MOND as a transformation between non-inertial reference frames via Sciama’s interpretation of Mach’s Principle

Very few authors have attempted a phenomenological explanation of the successful MOdified Newtonian Dynamics based on Mach’s principle. Milgrom’s MOND is already explained in one of our posts, see Does MOND work? and the origin of inertia. Mach’s principle is also explained in one of our posts, see Mach’s principle as an introduction to MOND and modified inertia. We briefly introduce Machian MOND here.

The baryonic mass in galaxies is observed through luminosity and mass-luminosity relationships, and rotational tangential velocities of stars and gas through the doppler effect. There are four important regularities or patterns found in galaxies, which we refer to as the new Kepler laws for galaxy dynamics. First, tangential velocities in galaxies are generally asymptotically flat in galaxy rotation curves, while Newtonian gravity predicts a Keplerian falloff. Secondly, these asymptotic flat velocities are proportional to the fourth root of the observed mass, a relationship called the Tully-Fisher law. Thirdly, features in observed mass strongly correlate with features in the velocity profiles, a relationship called Renzo’s rule. And finally, Newtonian gravity correctly predicts velocity profiles at the cores of galaxies, and mass discrepancies are observed only below a particular acceleration scale, a feature called the cuspy halo problem in dark matter models. These regularities do not have a simple explanation within the physical dark matter halo hypothesis.

Milgrom developed in 1983 a model called MOND, modified Newtonian dynamics, or Milgromian dynamics, as a correction to either Newtonian gravity or Newton’s second of inertia, through a correction factor called an interpolating factor. Milgrom has favored the modified inertia version instead of just modifying gravity. This interpolating function is precisely derived to match the observed patterns of galaxy rotation dynamics described earlier. The interpolating function interpolates between a Newtonian regime in which there is no correction to Newton’s laws (to give the right prediction for planetary dynamics), and the so called deep MOND regime through the total acceleration or Newotnian field, and an acceleration scale parameter a0, which is a new fundamental constant introduced in the MOND model. Below this particular data-fitted acceleration scale value of around 10^-10 m/s^2, the velocity of stars around galaxies is proportional to the fourth root of the observed mass, and above this acceleration scale (such as the case for the solar system), velocities are proportional to the Newtonian square root of mass, so that in the limit of a0 approaching zero, Newtonian gravity or Newtonian inertia is recovered.

Progress in fundamental physics has been made in the past by reducing the number of fundamental constants (such as the gravitational acceleration of the Earth and other planets for a universal gravitational constant, the unification of the speed of light with vacuum permittivity and permeability, and atomic constants for Planck’s constant). Therefore, it is of considerable interest to explore modifications of the laws of gravity without the need for more fundamental constants. Milgrom first suggested a possible coincidence between his new introduced acceleration constant with the cosmological constant, and briefly conjectured a possible connection with Mach’s principle in some of his papers, but he never developed this idea further. He stated that “an attractive possibility is that MOND results as a non-relativistic, small-scale expression of a fundamental theory by which inertia is a vestige of the interaction of a body with “the rest of the Universe”, in the spirit of Mach’s principle.”, that “The only system that is strongly general relativistic and in the MOND regime is the universe at large”, and that “It has been long suspected that local dynamics is strongly influenced by the universe at large, a la Mach’s principle, but MOND seems to be the first to supply a concrete evidence for such a connection”. In fact, the cosmological constant is related to the energy density of vacuum in General Relativity, but another coincidence in orders of magnitude is that of the energy density of the vacuum with the energy density of matter. Through this reasoning, together with Sciama’s relationship for the gravitational constant, Milgrom’s acceleration scale constant corresponds to something that loosk like a gravitational acceleration of the rest of the mass of the observable universe. Performing this substitution, we arrived at a so called Machian MOND formulation for Newtonian gravity or for inertia:

