Modified Newtonian dynamics


Modified Newtonian dynamics is a theory that proposes a modification of Newton's laws to account for observed properties of galaxies. Modifying Newton's law of gravity results in modified gravity, while modifying Newton's second law results in modified inertia. The latter has received little attention compared to the modified gravity version. Its primary motivation is to explain galaxy rotation curves without invoking dark matter, and is one of the most well-known theories of this class.
MOND was developed in 1982 and presented in 1983 by Israeli physicist Mordehai Milgrom. Milgrom noted that galaxy rotation curve data, which seemed to show that galaxies contain more matter than is observed, could also be explained if the gravitational force experienced by a star in the outer regions of a galaxy decays more slowly than predicted by Newton's law of gravity. MOND modifies Newton's laws for extremely small accelerations, which are common in galaxies and galaxy clusters. This provides a good fit to galaxy rotation curve data while leaving the dynamics of the Solar System with its strong gravitational field intact. However, the theory predicts that the gravitational field of the galaxy could influence the orbits of Kuiper Belt objects through the external field effect, which is unique to MOND.
Since Milgrom's original proposal, MOND has seen some successes. It is capable of explaining several observations in galaxy dynamics, a number of which can be difficult for Lambda-CDM to explain. However, MOND struggles to explain a range of other observations, such as the acoustic peaks of the cosmic microwave background and the matter power spectrum of the large scale structure of the universe. Furthermore, because MOND is not a relativistic theory, it struggles to explain relativistic effects such as gravitational lensing and gravitational waves. Finally, a major weakness of MOND is that all galaxy clusters, including the famous Bullet Cluster, show a residual mass discrepancy even when analyzed using MOND. As a result, it has not gained widespread acceptance.
In 2004, Jacob Bekenstein developed a relativistic generalization of MOND, TeVeS, which however had its own set of problems. Another notable attempt was by and in 2021, which proposed a relativistic model of MOND that is compatible with cosmic microwave background observations; this model requires multiple extra fields and is still unable to match observed gravitational lensing.

Overview

Missing mass problem

Several independent observations suggest that the visible mass in galaxies and galaxy clusters is insufficient to account for their dynamics, when analyzed using Newton's laws. This discrepancy – known as the "missing mass problem" – was identified by several observers, most notably by Swiss astronomer Fritz Zwicky in 1933 through his study of the Coma Cluster. This was subsequently extended to include spiral galaxies by the 1939 work of Horace Babcock on Andromeda.
These early studies were augmented and brought to the attention of the astronomical community in the 1960s and 1970s by the work of Vera Rubin, who mapped in detail the rotation velocities of stars in a large sample of spirals. While Newton's Laws predict that stellar rotation velocities should decrease with distance from the galactic centre, Rubin and collaborators found instead that they remain almost constant – the rotation curves are said to be "flat". This observation necessitates one of the following:
The first option leads to the dark matter hypothesis; the second option leads to alternative theories of gravitation like MOND.
MOND was proposed in 1983 and has been significantly revised ever since. By 2004 Jacob Bekenstein produced a relativistically correct version TeVeS that adds two additional fields and three free parameters to general relativity. The ΛCDM incorporating dark matter and MOND incorporating alternative gravity have different arenas of success. At the large scale of cosmology, galaxy clusters and galaxy formation, ΛCDM has been very successful. MOND is able to describe galaxy scale astronomical observations but fails as a model for cosmology.

