We analyze the weak-field limit of General Relativity with matter and its possible quantisations. This analysis aims towards a predictive quantum theory to provide a first-principles description of gravitational effects in macroscopic quantum systems. This includes recently proposed experiments on the generation of (Newtonian) gravitational forces from quantum distributions of matter, and phenomena like gravity-induced entanglement, gravitational cat states, gravity-induced Rabi oscillations, and quantum causal orderings of events. Our main results include: (i) The demonstration that these phenomena do not involve true gravitational degrees of freedom. (ii) We show that, unlike full general relativity, weak gravity with matter is a parameterised field theory, i.e., a theory obtained by promoting spacetime coordinates to `dynamical' variables. (iii) Quantisation via gauge-fixing leads to an effective field theory that account for some phenomena, but at the price of gauge dependence that manifests more strongly on spacetime observables. This ambiguity is a manifestation of the problem of time that persists even in weak gravity. (iv) A consistent quantisation of parameterised field theories is essential for a predictive and spacetime covariant theory of weak gravity that describes gravitational effects in macroscopic quantum systems. We also discuss the implication of our results to gravitational decoherence theories, the notion of locality in gravity vis-a-vis quantum information theory, and the intriguing possibility that proposed solutions to the problem of time can be tested in weak-gravity quantum experiments.
The existence of quantum correlations affects both microscopic and macroscopic systems. On macroscopic systems they are difficult to observe and usually irrelevant for the system's evolution due to the frequent energy exchange with the environment. The world-wide network of gravitational-wave (GW) observatories exploits optical as well as mechanical systems that are highly macroscopic and largely decoupled from the environment. The quasi-monochromatic light fields in the kilometre-scale arm resonators have photon excitation numbers larger than $10^{19}$, and the mirrors that are quasi-free fal…
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.
Through averaging the Einstein equations over transverse gravitational perturbations it is obtained a closed system of two ordinary differential equations describing macroscopic cosmological evolution of the isotropic space-flat Universe filled with gravitational radiation. It is found an asymptotic solution of evolution equation for gravitational perturbation amplitude. Making the substitution of this solution into Einstein equation averaged over gravitational perturbations, the single evolution non-linear ordinary differential second-order equation relative to macroscopic scale factor is obt…
We make an attempt to dynamically study, in four space-time dimensions, the classical gravitational collapse of a macroscopic circular fundamental string, by a truncation of the Einstein equations that suppresses retarded features but keeps the main self-gravity peculiarities of the relativistic string dynamics, and allows the investigation of a possible infinite red-shift. The numerical solution of the string evolution in the self-induced metric shows an infinite red shift at a macroscopic size of the string, when the string reaches the velocity of light. We further include the back-reaction …
We analyze the propagation of gravitational waves in a medium containing bounded subsystems ("molecules"), able to induce significant Macroscopic Gravity effects. We establish a precise constitutive relation between the average quadrupole and the amplitudes of a vacuum gravitational wave, via the geodesic deviation equation. Then we determine the modified equation for the wave inside the medium and the associated dispersion relation. A phenomenological analysis shows that anomalous polarizations of the wave emerge with an appreciable experimental detectability if the medium is identified with …
Several relatively small-scale experimental setups aimed on prototyping of future laser gravitational-wave detectors and testing of new methods of quantum measurements with macroscopic mechanical objects, are under development now. In these devices, not devoted directly to the gravitational-wave detection, Mach-Zehnder interferometer with pass-through Fabry-Perot cavities in the arms can be used instead of the standard Michelson/Fabry-Perot one. The advantage of this topology is that it does not contain high-reflectivity end mirrors with multilayer coatings, which Brownian noise could constitu…
A model for the static weak-field macroscopic medium is analyzed and the equation for the macroscopic gravitational potential is derived. This is a biharmonic equation which is a non-trivial generalization of the Poisson equation of Newtonian gravity. In case of the strong gravitational quadrupole polarization it essentially holds inside a macroscopic matter source. Outside the source the gravitational potential fades away exponentially. The equation is equivalent to a system of the Poisson equation and the nonhomogeneous modified Helmholtz equations. The general solution to this system is obt…
