Inequivalence between gravitational mass and energy of a composite quantum body in general relativity

Abstract

We consider the so-called semiclassical variant of general relativity, where gravitational field is not quantized but matter is quantized, for the simplest composite quantum body - a hydrogen atom. We create a stationary electron quantum state in the atom in the absence of gravitational field and study its time evolution in the presence of the field, using the local Lorentz invariance property of spacetime. It is shown that this state with a definite energy in the absence of gravitational field is not anymore a stationary state in the field. Therefore, quantum measurements of passive gravitational mass of electron in a hydrogen atom can give the following quantized values, $m_n = m_e + E_n/c^2$, where $m_e$ is the bare electron mass and $E_n$ is its energy level in the atom. We discuss some difficulties in the possible experimental observations of this mass quantization phenomenon.