfield, otherwise termed the Gravitational Zeeman Effect. These spinors are alsofound to acquire a geometric phase, in a similar way as they do in the presenceof magnetic fields. However, in a gravitational background, the Aharonov-Bohmtype effect, in addition to Berry-like phase, arises. Based on this result, weinvestigate geometric phases acquired by neutrinos propagating in a stronggravitational field. We also explore entanglement of neutrino states due togravity, which could induce neutrino-antineutrino oscillation in the firstplace. We show that entangled states also acquire geometric phases which aredetermined by the relative strength between gravitational field and neutrinomasses.

Gravity-Induced Geometric Phases and Entanglement in Spinors and Neutrinos: Gravitational Zeeman Effect