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 equat…
The problem of construction of a continuous (macroscopic) matter model for a given point-like (microscopic) matter distribution in general relativity is formulated. The existing approaches are briefly reviewed and a physical analogy with the similar problem in classical macroscopic electrodynamics is pointed out. The procedure by Szekeres in the linearized general relativity on Minkowski background to construct a tensor of gravitational quadruple polarization by applying Kaufman's method of molecular moments for derivation of the polarization tensor in macroscopic electrodynamics and to derive…
A quantum measurement-like event can produce any of a number of macroscopically distinct results, with corresponding macroscopically distinct gravitational fields, from the same initial state. Hence the probabilistically evolving large-scale structure of space-time is not precisely or even always approximately described by the deterministic Einstein equations. Since the standard treatment of gravitational wave propagation assumes the validity of the Einstein equations, it is questionable whether we should expect all its predictions to be empirically verified. In particular, one might expect th…
We discuss some general and relevant features of longitudinal gravitational modes in Horndeski gravity and their interaction with matter media. Adopting a gauge-invariant formulation we clarify how massive scalar and vector fields can induce additional transverse and longitudinal excitations, resulting in breathing, vector and longitudinal polarizations. We review, then, the interaction of standard gravitational waves with a molecular medium, outlining the emergence of effective massive gravitons, induced by the net quadrupole moment due to the molecule deformation. Finally, we investigate the…
We review experiments and theoretical models about the possible mutual interplay between the gravitational field and materials in the superconducting state or other macroscopic quantum states. More generally, we focus on the possibility for quantum macrosystems in a coherent state to produce local alterations of the gravitational field in which they are immersed. This fully interdisciplinary research field has witnessed a conspicuous progress in the last decades, with hundreds of published papers, and yet several questions are still completely open.
The averaging problem in cosmology and the approach of macroscopic gravity to resolve the problem is discussed. The averaged Einstein equations of macroscopic gravity are modified on cosmological scales by the macroscopic gravitational correlation tensor terms as compared with the Einstein equations of general relativity. This correlation tensor satisfies a system of structure and field equations. An exact cosmological solution to the macroscopic gravity equations for a constant macroscopic gravitational connection correlation tensor for a flat spatially homogeneous, isotropic macroscopic spac…
Measurements of the tunneling time are briefly reviewed. Next, time and matter in general relativity and quantum mechanics is examined. In particular, the question arises: How does gravitational radiation interact with a coherent quantum many-body system (a ``quantum fluid'')? A minimal coupling rule for the coupling of the electron spin to curved spacetime in general relativity implies the possibility of a coupling between electromagnetic (EM) and gravitational (GR) radiation mediated by a quantum Hall fluid. This suggests that quantum transducers between these two kinds of radiation fields m…
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 masses operators are equivalent to the expectation value of energy for electron stationary quantum states in a 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 …
Gravitationally Induced Quantum Decoherence of Macroscopic Objects
The Hakubi Center for Advanced Research, Kyoto University（京都大学白川中心）；Department of Physics, Kyoto University（京都大学物理系）；Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）
In their original study of conformal gravity, a candidate alternate gravitational theory, Mannheim and Kazanas showed that in any empty vacuum region exterior to a localized static spherically symmetric gravitational source, the geometry would reduce to the standard attractive gravity Schwarzschild geometry on solar system distance scales. In a recent paper Flanagan has argued that this would not be the case if the source has associated with it a macroscopic scalar field which makes a non-zero contribution to the energy-momentum tensor in the otherwise empty exterior region. In this paper we e…
Based on the macroscopic equations of cosmological evolution obtained earlier by the Author, a closed system of macroscopic Einstein equations in the short-wave approximation for perturbations of the scalar Higgs and gravitational fields has been obtained and examined. The obtained exact solutions of macroscopic equations are determined by three microscopic parameters, depending on the spectrum of perturbations.
