Modified Newtonian dynamics

Modified Newtonian dynamics (MOND) is a theory that proposes a modification of

Since Milgrom's original proposal, MOND has seen scattered successes. It is capable of explaining several observations in galaxy dynamics,

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 first identified for clusters by Swiss astronomer

The basic premise of MOND is that while Newton's laws have been extensively tested in high-acceleration environments (in the Solar System and on Earth), they have not been verified for objects with extremely low acceleration, such as stars in the outer parts of galaxies. This led Milgrom to postulate a new effective gravitational force law (sometimes referred to as "Milgrom's law") that relates the true acceleration of an object to the acceleration that would be predicted for it on the basis of Newtonian mechanics.^[3] This law, the keystone of MOND, is chosen to reproduce the Newtonian result at high acceleration but leads to different ("deep-MOND") behavior at low acceleration:

${\displaystyle F_{\text{N}}=m\,\mu \left({\frac {a}{a_{0}}}\right)\,a~.}$

(1)

Here F_N is the Newtonian force, m is the object's (gravitational) mass, a is its acceleration, μ(x) is an as-yet unspecified function (called the interpolating function), and a₀ is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires

${\displaystyle {\begin{aligned}\mu (x)\longrightarrow 1&&{\text{ for }}x\gg 1\end{aligned}}~,}$

and consistency with astronomical observations requires

${\displaystyle {\begin{aligned}\mu (x)\longrightarrow x&&{\text{ for }}x\ll 1\end{aligned}}~.}$

Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically.^[15][16] Two common choices are the "simple interpolating function":

${\displaystyle \mu \left({\frac {a}{a_{0}}}\right)={\frac {1}{1+{\frac {a_{0}}{a}}}}~,}$

and the "standard interpolating function":

${\displaystyle \mu \left({\frac {a}{a_{0}}}\right)={\sqrt {{\frac {1}{1+\left({\frac {a_{0}}{a}}\right)^{2}}}~}}~.}$

Thus, in the deep-MOND regime (a ≪ a₀):

${\displaystyle F_{\text{N}}=m{\frac {\,a^{2}\,}{\,a_{0}\,}}~.}$

Applying this to a star or other object of mass m in circular orbit around mass M (the total baryonic mass of the galaxy), produces

${\displaystyle {\frac {GMm}{r^{2}}}=m{\frac {\left({\frac {v^{2}}{r}}\right)^{2}}{a_{0}}}\quad \Longrightarrow \quad v^{4}=GMa_{0}~}$

(2)

By fitting his law to rotation curve data, Milgrom found a₀ ≈ 1.2 × 10⁻¹⁰ m/s² to be optimal.

MOND holds that for accelerations smaller than an a₀ value of roughly 1.2 × 10⁻¹⁰ m/s², accelerations increasingly depart from the standard M · G / r ² Newtonian relationship of mass and distance, wherein gravitational strength is proportional to mass and the inverse square of distance. Specifically, the theory holds that when gravity is well below the a₀ value, its rate of change—including the curvature of spacetime—increases with the square root of mass (rather than linearly as per Newtonian law) and decreases linearly with distance (rather than distance squared).

Whenever a small mass, m is near a much larger mass, M, whether it be a star near the center of a galaxy or an object near or on Earth, MOND yields dynamics that are indistinguishably close to those of Newtonian gravity. This 1-to-1 correspondence between MOND and Newtonian dynamics is retained down to accelerations of about 1.2 × 10⁻¹⁰ m/s² (the a₀ value); as accelerations decline below a₀, MOND's dynamics rapidly diverge from the Newtonian description of gravity. For instance, there is a certain distance from the center of any given galaxy at which its gravitational acceleration equals a₀; at ten times that distance, Newtonian gravity predicts a hundredfold decline in gravity whereas MOND predicts only a tenfold reduction.

It is important to note that the Newtonian component of MOND's dynamics remains active at accelerations well below the a₀ value of 1.2 × 10⁻¹⁰ m/s²; the equations of MOND assert no minimum acceleration for the Newtonian component. However, because the residual Newtonian-like dynamics of MOND continue to decline as the inverse square of distance below a₀—just as they do above—they comparatively vanish as they become overwhelmed by the stronger “deep-MOND” linear dynamics of the theory.

