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Ultracold neutrons are great experimental tools to explore the gravitationalinteraction in the regime of quantized states. From a theoretical perspective,starting from a Dirac equation in curved spacetime, we applied a perturbativescheme to systematically derive the non-relativistic Schr\"odinger equationthat governs the evolution of the neutron's wave function in the Earth'sgravitational field. At the lowest order, this procedure reproduces aSchr\"odinger system affected by a linear Newtonian potential, but correctionsdue to both curvature and relativistic effects are present. Here, we argue thatone should be very careful when going one step further in the perturbativeexpansion. Proceeding methodically with the help of the Foldy-Wouthuysentransformation and a formal post-Newtonian $c^{-2}-$expansion, we derive thenon-relativistic Hamiltonian for a generic static spacetime. By employing Fermicoordinates within this framework, we calculate the next-to-leading ordercorrections to the neutron's energy spectrum. Finally, we evaluate them fortypical experimental configurations, such as that of qBOUNCE, and note that,while the current precision for observations of ultracold neutrons may not yetenable to probe them, they could still be relevant in the future or inalternative circumstances.

Ultracold Neutrons in the Low Curvature Limit: Remarks on the post-Newtonian effects