In a previous post, we explained how Modified Newtonian Dynamics (MOND), a modification to Newtonian laws that correctly predicts galaxy rotation curves without dark matter by introducing a single universal constant, falls short in galaxy clusters. We attempted to solve this mismatch, but ended up rediscovering an attempt suggested by Bekenstein, and later developed by others under the name of EMOND, extended MOND. In this one, we will show how a small modification to MOND inspired by Mach’s principle seems to fix this shortcoming of MOND, while retaining its successes in galaxies and the solar system.
The typical inferred accelerations of galaxies in galaxy clusters at their cores is around a0, 10^-10 m/s^2. This implies that MOND, a correction to Newtonian gravity or inertia kicking in at accelerations of a0 and smaller, predicts only a small correction in them. MOND leaves a mass discrepancy (between observed and required to explain its dynamics) at the cores of galaxy clusters of around 2-5, which decreases with increasing distance from the core. Thus, pristine MOND still requires dark matter at galaxy clusters to fit observations, even though it helps to reduce the amount of dark matter required with Newtonian laws.
It is well known that a boost of a0 by a factor of a few at the cores generally resolves the discrepancy that MOND leaves in galaxy clusters. It seems that a boost in terms of potentials is not natural, so we focused on field intensities instead.
We explained Mach’s principle before, and how it could be behind MOND. One of the conclusions drawn from those posts is that a0 matches what seems to be the ‘gravitational acceleration’ of the universe at large, GMu/Ru^2, in order orders of magnitude. But this can’t be a gravitational acceleration, since accelerations vanish inside a spherically symmetric mass distribution, be that the observable universe at large. Such coincide can only come from the inverse-squared distance sum of every mass element, a scalar field which goes with 1/r^2 instead of the usual 1/r. Let’s assume a0 is that scalar field. What are the consequences in local dynamics?
In the solar system, such scalar field gives GMu/Ru^2 + GM/r^2 with M the mass of the sun and r our distance to it. This does not give the required a0 that allows MOND to fit the solar system by not modifying Newtonian laws. It is straightforward that the new a0, lets denote it a’0, must be such scalar field minus the modulus of the Newtonian acceleration. That gives a MONDian a0 in galaxies, Newtonian dynamics (MOND in high accelerations) in the solar system, and will naturally give a boost to MOND at the cores of galaxy clusters via a position dependent a0.
In order to see what are we really doing by introducing such scalar field, the difference between the scalar field and the modulus of the Newtonian acceleration is that the Newtonian acceleration accounts for vectorial cancelling. If we imagine the observable universe to be a hollow sphere with mass Mu and radius Ru, the Newtonian acceleration inside is always zero due to Newton’s shell theorem, but our field will give Mu/Ru^2 at the center. Inside a galaxy cluster, we can regard the mass external to a certain radius as spherical shell of mass Ms at a radius Rs (so that it gives the same potential inside). Then our a’0 gives GMu/Ru^2 + GMs/Rs^2, a boost that is the greatest at the core and vanishes far away from it. Thus, the external cluster mass contributes to a0 through the directionless scalar sum of inverse-square mass contributions rather than through the Newtonian vector field.
How much of a boost is that? It is roughly what it is required to fit observations (in the paper, we used a slightly different version of the a’0 explained here). How much does the boost change the dynamics inside individual galaxies, where MOND works well? Not a lot, given the uncertainties in the mass to light ratios. And we have done that without introducing extra data-fitted free parameters, as EMOND requires (at least one extra free parameter). Moreover, our boost does not require any tweaking to avoid high boosts at deep potential regimes, e.g., for neutron stars, in the potential-based approaches such as MOND.
This proposal opens the door to a Lagrangian formulation with an extra scalar field satisfying conservation laws, which we will develop in the near future. This work was developed together with Juan David Santander.
Publication: Machian MOND: a variable a0 in galaxy clusters, Int. J. Mod. Phy. D., Vol. 35, 11, 2650043, Q2 (2026): https://doi.org/10.1142/S0218271826500434
Preprint available for free in ArXiv: https://arxiv.org/abs/2608.04894