We determine active gravitational mass operator of the simplest composite quantum body - a hydrogen atom - within the semiclassical approach to the Einstein equation for a gravitational field. We show that the expectation value of the mass is equivalent to energy for stationary quantum states. On the other hand, it occurs that, for quantum superpositions of stationary states with constant expectation values of energy, the expectation values of the gravitational mass exhibit time-dependent oscillations. This breaks the equivalence between active gravitational mass and energy and can be observed…
We consider the gravitational potential and the gravitational rotation field generated by an spherical mass distribution with exponential density, when the force between any two mass elements is not the usual Newtonian one, but some general central force. We invert the usual integral relations and obtain the elemental interaction (between two point-like masses) as a function of the macroscopic gravitational field (the one generated by the distribution). Thus, we have a direct way for testing the possibility of finding a correction to the Newtonian law of gravity that can explain the observed d…
The averaging problem in general relativity is briefly discussed. A new setting of the problem as that of macroscopic description of gravitation is proposed. A covariant space-time averaging procedure is described. The structure of the geometry of macroscopic space-time, which follows from averaging Cartan's structure equations, is described and the correlation tensors present in the theory are discussed. The macroscopic field equations (averaged Einstein's equations) derived in the framework of the approach are presented and their structure is analysed. The correspondence principle for macros…
We discuss a limit for sensitivity of length measurements which is due to the effect of vacuum fluctuations of gravitational field. This limit is associated with irreducible quantum fluctuations of geodesic distances and it is characterized by a noise spectrum with an order of magnitude mainly determined by Planck length. The gravitational vacuum fluctuations may (in an analysis restricted to questions of principle and when the measurement strategy is optimized) dominate fluctuations added by the measurement apparatus if macroscopic masses, i.e. masses larger than Planck mass, are used.
A significant fraction of cosmological dark matter can be formed by very dense macroscopic objects, for example primordial black holes. Gravitational waves offer a promising way to probe these kinds of dark-matter candidates, in a parameter space region that is relatively untested by electromagnetic observations. In this work we consider an ensemble of macroscopic dark matter with masses in the range $10^{-13}$ - $1\,M_{\odot}$ orbiting a super-massive black hole. While the strain produced by an individual dark-matter particle will be very small, gravitational waves emitted by a large number o…
New theoretical approaches developed in the last years predict that macroscopic quantum gravity effects in black holes should lead to modifications of the gravitational wave signals expected in the framework of classical general relativity, with these modifications being characterized by the existence of dampened repetitions of the primary signal. Here we use the fact that non-perturbative corrections to the near-horizon external geometry of black holes are necessary for these modifications to exist, in order to classify different proposals and paradigms with respect to this criterion and stud…
It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics may turn to be relevant to the derivation of the second law of thermodynamics. In particular, the spreading of the probability density of the center of mass of an isolated macroscopic body does not imply delocalization of the wave function, but on the contrary it corresponds to an entropy growth.
We propose a Newtonian semiclassical gravity theory based on the GRW collapse theory with matter density ontology (GRWm), which we term GRWmN. The theory is proposed because, as we show, the standard Newtonian semiclassical gravity theory based on the Schroedinger-Newton equations does not have a consistent Born rule probability interpretation for gravitationally self-interacting particles and implies gravitational cat states for macroscopic mass superpositions. By contrast, we show that GRWmN has a consistent statistical description of gravitationally self-interacting particles and adequately…
The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained …
In the macroscopic gravity approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We present exact cosmological solutions to the equations of macroscopic gravity for a spatially homogeneous and isotropic macroscopic space-time and find that the correlation tensor is of the form of a spatial curvature term. We briefly discuss the physical consequences of these results.
A space with deformed commutation relations for coordinates and momenta leading to generalized uncertainty principle (GUP) is studied. We show that GUP causes great violation of the weak equivalence principle for macroscopic bodies, violation of additivity property of the kinetic energy, dependence of the kinetic energy on composition, great corrections to the kinetic energy of macroscopic bodies. We find that all these problems can be solved in the case of arbitrary deformation function depending on momentum if parameter of deformation is proportional inversely to squared mass.