So why is MOND Mach’s principle? This Machian MOND formulation, which is equivalent to MOND, contains the same parameters as the Machian modified theories of inertia of Sciama and Treder, which we covered in a previous post. MOND can be thought as a stronger dependence of local dynamics on global parameters, because MOND is nonlinear due to its external field effect (in contrast to Sciama’s or Treder’s models, which are linear). A decrease in the local field intensity is equivalent to a decrease in inertia, which aligns with the Machian consequence by which ‘the inertia of a body decreases when masses are removed from its neighbourhood’. Also, decreasing the mass of the universe is equivalent to a decrease of inertia, through Sciama’s relationship for the gravitational constant. Machian MOND contains no fundamental constants or parameters except for the speed of light. The velocities of stars around below the universe’s field intensity are computed directly out of a factor of the speed of light through the fourth root of their visible mass, equivalent to the deep MOND regime. The Machian MOND interpolating function respects the global scale-invariance of mass, length and time of the Sciama’s relationship in all regimes of application, unlike classical Newtonian gravity. This means that observables such as velocities do not depend on global re-scaling of all masses or all distances, if we double all masses of the universe, the velocities take the same value. MOND is only length scale-invariant in the deep MOND regime, because in the Newtonian regime it reduces to the Newtonian framework. It also respects the Machian effect of Sciama’s relationship by which in absence of a background universe, substituting the mass and radius of the universe for the local mass and radius, the interpolating function reduces to a simple number which can be set to unity with the appropriate choice of integers within the interpolating function, so that rotation is undefined up to the speed of light. The analogous formal limit in which a0 goes to zero cannot be made in Machian MOND, because the mass of the universe can never be zero in the system under study, it can only be the very same mass of the system itself. This agrees with the though experiment of having two masses rotating around each other in empty space, in which from a Machian point of view without absolute space, the two masses define the inertial frame and one cannot speak about rotation of any kind. The system can choose between any rotation up to the speed of light.

In summary, Machian MOND is a reformulation of MOND in terms of Mach’s principle, which hints that the origin of MOND is the relativity of inertia of Mach’s principle. It shows how Newtonian gravity and General Relativity are incomplete due to not incorporating Mach’s principle and being unable to explain galaxy rotation dynamics. The Machian MOND interpolating function can be thought to apply to inertial mass or just to gravity as a correction to the gravitational constant. Special Relativity relativizes velocities through the Lorentz correction factor with a factor of velocities and the speed of light, but acceleration is required to be absolute for consistency. Machian MOND seems to come from relativizing accelerations through a correction factor in a similar way, because it contains a fraction of field intensity divided by the universe’s field intensity. The unanswered question is where does the Machian MOND interpolating function comes from. We know it must come from a nonlinear theory, but currently only non-relativistic linear theories of modified inertia à la Mach exist, such as Sciama’s or Treder’s, which are like gravitoelectromagnetism: linear and not Lorentz invariant. So, the interpolating function is probably an effective correction in the weak or linearized limit. This means that in the same way General Relativity reduces to the Newtonian gravity law in the weak limit, a complete nonlinear and Machian theory of gravity should reduce to Machian MOND in its weak limit. The interpolating function may not appear explicitly in this theory, but the acceleration scale of the rest of the universe should appear somehow, in contrast to General Relativity, in which solutions to local dynamics do not depend on the mass or size of the observable universe, because they assume asymptotic flat spacetime at infinity. The solution should then come from making General Relativity fully Machian.

As final remarks, MOND generally solves dark matter effects in galaxy rotation curves, but falls short in galaxy clusters. MOND does reduce the mass discrepancy observed in clusters, but a mass discrepancy of around 3:1 or 2:1 is still observed. The mass discrepancies in clusters when using MOND curiously appear at the core of the clusters, unlike in galaxy rotation curves, which appear far away from their centers when not using MOND. Thus, the physical dark matter hypothesis makes more sense in clusters than in galaxies, when using MOND. In the same way Milgrom identified that the mass discrepancy in galaxies occur at a certain acceleration scale, Jakob Bekenstein identified that the mass discrepancy in clusters when using MOND occurs at a certain potential scale. A modification based on potentials should only affect the low acceleration regime where the correction of Machian MOND applies. Otherwise, it would result in stronger gravity in high-potential high-acceleration regime such as neutron stars, whose dynamics constrain a possible modification of gravity in them.

Can this Machian MOND expression be derived from a nonlinear relativistic generalized theory of modified inertia or modified gravity that satifies Mach’s principle?

Research published in MOND as a Transformation Between Non-inertial Reference Frames Via Sciama’s Interpretation of Mach’s Principle, International Journal of Theoretical Physics, Volume 63, article number 271, (2024)

Preprint available at: ArXiv /2410.19007 and Researchgate

Mentioned by Alexander Unzicker: https://www.youtube.com/watch?v=5yw7m0xHTCk