Inequivalence between gravitational mass and energy due to quantum effects at microscopic and macroscopic levels

We review recent theoretical results, demonstrating breakdown of the equivalence between active and passive gravitational masses and energy due to quantum effects in General Relativity. In particular, we discuss the simplest composite quantum body - a hydrogen atom - and define its gravitational masses operators. Using Gedanken experiment, we show that the famous Einstein's equation, $E =mc^2$, is broken with small probability for passive gravitational mass of the atom. It is important that the expectation values of both active and passive gravitational masses satisfy the above mentioned equation for stationary quantum states. Nevertheless, we stress that, for quantum superpositions of stationary states in a hydrogen atom, where the expectation values of energy are constant, the expectation values of the masses oscillate in time and, thus, break the Einstein's equation. We briefly discuss experimental possibility to observe the above-mentioned time-dependent oscillations. In this review, we also improve several drawbacks of the original pioneering works.

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