McCullochs’s Quantized Inertia theory attempts to phenomenologically explain the success of modified Newtonian dynamics MOND. We explained MOND in a previous post, and how impressive it is that with a simple modification to Newton’s law, one can predict the rotation curves of galaxies without the need for any physical dark matter, and with a single new constant. The interpolating functions in MOND and Quantized inertia are not exactly the same but behave very similarly.
Here a is the true measured acceleration (for instance of a star around the galaxy) and a0 is the acceleration scale. In MOND, this parameter is a constant, and in Quantized inertia, it is a varying parameter with the radius of the comoving horizon, i.e., the particle horizon or the size of the observable universe. The reason behind this choice is that McCulloch interprets this radius of the universe as a Rindler horizon that appears for an accelerated body, and has a corresponding Unruh radiation. He then argues that it is this radiation from the quantum vacuum, the one that modifies inertia in galaxy rotation curves, because at a particular low acceleration scale, the Rindler horizon becomes larger than the radius of the comoving horizon. For a detailed explanation of this mechanism, I will leave in the description more information.
As far as I know, McCulloch does not derive MOND’s expression or its function from his proposed quantum vacuum mechanism. But this is not a problem, since no one has been able to derive MOND from some fundamental principle or mechanism. In other words, no one has shown where MOND comes from.
And apart from the fact that I would expect Planck’s constant to appear in the formulation if MOND had an origin in the quantum vacuum, there is another problem that Quantized Inertia presents. Stacy McGaugh has measured that the Tully-Fisher law and the acceleration scale constant is the same for nearby galaxies and for very far away galaxies at redshift z=2,5, i.e., that the acceleration scale hasn’t changed during the expansion of the universe. Quantized inertia implies a dependence of the acceleration scale with the size of the universe as a Rindler horizon, and if our cosmic expansion model is correct, Stacy claims that Quantized inertia is ruled out.
If the acceleration scale depended on the size of the universe according to Quantized Inertia, this would make galaxies show greater mass discrepancies starting at a greater acceleration scale, and thus, overall faster spins. McCulloch claims that this is observed in a few high redshift galaxies, and we think that this disagreement between McGaugh and McCulloch should be easy to settle down by the evidence in the next years, with better measurements of high redshift galaxy rotation curves.
Another difference between MOND and Quantized Inertia is that the latter is claimed to not have an external field effect which is present in MOND. The external field effect is essential in MOND to describe systems which are apparently dark matter free, by taking into account the externally imposed acceleration by a mother system, by which it sets the local system in the Newtonian regime.
In contrast, I have been supporting in this website a different attempt to phenomenologically explain MOND. You can think about my version as the Machian approach and McCulloch’s as the quantum vacuum (Unruh radiation and Rindler horizon) approach. In my Machian MOND, I hypothesize that a0 is not constant or the speed of light squared divided by the size of the universe, but the gravitational field intensity of the observable universe. This also matches in orders of magnitude. And while Quantized Inertia directly predicts a varying acceleration scale along the universe expansion, my Machian version does not, if the gravitational field intensity of the observable universe hasn’t changed along the cosmic expansion. And the question is, has it? Well there are many uncertainties at play here. The radius of the universe depends on the Hubble tension, and the mass of the universe should really be the energy density of the universe, since all forms of energy gravitate. The universe at large is a relativistic system (you cannot do cosmology with Newton), and perhaps a relativistic contribution to the mass of the universe is needed. Moreover, there could be missing mass in clusters even if MOND solves the need for dark matter in galaxies. I will try in the future to work this out in a future paper.
But don’t get me wrong, the intention behind Quantized Inertia is good, and even Mordehai Milgrom, the author of MOND, favors the idea that MOND comes from the quantum vacuum modifying inertia. In this post I present why I disagree with both McCulloch and Milgrom in this idea of MOND being related to the quantum vacuum. But I thank McCulloch for putting all his effort in trying to explain MOND, a challenge that I have also accepted but with a Machian interpretation.
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