Post-Newtonian Gravitational Collapse: How Rotation Destroys Quantum Superposition ✦
A general collapse mechanism tied to gravity is proposed, grounded in linearized gravity. From the weak-field limit of general relativity, gravitoelectromagnetism yields an effective coupling between the gravitoelectric potential and mass density, as well as between the gravitomagnetic vector potential and mass current. Within a hybrid (classical-quantum) framework, these couplings lead to a master equation whose non-unitary part is determined by the mass distribution and currents. Considering only the gravitoelectric potential reproduces the Diósi–Penrose collapse model, acting on positional degrees of freedom. Incorporating the gravitomagnetic potential introduces additional collapse mechanisms for rotational degrees of freedom and mixed mass-rotation contributions.
Context
The problem of measurement in quantum mechanics remains one of the deepest mysteries. Wave function collapse models offer a phenomenological way out, parameterizing deviations from unitary evolution at macroscales. Among them, a special place is occupied by the Diósi–Penrose model, where collapse is linked to gravity in the Newtonian limit. However, until now it accounted only for mass density distribution, ignoring mass currents—and thus rotation. But modern experiments, from levitating nanoparticles to astrophysical observations of pulsars, increasingly involve rotational degrees of freedom, demanding an extension of the theory.
Methods
The authors turned to the linearized general theory of relativity of Einstein in the weak field and slow motion limit. Here, gravitoelectromagnetism emerges—a formal analogy with electrodynamics, where the gravitoelectric potential (analogous to the scalar potential) and the gravitomagnetic vector potential (analogous to the vector potential) obey equations similar to Maxwell’s. Following Diósi’s hybrid classical-quantum dynamics, they linked these potentials to quantized four-dimensional mass currents. To ensure positivity of evolution and preserve the uncertainty relations of Heisenberg, Gaussian white noise was added to the Hamiltonian, and the resulting master equation was averaged over the noise. This led to the emergence of decoherence with three types of channels.
Results
The first channel reproduces the standard Diósi–Penrose model acting on positional degrees of freedom. The second is purely rotational, quadratic in the mass current. For rigid rotors, it reduces to decoherence in the angular momentum basis, remarkably affecting even perfectly symmetric bodies rotating around their symmetry axis—a case where the previous model shows no effect. The third consists of mixed terms linking density and current. The relative magnitude of the channels scales as v^2/c^2: for 6-GHz levitating nanospheres (surface v/c ~ 10^-5), the rotational channel is suppressed by 10 orders of magnitude, but its strength is determined by independent parameters. For millisecond pulsars—neutron stars with rotation frequencies up to 716 Hz—v/c reaches 0.15, and all channels are comparable. Estimates for binary pulsars give a rotational decoherence rate of order 10^78 s^-1, which is comparable to the positional decoherence for a superposition with a separation of ~300 m. Such decoherence should suppress gravitationally mediated entanglement between rotating masses.
Implications
The proposed formalism provides for the first time a systematic language for describing collapse that includes mass currents. It predicts fundamentally new decoherence channels that are not reducible to positional ones and opens access to an independent sector of the theory—gravitomagnetic noise, which is completely unconstrained by existing experiments testing the Diósi–Penrose model.
Future development
In the near future, the explicit form of the noise correlators D^μν must be found to connect them with known collapse models and identify specific rotational signatures. The formalism can also be adapted for mesoscopic devices, where both translational and rotational degrees of freedom are coherently controlled, allowing the computation of concrete decoherence rates and the imposition of constraints from current experimental data.
Impact
The work will have an impact on fundamental tests of quantum mechanics, optomechanics with levitating particles, astrophysics of neutron stars, and the search for quantum effects of gravity in laboratory settings.
Next steps
It is necessary to determine the functional form of the gravitomagnetic noise kernels D^kl_A from theoretical considerations and conduct targeted experiments with rapidly rotating levitating nanoparticles, as well as analyze pulsar timing data for anomalous loss of coherence.
Key open problems
The model is directly connected to the problem of quantum measurement and the unification of quantum mechanics with gravity. It offers a testable mechanism that could manifest in an as-yet unexplored domain—rotational degrees of freedom—bringing us closer to understanding how gravity modifies quantum theory.
🎯 The fastest man-made rotors—levitating nanospheres—spin at a frequency of 6 GHz, corresponding to a linear surface speed of about 0.0016% of the speed of light. By comparison, the record-holder among natural rotors—the millisecond pulsar PSR J1748-2446ad—completes 716 revolutions per second, and points on its surface move at nearly 15% of the speed of light. Truly, nature provides us with ready-made quantum laboratories on a cosmic scale!
🎬 In Robert L. Forward's novel Dragon’s Egg, life on the surface of a neutron star is described, where monstrous gravity and rapid rotation play a key role. Perhaps such worlds are natural testing grounds for the post-Newtonian quantum effects predicted in the article.
Key numbers
- surface speed of levitating nanosphere (6 GHz): 0.0016% of light speed (v/c ≈ 1.6×10⁻⁵)
- surface speed of millisecond pulsar: ~15% of light speed
- estimated rotational decoherence rate for binary pulsar system: ~10⁷⁸ s⁻¹
- comparable superposition separation in DP model for the same system: ~300 m