MOND predicts stellar velocities that closely match observations for an extraordinarily wide range of distances from galactic centers of mass. The 1.2 × 10⁻¹⁰ magnitude of a₀ establishes not only the distance from the center of the galaxy at which Newtonian and MOND dynamics diverge, but a₀ also establishes the angle (when not plotted with log/log scales) of the non-Newtonian linear slope on velocity/radius graphs like Fig. 1.

MOND-compliant gravity, which explains galactic-scale observations, was not previously detected closer to Earth, such as in national laboratories or the trajectories of interplanetary spacecraft, because the a₀ acceleration, 1.2 × 10⁻¹⁰ m/s², at which the dynamics of MOND begin diverging from Newtonian dynamics, is—as a practical matter—indistinguishably close to perfect weightlessness. Within the Solar System, the v⁴ = GMa₀ equation makes the effect of the a₀ term virtually nonexistent; it is overwhelmed by the enormous—and highly Newtonian—gravitational influence of the Sun as well as the variability of Earth's surface gravity.

On Earth's surface—and in national laboratories when performing ultra-precise gravimetry—the a₀ value is equal to 0.012 microgal (μGal), which is only twelve-trillionths the strength of Earth's gravity. A change in the laws of gravity below this acceleration is far too small to be resolved with even the most sensitive free-fall-style absolute gravimeters available to national labs, like the FG5-X, which is accurate to just ±2 μGal. When considering why MOND's effects aren't detectable with precision gravimetry on Earth, it is important to remember that a₀ doesn't represent a spurious force; it is the gravitational strength at which MOND is theorized to significantly begin departing from the Newtonian dynamic. Moreover, the a₀ strength is equivalent to the change in Earth's gravity brought about by an elevation difference of 0.04 mm—the width of a fine human hair. Such subtle gravitational details, besides being unresolvable with current gravimeters, are overwhelmed by twice-daily distortions in Earth's shape due to lunar gravitational tides, which can cause local elevation changes nearly 10,000 times greater than 0.04 mm. Such disturbances in local gravity due to tidal distortions are even detectable as variations in the rate of a Shortt double-pendulum clock, which was a national timekeeping standard in the late 1920s.

Even at the edge of the Solar System, the a₀ point at which MOND dynamics significantly diverge from Newtonian dynamics is overwhelmed and masked by the much stronger gravitational fields from the Sun and planets, which follow Newtonian gravity. To give a sense of scale to a₀, a free-floating mass in space that was exposed for one hour to 1.2 × 10⁻¹⁰ m/s² would "fall" by just 0.8 millimeter—roughly the thickness of a credit card. An interplanetary spacecraft on a free-flying inertial path well above the Solar System's

Despite its vanishingly small and undetectable effects on bodies that are on Earth, within the Solar System, and even in proximity to the Solar System and other planetary systems, MOND successfully explains significant observed galactic-scale rotational effects without invoking the existence of as-yet undetected dark matter particles lying outside of the highly successful Standard Model of particle physics. This is in large part due to MOND holding that exceedingly weak galactic-scale gravity holding galaxies together near their perimeters declines as a very slow linear relationship to distance from the center of a galaxy rather than declining as the inverse square of distance.

Milgrom's law can be interpreted in two ways:

• 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 a but rather to ${\textstyle \mu \left({\frac {a}{a_{0}}}\right)a.}$ In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other
  forces, for example electromagnetism.^[17]
• Alternatively, Milgrom's law can be viewed as leaving
  Newton's Second Law
  intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass m due to another of mass M is roughly of the form ${\textstyle {\frac {GMm}{\mu \left({\frac {a}{a_{0}}}\right)r^{2}}}.}$ In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena.

Since MOND was specifically designed to produce flat rotation curves, these do not constitute evidence for the hypothesis, but every matching observation adds to support of the empirical law. Nevertheless, proponents claim that a broad range of astrophysical phenomena at the galactic scale are neatly accounted for within the MOND framework.[18]^[21] Many of these came to light after the publication of Milgrom's original papers and are difficult to explain using the dark matter hypothesis. The most prominent are the following:

