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arXiv:1404.4044 · Journal of Physics Conference Series

Breakdown of the equivalence between gravitational mass and energy for a composite quantum body

A. G. Lebed

Published Mar 11, 2014• 7 citations
arXivHTMLPDF

Abstract

The simplest quantum composite body, a hydrogen atom, is considered in the presence of a weak external gravitational field. We define an operator for the passive gravitational mass of the atom in the post-Newtonian approximation of the general relativity and show that it does not commute with its energy operator. Nevertheless, the equivalence between the expectation values of the mass and energy is shown to survive at a macroscopic level for stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiation, emitted by the atoms, supported by and moving in the Earth's gravitational field with constant velocity, using spacecraft or satellite.

DOI
10.1088/1742-6596/490/1/012154

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The simplest quantum composite body, a hydrogen atom, is considered in the presence of a weak external gravitational field. We define an operator for the passive gravitational mass of the atom in the post-Newtonian approximation of the general relativity and show that it does not commute with its energy operator. Nevertheless, the equivalence between the expectation values of the mass and energy is shown to survive at a macroscopic level for stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiation, emitted by the atoms, supported by and moving in the Earth's gravitational field with constant velocity, using spacecraft or satellite.
It is shown that weight operator of a composite quantum body in a weak external gravitational field in the post-Newtonian approximation of the General Relativity does not commute with its energy operator, taken in the absence of the field. Nevertheless, the weak equivalence between the expectations values of weight and energy is shown to survive at a macroscopic level for stationary quantum states for the simplest composite quantum body - a hydrogen atom. Breakdown of the weak equivalence between weight and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity, using spacecraft or satellite. For superpositions of stationary quantum states, a breakdown of the above mentioned equivalence at a macroscopic level leads to time dependent oscillations of the expectation values of weight, where the equivalence restores after averaging over time procedure.
Abstract We define passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of general relativity and show that it does not commute with energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between passive gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of passive gravitational mass for superpositions of stationary quantum states, where the equivalence restores after averaging over time. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite, which can be experimentally measured.
We define passive gravitational mass operator of the simplest composite quantum body - a hydrogen atom - to be proportional to its weight operator in a weak gravitational field. Although it does not commute with energy operator, taken in the absence of the field, the equivalence between the expectation value of passive gravitational mass and energy is shown to survive for stationary quantum states. All the so-called relativistic corrections to electron energy in a hydrogen atom are taken into account in the calculations.
Recently, we have started to investigate behavior of a composite quantum body in an external gravitational field in the framework of General Relativity [see, for a review, A. G. Lebed, Mod. Phys. Lett. A, {\bf 35}, 2030010 (2020)]. As the simplest example, we have considered a hydrogen atom in a weak gravitational field. Our results are the following. The Einstein's Equivalence Principle survives for the most of macroscopic ensembles of the atoms, containing the stationary quantum states. On the other hand, we have demonstrated that this principle is sometimes broken. In particular, it is broken for the so-called Gravitational demons, which are the coherent macroscopic ensembles of two or more stationary quantum states in the hydrogen atoms. In the above cited paper we have considered the Gedanken experiment, where the gravitational field is suddenly switched on in a free from gravitation space. In the current paper we consider the much more realistic from experimental point of view Gedanken experiment and come to the same conclusion about violations of the Einstein's Equivalence Principle for the Gravitational demons.
Semiclassical gravity, in which a classical spacetime is sourced by the quantum expectation value of the stress-energy tensor, is a standard framework for describing the gravitational interaction of quantum matter. In the nonrelativistic limit this approach leads to the Schrödinger-Newton equation, which is often assumed to be consistent at least in the weak-field regime. In this work, we reexamine this assumption for spatial quantum superpositions of massive particles. We show that, when the quantum state is properly normalized, no modification of the Newtonian gravitational potential arises at leading order. However, at first post-Newtonian order the semiclassical coupling generically produces state-dependent contributions involving the mass density and the mass current of the superposition. These terms have a parametric scaling which is different from that of the corresponding relativistic corrections and which does not have Planck mass suppression. Our results therefore impose a strong post-Newtonian consistency constraint on deterministic semiclassical gravity, indicating that sourcing the metric solely by expectation values is insufficient to recover a consistent relativistic weak-field expansion.
We extend the systematic calculation of an approximately relativistic Hamiltonian for center of mass and internal dynamics of an electromagnetically bound two-particle system by Sonnleitner and Barnett [Phys. Rev. A 98, 042106 (2018)] to the case including a weak post-Newtonian gravitational background field, described by the Eddington-Robertson parametrized post-Newtonian metric. Starting from a proper relativistic description of the situation, this approach allows us to systematically derive the coupling of the model system to gravity, instead of ``guessing'' it by means of classical notions of relativistic effects. We embed this technical result into a critical discussion concerning the problem of implementing and interpreting general couplings to the gravitational field and the connected problem of how to properly address the question concerning the validity of the Equivalence Principle in Quantum Mechanics.