Abstract We probe the dynamics of a modified form of the Schrödinger–Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of ‘collapse’ (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of . At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the SN system. Unlike the SN dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.
Abstract We probe the dynamics of a modified form of the Schrödinger–Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of ‘collapse’ (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of . At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the SN system. Unlike the SN dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.
It has been suggested that the nonlinear Schrödinger–Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the Møller–Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than 1.14(ℏ 2 /( G σ)) 1/3 , a wave packet of width σ partially 'collapses' to a groundstate solution found by Moroz, Penrose and Tod, with excess probability dispersing away. However, for a mass less than 1.14(ℏ 2 /( G σ)) 1/3 , the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.
Abstract The Schrödinger–Newton (SN) model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the SN interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the SN model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.
We reconsider the time-dependent Schrödinger–Newton equation as a model for the self-gravitational interaction of a quantum system. We numerically locate the onset of gravitationally induced inhibitions of dispersion of Gaussian wave packets and find them to occur at mass values more than six orders of magnitude higher than reported by Salzman and Carlip (Salzman and Carlip 2006, arXiv:gr-qc/0606120, Carlip 2008 Class. Quantum Grav. 25 107–44), namely at about 10 10 u. This fits much better to simple analytical estimates but unfortunately also questions the experimental realizability of the proposed laboratory test of quantum gravity in the foreseeable future, not just because of large masses, but also because of the need to provide sufficiently long coherence times.
The Schr\"odinger-Newton equation is a proposed model to explain the localization of macroscopic particles by suppressing quantum dispersion with the particle's own gravitational attraction. On cosmic scales, however, dark energy also acts repulsively, as witnessed by the accelerating rate of universal expansion. Here, we introduce the effects of dark energy in the form of a cosmological constant $\mathrm{\ensuremath{\Lambda}}$, that drives the late-time acceleration of the Universe, into the Schr\"odinger-Newton approach. We then ask in which regime dark energy dominates both canonical quantum diffusion and gravitational self-attraction. It turns out that this happens for sufficiently delocalized objects with an arbitrary mass and that there exists a minimal delocalization width of about 67 m. While extremely macroscopic from a quantum perspective, the value is in principle accessible to laboratories on Earth. Hence, we analyze, numerically, how the dynamics of an initially spherical Gaussian wave packet is modified in the presence of $\mathrm{\ensuremath{\Lambda}}>0$. A notable feature is the gravitational collapse of part of the wave packet, in the core region close to the center of mass, accompanied by the accelerated expansion of the more distant shell surrounding it. The order of magnitude of the distance separating collapse from expansion matches analytical estimates of the classical turnaround radius for a spherically symmetric body in the presence of dark energy. However, the time required to observe these modifications is astronomical. They can potentially be measured only in physical systems simulating a high effective cosmological constant, or, possibly, via their effects on the inflationary universe.
We study the collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction. Equilibrium states in which the gravitational attraction and the attraction due to the self-interaction are counterbalanced by the quantum pressure (Heisenberg's uncertainty principle) exist only below a maximum mass ${M}_{\mathrm{max}}=1.012\ensuremath{\hbar}/\sqrt{Gm|{a}_{s}|}$ where as<0 is the scattering length of the bosons and m is their mass [P. H. Chavanis, Phys. Rev. D 84, 043531 (2011)]. For M>Mmax the system is expected to collapse and form a black hole. We study the collapse dynamics by making a Gaussian ansatz for the wave function and reducing the problem to the study of the motion of a particle in an effective potential. We find that the collapse time scales as $(M/{M}_{\mathrm{max}}\ensuremath{-}1{)}^{\ensuremath{-}1/4}$ for $M\ensuremath{\rightarrow}{M}_{\mathrm{max}}^{+}$ and as ${M}^{\ensuremath{-}1/2}$ for $M\ensuremath{\gg}{M}_{\mathrm{max}}$. Other analytical results are given above and below the critical point corresponding to a saddle-node bifurcation. We apply our results to QCD axions with mass $m={10}^{\ensuremath{-}4}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}5.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}53}\text{ }\text{ }\mathrm{m}$ for which ${M}_{\mathrm{max}}=6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ and $R=3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{R}_{\ensuremath{\bigodot}}$. We confirm our previous claim that bosons with attractive self-interaction, such as QCD axions, may form low mass stars (axion stars or dark matter stars) but cannot form dark matter halos of relevant mass and size. These mini axion stars could be the constituents of dark matter. They can collapse into mini black holes of mass $\ensuremath{\sim}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ in a few hours. In that case, dark matter halos would be made of mini black holes. We also apply our results to ultralight axions with mass $m=1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}8.29\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}60}\text{ }\text{ }\mathrm{fm}$ for which ${M}_{\mathrm{max}}=0.39\ifmmode\times\else\texttimes\fi{}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ and R=33pc. These ultralight axions could cluster into dark matter halos. Axionic dark matter halos with attractive self-interaction can collapse into supermassive black holes of mass $\ensuremath{\sim}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ (similar to those reported at the center of galaxies) in about one million years. We point out the limitations of the Gaussian ansatz to describe the late stages of the collapse dynamics. We also mention the possibility that, instead of forming a black hole, the collapse may be accompanied by a burst or relativistic axions (bosenova) leading to a cycle of collapses and explosions as observed for nongravitational Bose-Einstein condensates with attractive self-interaction.
We propose an effective non-relativistic framework in which wave-function collapse emerges as a deterministic dynamical instability induced by gravitational self-interaction and regulated by short-distance repulsion. The dynamics is described by a nonlinear Schrödinger equation supplemented by a phenomenological repulsive sector ensuring regularity at high densities. Using a variational Gaussian ansatz, we derive an explicit effective energy functional and show that extended quantum states lose stability beyond a critical mass scale. This loss of stability is associated with a bifurcation in the reduced dynamical system governing the wave-function width, leading to the emergence of stable localized configurations. Within this picture, collapse corresponds to the dynamical selection of one of these localized attractors, driven by infinitesimal asymmetries in the initial state and occurring without stochastic noise or environmental coupling. The mechanism provides a controlled and quantitative realization of gravity-induced localization, extending Schrödinger--Newton-type models while avoiding their pathological short-distance behavior. Possible implications for mesoscopic systems probing the quantum-to-classical transition are briefly discussed.