We consider a thought experiment where the preparation of a macroscopically massive or charged particle in a quantum superposition and the associated dynamics of a distant test particle apparently allow for superluminal communication. We give a solution to the paradox which is based on the following fundamental principle: any local experiment, discriminating a coherent superposition from an incoherent statistical mixture, necessarily requires a minimum time proportional to the mass (or charge) of the system. For a charged particle, we consider two examples of such experiments, and show that th…
The detectability of gravitational waves originating from primordial black holes or other large macroscopic dark-matter candidates inspiraling into Sagittarius ${\rm A}^{\!*}$ is investigated. It is shown that LISA should be a formidable machine to detect the stochastic signal from the accumulated collection of such objects that have been gravitationally captured by Sgr$\,{\rm A}^{\!*}$, provided they have masses above $\sim 10^{-20}\,M_{\odot}$. This is especially so, if they are the main component of the dark matter, and the dark-matter density at Sagittarius ${\rm A}^{\!*}$ exceeds its valu…
A model quantum system is proposed to describe position states of a massive body in flat space on large scales, excluding all standard quantum and gravitational degrees of freedom. The model is based on standard quantum spin commutators, with operators interpreted as positions instead of spin, and a Planck-scale length $\ell_P$ in place of Planck's constant $\hbar$. The algebra is used to derive a new quantum geometrical uncertainty in direction, with variance given by $\langle Δθ^2\rangle = \ell_P/L$ at separation $L$, that dominates over standard quantum position uncertainty for bodies great…
Macroscopic quantum vacuum and modern theories of gravitation share the strong interplay between geometry and physical phenomena. We review selected issues related to the accuracy of the measurement of Casimir forces with particular emphasis on the implications for the search of non-Newtonian gravitational forces in the micrometer range. We then discuss the interplay of the Higgs particle with gravitation, arguing that spectroscopic shifts in atomic transitions due to the modifications of the vacuum expectation value of the Higgs field in regions with strong curvature of space-time may be of c…
A theory recently proposed by the author aims to explain decoherence and the thermodynamical behaviour of closed systems within a conservative, unitary, framework for quantum gravity by assuming that the operators tied to the gravitational degrees of freedom are unobservable and equating physical entropy with matter-gravity entanglement entropy. Here we obtain preliminary results on the extent of decoherence this theory predicts. We treat first a static state which, if one were to ignore quantum gravitational effects, would be a quantum superposition of two spatially displaced states of a sing…
We investigate leading order deviations from general relativity that violate the Einstein equivalence principle in the gravitational standard model extension. We show that redshift experiments based on matter waves and clock comparisons are equivalent to one another. Consideration of torsion balance tests, along with matter wave, microwave, optical, and Mössbauer clock tests, yields comprehensive limits on spin-independent Einstein equivalence principle-violating standard model extension terms at the $10^{-6}$ level.
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 detec…
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…
As a consequence of the extreme precision of the measurements it performs, an interferometric gravitational wave detector is a macroscopic apparatus for which quantum effects are not negligible. I observe that this property can be exploited to probe some aspects of the interplay between Quantum Mechanics and Gravity.
The article discusses and substantiates a self-consistent approach to the macroscopic description of systems with gravitational interaction. Corrections to the equation of state of the fluid are found based on macroscopic Einstein equations which were obtained by averaging over microscopic spherically symmetric metric fluctuations created by the primary Black Holes in a fluid medium. It is shown that these corrections are effectively equivalent to addition of a fluid to the system with the equation of state $p=-\varepsilon$. In addition, it is shown that, in this case, the mass of Black Holes …
The higher derivative gravitational theories exhibit new phenomena absent in General Relativity. One of them is the possible formation of the so called double layer which is the pure gravitational phenomenon and can be interpreted, in a sense, as the gravitational shock wave. In this paper we show how some very important features of the double layer equations of motion can be extracted straight from the least action principle.