• In addition to demonstrating that rotation curves in MOND are flat, equation 2 provides a concrete relation between a galaxy's total baryonic mass (the sum of its mass in stars and gas) and its asymptotic rotation velocity. This predicted relation was called by Milgrom the mass-asymptotic speed relation (MASSR); its observational manifestation is known as the baryonic Tully–Fisher relation (BTFR),^[22] and is found to conform quite closely to the MOND prediction.^[23]
• Milgrom's law fully specifies the rotation curve of a galaxy given only the distribution of its baryonic mass. In particular, MOND predicts a far stronger correlation between features in the baryonic mass distribution and features in the rotation curve than does the dark matter hypothesis (since dark matter dominates the galaxy's mass budget and is conventionally assumed not to closely track the distribution of baryons). Such a tight correlation is claimed to be observed in several spiral galaxies, a fact which has been referred to as "Renzo's rule".^[18]
• Since MOND modifies Newtonian dynamics in an acceleration-dependent way, it predicts a specific relationship between the acceleration of a star at any radius from the centre of a galaxy and the amount of unseen (dark matter) mass within that radius that would be inferred in a Newtonian analysis. This is known as the mass discrepancy-acceleration relation, and has been measured observationally.^[24][25] One aspect of the MOND prediction is that the mass of the inferred dark matter goes to zero when the stellar centripetal acceleration becomes greater than a₀, where MOND reverts to Newtonian mechanics. In a dark matter hypothesis, it is a challenge to understand why this mass should correlate so closely with acceleration, and why there appears to be a critical acceleration above which dark matter is not required.^[5]
• Both MOND and dark matter halos stabilize disk galaxies, helping them retain their rotation-supported structure and preventing their transformation into elliptical galaxies. In MOND, this added stability is only available for regions of galaxies within the deep-MOND regime (i.e., with a < a₀), suggesting that spirals with a > a₀ in their central regions should be prone to instabilities and hence less likely to survive to the present day.^[26] This may explain the "Freeman limit" to the observed central surface mass density of spiral galaxies, which is roughly a₀/G.^[27] This scale must be put in by hand in dark matter-based galaxy formation models.^[28]
• Particularly massive galaxies are within the Newtonian regime (a > a₀) out to radii enclosing the vast majority of their baryonic mass. At these radii, MOND predicts that the rotation curve should fall as 1/r, in accordance with
  Kepler's Laws. In contrast, from a dark matter perspective one would expect the halo to significantly boost the rotation velocity and cause it to asymptote to a constant value, as in less massive galaxies. Observations of high-mass ellipticals bear out the MOND prediction.^[29][30]
• In MOND, all gravitationally bound objects with a < a₀ – regardless of their origin – should exhibit a mass discrepancy when analyzed using Newtonian mechanics, and should lie on the BTFR. Under the dark matter hypothesis, objects formed from baryonic material ejected during the merger or tidal interaction of two galaxies ("tidal dwarf galaxies") are expected to be devoid of dark matter and hence show no mass discrepancy. Three objects unambiguously identified as Tidal Dwarf Galaxies appear to have mass discrepancies in close agreement with the MOND prediction.^[31][32][33]
• Recent work has shown that many of the dwarf galaxies around the Milky Way and Andromeda are located preferentially in a single plane and have correlated motions. This suggests that they may have formed during a close encounter with another galaxy and hence be Tidal Dwarf Galaxies. If so, the presence of mass discrepancies in these systems constitutes evidence for MOND. In addition, it has been claimed that a gravitational force stronger than Newton's (such as Milgrom's) is required for these galaxies to retain their orbits over time.^[34]
• In 2020, a group of astronomers analyzing data from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample together with estimates of the large-scale external gravitational field from an all-sky galaxy catalog, concluded that there was highly statistically significant evidence of violations of the
  strong equivalence principle in weak gravitational fields in the vicinity of rotationally supported galaxies.^[35] They observed an effect consistent with the external field effect of modified Newtonian dynamics and inconsistent with tidal effects in the Lambda-CDM model
  paradigm commonly known as the Standard Model of Cosmology.
• In a 2022 published survey of dwarf galaxies from the Fornax Deep Survey (FDS) catalogue, a group of astronomers and physicists conclude that 'observed deformations of dwarf galaxies in the Fornax Cluster and the lack of low surface brightness dwarfs towards its centre are incompatible with ΛCDM expectations but well consistent with MOND.'[36]
• In 2022, Kroupa et al. published a study of open star clusters, arguing that asymmetry in the population of leading and trailing tidal tails, and the observed lifetime of these clusters, are inconsistent with Newtonian dynamics but consistent with MOND.^[37][38]
• In 2023, a study claimed that cold dark matter cannot explain galactic rotation curves, while MOND can.^[39]
• In 2023, a study measured the acceleration of 26,615 wide binaries within 200 parsecs. The study showed that those binaries with accelerations less than 1 nanometer per second squared systematically deviate from Newtonian dynamics, but conform to MOND predictions, specifically to AQUAL.^[40] The results are disputed, with some authors arguing that the detection is caused by poor quality controls,^[41] while the original authors claimed that the added quality controls do not significantly affect the results.^[42]