Milgrom's law

The basic premise of MOND is that while Newton's laws have been extensively tested in high-acceleration environments, they have not been verified for environments with extremely low acceleration, such as orbits in the outer parts of galaxies. This led Milgrom to postulate a new effective gravitational force law that relates the true acceleration of an object to the acceleration that would be predicted for it on the basis of Newtonian mechanics. This law, the keystone of MOND, is chosen to reproduce the Newtonian result at high acceleration but leads to different behavior at low acceleration:
Here is the Newtonian force, is the object's mass, is its acceleration, is an as-yet unspecified function, and is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires
and consistency with astronomical observations requires
Beyond these limits, the interpolating function is not specified by the hypothesis.
Milgrom's law can be interpreted in two ways:
  • Modified inertia: One possibility is to treat it as a modification to Newton's second law, so that the force on an object is not proportional to the particle's acceleration but rather to In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example electromagnetism. This interpretation is experimentally disfavoured by laboratory experiments.
  • Modified gravity: Alternatively, Milgrom's law can be viewed as modifying Newton's universal law of gravity instead, so that the true gravitational force on an object of mass due to another of mass is roughly of the form In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena. This interpretation has received more attention between the two.
Milgrom's law states that for accelerations smaller than a0 accelerations increasingly depart from the standard Newtonian relationship of mass and distance, wherein gravitational strength is linearly proportional to mass and the inverse square of distance. Instead, the theory holds that the gravitational field below the a0 value, increases with the square root of mass and decreases linearly with distance. Whenever the gravitational field is larger than a0, whether it be near the center of a galaxy or an object near or on Earth, MOND yields dynamics that are nearly indistinguishable from those of Newtonian gravity. For instance, if the gravitational acceleration equals a0 at a distance from a mass, at ten times that distance, Newtonian gravity predicts a hundredfold decline in gravity whereas MOND predicts only a tenfold reduction. In 1991 Begeman et al. found to be optimal by fitting Milgrom's law to rotation curve data. The value of Milgrom's acceleration constant has not varied meaningfully since then. The value of a0 also establishes the distance from a mass at which Newtonian and MOND dynamics diverge.
By itself, Milgrom's law is not a complete and self-contained physical theory, but rather an empirically motivated variant of an equation in classical mechanics. Its status within a coherent non-relativistic hypothesis of MOND is akin to Kepler's Third Law within Newtonian mechanics. Milgrom's law provides a succinct description of observational facts, but must itself be grounded in a proper field theory. Several complete classical hypotheses have been proposed. These generally yield Milgrom's law exactly in situations of high symmetry and otherwise deviate from it slightly. For MOND as modified gravity two complete field theories exist called AQUAL and QUMOND. A subset of these non-relativistic hypotheses have been further embedded within relativistic theories, which are capable of making contact with non-classical phenomena and cosmology. Distinguishing both theoretically and observationally between these alternatives is a subject of current research.

Interpolating function

Milgrom's law uses an interpolation function to join its two limits together. It represents a simple algorithm to convert Newtonian gravitational accelerations to observed kinematic accelerations and vice versa. Many functions have been proposed in the literature although currently there is no single interpolation function that satisfies all constraints. Two common choices are the "simple interpolating function" and the "standard interpolating function". Each has a and a direction to convert the Milgromian gravitational field to the Newtonian and vice versa such that:
The simple interpolation function is:
The standard interpolation function is:
Thus, in the deep-MOND regime :
Data from spiral and elliptical galaxies favour the simple interpolation function, whereas data from lunar laser ranging and radio tracking data of the Cassini spacecraft towards Saturn require interpolation functions that converge to Newtonian gravity faster.

External field effect

In Newtonian mechanics, an object's acceleration can be found as the vector sum of the acceleration due to each of the individual forces acting on it. This means that a subsystem can be decoupled from the larger system in which it is embedded simply by referring the motion of its constituent particles to their centre of mass; in other words, the influence of the larger system is irrelevant for the internal dynamics of the subsystem. Since Milgrom's law is non-linear in acceleration, MONDian subsystems cannot be decoupled from their environment in this way, and in certain situations this leads to behaviour with no Newtonian parallel. This is known as the "external field effect", for which there exists observational evidence.
The external field effect is best described by classifying physical systems according to their relative values of ain, aex, and a0:
Image:TheFourRegimesOfMOND.png|thumb|upright 1.6|The four different limits of MOND. The black line connecting the first and second regimes is the simple interpolation function. The quasi-Newtonian regime can follow any line parallel to the green ones in between the Deep-MOND and forced-Newtonian limits, depending on the strength of the external field.
  • : Newtonian regime
  • : Deep-MOND regime
  • : The external field is dominant and the behavior of the system is Newtonian.
  • : The external field is larger than the internal acceleration of the system, but both are smaller than the critical value. In this case, dynamics is Newtonian but the effective value of G is enhanced by a factor of a0/aex.
The external field effect implies a fundamental break with the strong equivalence principle. The effect was postulated by Milgrom in the first of his 1983 papers to explain why some open clusters were observed to have no mass discrepancy even though their internal accelerations were below a0. It has since come to be recognized as a crucial element of the MOND paradigm.
The dependence in MOND of the internal dynamics of a system on its external environment is strongly reminiscent of Mach's principle, and may hint towards a more fundamental structure underlying Milgrom's law. In this regard, Milgrom has commented:
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 concrete evidence for such a connection. This may turn out to be the most fundamental implication of MOND, beyond its implied modification of Newtonian dynamics and general relativity, and beyond the elimination of dark matter.