Long before the general theory of relativity was finally formulated in 1916, arguments based entirely on Einstein's equivalence principle predicted the well known phenomenon of the gravitational red shift. Precisely the same arguments are widely being used today to derive the same phenomenon. Accordingly, it is often claimed that the observed gravitational red shift is a verification of the equivalence principle rather than a verification of the full theory of general relativity. Here we show that, contrary to these claims, the arguments based on the equivalence principle are false and that \i…
For a given space-time and for an arbitrary time-like geodesic, we analyze the conditions for the construction of Fermi coordinates so that they are also rigid covariant. We then apply these conditions to linear plane gravitational waves.
We will use Fisher information to properly analyze the quantum weak equivalence principle. We argue that gravitational waves will be partially reflected by superconductors. This will occur as the violation of the weak equivalence principle in Cooper pairs is larger than the surrounding ionic lattice. Such reflections of virtual gravitational waves by superconductors can produce a gravitational Casimir effect, which may be detected using currently available technology.
A scheme of an optical detector is proposed for checking Einsteins equivalence principle (EEP) in a null gravitational redshift experiment and for testing methods for calculating the length of a resonator in a weak variable gravitational field by recording the variations of the difference frequency of resonators caused by lunisolar variations of the geopotential in a double or a two-resonator laser system.
We review the hypothesis of the existence of gravitational magnetic monopoles (H-pole for short) defined in analogy with the Dirac's hypothesis of magnetic monopoles in electrodynamics. These hypothetical dual particles violate the equivalence principle and are accelerated by a gravitational field. We propose an expression for the gravitational force exerted upon an H-pole. According to GR ordinary matter (which we call E-poles) follows geodesics in a background metric. The dual H-poles follows geodesics in an effective metric.
We analyze propagation equations for the polar modes of gravitational waves in cosmological space-times. We prove that polar gravitational waves must perturb the density and non-azimuthal components of the velocity of material medium of the Friedman-Lemaitre-Robertson-Walker spacetimes. Axial gravitational waves can influence only the azimuthal velocity, leading to local cosmological rotation. The whole gravitational dynamics reduces to the single "master equation" that has the same form for polar and axial modes. That allows us to conclude that the status of the Huygens principle is the same …
It is shown that the gravitational ultrarelativistic spin-orbit interaction violates the weak equivalence principle in the traditional sense. This fact is a direct consequence of the Mathisson-Papapetrou equations in the frame of reference comoving with a spinning test particle. The widely held assumption that the deviation of a spinning test body from a geodesic trajectory is caused by tidal forces is not correct
Take the gravitational field, a number of macroscopic observers and put them in a highly-quantum state, made by a coherent superposition of different classical configurations. How would that look like? I try to study this system by introducing a spatial distance operator among the observers and looking at the properties of its expectation values.
The Strong Equivalence Principle (SEP) holds the full essence and meaning of the General Theory of Relativity as the nonlinear relativistic theory of gravitation. It asserts the universal coupling of gravity to all matter and its interactions including the gravitational interaction and the gravitational self energy. We point out that confirming the gravitational coupling to gravitons, and hence to the gravitational waves, is the direct test of the SEP. We show that the near simultaneous detection of gravitational waves and gamma rays from the merger of binary neutron stars provides a unique an…
In this work, the experiment is discussed on the verification of the principle of universality of gravitational interactions and some related problems of gravity theory and physics of elementary particles. The meaning of this proposal lies in the fact that the self-consistency of General Relativity, as it turns out, presuppose the existence of the nongravitating form of energy. Theory predicts that electrons are particles that transfer the nongravitating form of energy.
Thermodynamics plays an important role in gravitational theories. It is a principle independent of the gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system in a general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of perfect fluid inside a…
The article proposes generalizations of the macroscopic model of plasma of scalar charged particles to the cases of inter-particle interaction with multiple scalar fields and negative effective masses of these particles. The model is based on the microscopic dynamics of a particle at presence of scalar fields. The theory is managed to be generalized naturally having strictly reviewed a series of its key positions depending on the sign of particle masses. Thereby, it is possible to remove the artiicial restriction contradicting the more fundamental principle of action functional additivity.
