Breakdown of the Einstein’s Equivalence Principle for a quantum body

We review our recent theoretical results about inequivalence between passive gravitational mass and energy for a composite quantum body at a macroscopic level. In particular, we consider macroscopic ensembles of the simplest composite quantum bodies — hydrogen atoms. Our results are as follows. For the most ensembles, the Einstein’s Equivalence Principle is valid. On the other hand, we discuss that for some special quantum ensembles — ensembles of the coherent superpositions of the stationary quantum states in the hydrogen atoms (which we call Gravitational demons) — the Equivalence Principle between passive gravitational mass and energy is broken. We show that, for such superpositions, the expectation values of passive gravitational masses are not related to the expectation values of energies by the famous Einstein’s equation, i.e. [Formula: see text]. Possible experiments at the Earth’s laboratories are briefly discussed, in contrast to the numerous attempts and projects to discover the possible breakdown of the Einstein’s Equivalence Principle during the space missions.

Breakdown of the equivalence between gravitational mass and energy due to quantum effects

We review our recent theoretical results about inequivalence between passive and active gravitational masses and energy in the semiclassical variant of general relativity, where the gravitational field is not quantized but matter is quantized. To this end, we consider the simplest quantum body with internal degrees of freedom — a hydrogen atom. We concentrate our attention on the following physical effects, related to electron mass. The first one is the inequivalence between passive gravitational mass and energy at the microscopic level. Indeed, the quantum measurement of gravitational mass can give a result which is different from the expected one, [Formula: see text], where the electron is initially in its ground state; [Formula: see text] is the bare electron mass. The second effect is that the expectation values of both the passive and active gravitational masses of stationary quantum states are equivalent to the expectation value of the energy. The most spectacular effects are the inequivalence of the passive and active gravitational masses and the energy at the macroscopic level for an ensemble of coherent superpositions of stationary quantum states. We show that, for such superpositions, the expectation values of passive and active gravitational masses are not related to the expectation value of energy by Einstein’s famous equation, [Formula: see text]. In this paper, we also improve several drawbacks of the original pioneering works.

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

In this paper, 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, [Formula: see text], 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.

Response to: Comment on “Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?”

We have recently shown that both passive and active gravitational masses of a composite body are not equivalent to its energy due to some quantum effects. We have also suggested idealized and more realistic experiments to detect the above-mentioned inequivalence for a passive gravitational mass. The suggested idealized effect is as follows. A spacecraft moves protons of a macroscopic ensemble of hydrogen atoms with constant velocity in the Earth’s gravitational field. Due to nonhomogeneous squeezing of space by the field, electron ground state wave function experiences time-dependent perturbation in each hydrogen atom. This perturbation results in the appearance of a finite probability for an electron to be excited at higher energy levels and to emit a photon. The experimental task is to detect such photons from the ensemble of the atoms. More realistic variants of such experiment can be realized in solid crystals and nuclei, as first mentioned by us. In his recent comment on our paper, Crowell has argued that the effect, suggested by us, contradicts the existing experiments and, in particular, astronomic data. We show here that this conclusion is incorrect and based on the so-called “free fall” experiments, where our effect does not have to be observed.

Breakdown of the equivalence between gravitational mass and energy for a quantum body: Theory and suggested experiments

In this paper, we review recent theoretical results, obtained for the equivalence between gravitational mass and energy of a composite quantum body as well as for its breakdown at macroscopic and microscopic levels. In particular, we discuss that the expectation values of passive and active gravitational mass operators are equivalent to the expectation value of energy for electron stationary quantum states in hydrogen atom. On the other hand, for superpositions of the stationary quantum states, inequivalence between the gravitational masses and energy appears at a macroscopic level. It reveals itself as time-dependent oscillations of the expectation values of passive and active gravitational masses, which can be, in principle, experimentally measured. Inequivalence between passive gravitational mass and energy at a microscopic level can be experimentally observed as unusual electromagnetic radiation, emitted by a macroscopic ensemble of the atoms. We propose the corresponding experiment, which can be done on the Earth's orbit, using small spacecraft. If such experiment is done it would be the first direct observation of quantum effects in general relativity.

Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?

We define passive and active gravitational mass operators of the simplest composite quantum body—a hydrogen atom. Although they do not commute with its energy operator, the equivalence between the expectation values of passive and active gravitational masses and energy is shown to survive for stationary quantum states. In our calculations of passive gravitational mass operator, we take into account not only kinetic and Coulomb potential energies but also the so-called relativistic corrections to electron motion in a hydrogen atom. Inequivalence between passive and active gravitational masses and energy at a macroscopic level is demonstrated to reveal itself as time-dependent oscillations of the expectation values of the gravitational masses for superpositions of stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by macroscopic ensemble of hydrogen atoms, moved by small spacecraft with constant velocity in the Earth’s gravitational field. We suggest the corresponding experiment on the Earth’s orbit to detect this radiation, which would be the first direct experiment where quantum effects in general relativity are observed.

Is gravitational mass of a composite quantum body equivalent to its energy?

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.

EINSTEIN'S REAL "BIGGEST BLUNDER"

Albert Einstein's real "biggest blunder" was not the 1917 introduction into his gravitational field equations of a cosmological constant term Λ, rather was his failure in 1916 to distinguish between the entirely different concepts of active gravitational mass and passive gravitational mass. Had he made the distinction, and followed David Hilbert's lead in deriving field equations from a variational principle, he might have discovered a true (not a cut and paste) Einstein–Rosen bridge and a cosmological model that would have allowed him to predict, long before such phenomena were imagined by others, inflation, a big bounce (not a big bang), an accelerating expansion of the universe, dark matter, and the existence of cosmic voids, walls, filaments and nodes.