Abstract: This short note takes Roger Penrose's idea that entropy and quantum state vector reduction must be associated with gravity, along with an additional assumption that momentum or velocity be quantized rather than continuous, and shows that very weak gravitational fields must be modified in a manner that matches the MOND gravitational anomaly.
Roger Penrose has written some compelling arguments for associating gravitational fields with state vector reduction and entropy. In short, the reasoning goes that Stephen Hawking's demonstration that black holes are associated with entropy, along with the fact that black holes violate Liouville's theorem about the constancy of phase space volume, imply that entropy must be associated with a quantum gravity effect that is due to gravity in general, and not just black holes. But his argument implies that there might also be an effect in the reverse direction. That is, that wave function collapse may modify the effective gravitational field. Furthermore, since it appears to be evident that entropy is as active in the weak gravitational field limit as in the strong (where Penrose makes his case) the place to look for gravitational anomalies due to wave function collapse is in the weak gravitational limit. But there is already experimental evidence of something odd going on with weak gravitational fields due to Mordecai Milgrom's MOND (MOdified Newtonian Dynamics) explanation for the gravitational anomalies of galactic rotation data usually attributed to dark matter. The purpose of this note is to suggest a connection between these theories.
One of the most intriguing effects of state vector reduction is the Quantum Zeno Effect, where repeated measurements of a quantum system are shown to result in a suppression of the time evolution of the wave function. The effect is due to the fact that perturbations in quantum mechanics always show up as the square of the wave function. In order to calcuate the perturbation, one needs to have a discrete set of eigenstates, rather than continuous. In this condition, a perturbational analysis of the time evolution of a wave state that begins as an eigenstate, must show that the eigenstate is abandoned with a time dependency proportional to the square of time rather than (as exponential decay implies) simply proportional to time. It can be shown that as time passes, the perturbational approximation eventually fails and exponential decay follows. In short, the effect depends on (a) discrete eigenstates, and (b) wave function collapse occuring at a rate faster than the rate at which perturbational analysis fails.
To assist the calculation, it helps to imagine an experiment where a test particle is used to calculate the gravitational acceleration of a spherical massive body. If we assume that the test particle is negatively charged, we can apply a negative charge to the massive body that will balance the gravitational acceleration, and then from our knowledge of electricity, we can calculate the acceleration of the massive body. Furthermore, since the massive body is spherical, the cancellation of the gravitaional and electrical force must extend over all space. That is, the cancellation must occur in both the strong field regions of the gravitational field and the weak field regions.
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brannenworks.com/PenGrav.html