Complete MOND hypotheses

Milgrom's law requires incorporation into a complete hypothesis if it is to satisfy

Nonrelativistic

The first hypothesis of MOND (dubbed AQUAL) was constructed in 1984 by Milgrom and Jacob Bekenstein.^[43] AQUAL generates MONDian behavior by modifying the gravitational term in the classical Lagrangian from being quadratic in the gradient of the Newtonian potential to a more general function. (AQUAL is an acronym for A QUAdratic Lagrangian.) In formulae:

${\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{Newton}}&=-{\frac {1}{8\pi G}}\cdot \|\nabla \phi \|^{2}\\[6pt]{\mathcal {L}}_{\text{AQUAL}}&=-{\frac {1}{8\pi G}}\cdot a_{0}^{2}F\left({\tfrac {\|\nabla \phi \|^{2}}{a_{0}^{2}}}\right)\end{aligned}}}$

where ${\displaystyle \phi }$ is the standard Newtonian gravitational potential and F is a new dimensionless function. Applying the

This can be solved given suitable boundary conditions and choice of F to yield Milgrom's law (up to a curl field correction which vanishes in situations of high symmetry).

An alternative way to modify the gravitational term in the lagrangian is to introduce a distinction between the true (MONDian) acceleration field a and the Newtonian acceleration field a_N. The Lagrangian may be constructed so that a_N satisfies the usual Newton-Poisson equation, and is then used to find a via an additional algebraic but non-linear step, which is chosen to satisfy Milgrom's law. This is called the "quasi-linear formulation of MOND", or QUMOND,^[44] and is particularly useful for calculating the distribution of "phantom" dark matter that would be inferred from a Newtonian analysis of a given physical situation.^[18]

Both AQUAL and QUMOND propose changes to the gravitational part of the classical matter action, and hence interpret Milgrom's law as a modification of Newtonian gravity as opposed to Newton's second law. The alternative is to turn the kinetic term of the action into a functional depending on the trajectory of the particle. Such "modified inertia" theories, however, are difficult to use because they are time-nonlocal, require energy and momentum to be non-trivially redefined to be conserved, and have predictions that depend on the entirety of a particle's orbit.^[18]

Relativistic

In 2004, Jacob Bekenstein formulated

• ${\displaystyle a_{\mathrm {in} }>a_{0}}$ : Newtonian regime
• ${\displaystyle a_{\mathrm {ex} }<a_{\mathrm {in} }<a_{0}}$ : Deep-MOND regime
• ${\displaystyle a_{\mathrm {in} }<a_{0}<a_{\mathrm {ex} }}$ : The external field is dominant and the behavior of the system is Newtonian.
• ${\displaystyle a_{\mathrm {in} }<a_{\mathrm {ex} }<a_{0}}$ : 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/a_ex.[51]

The external field effect implies a fundamental break with the

The dependence in MOND of the internal dynamics of a system on its external environment (in principle, the rest of the universe) 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:^[52]

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.

Indeed, the potential link between MONDian dynamics and the universe as a whole (that is, cosmology) is augmented by the observation that the value of a₀ (determined by fits to internal properties of galaxies) is within an order of magnitude of cH₀, where c is the

The most serious problem facing Milgrom's law is that it cannot eliminate the need for dark matter in all astrophysical systems: galaxy clusters show a residual mass discrepancy even when analyzed using MOND.[5] The fact that some form of unseen mass must exist in these systems detracts from the adequacy of MOND as a solution to the missing mass problem, although the amount of extra mass required is a fifth that of a Newtonian analysis, and there is no requirement that the missing mass be non-baryonic. It has been speculated that 2 eV neutrinos could account for the cluster observations in MOND while preserving the hypothesis's successes at the galaxy scale.^[59][60] Indeed, analysis of sharp lensing data for the galaxy cluster Abell 1689 shows that MOND only becomes distinctive at Mpc distance from the center, so that Zwicky's conundrum remains,^[61] and 1.8 eV neutrinos are needed in clusters.^[62]

The 2006 observation of a pair of colliding galaxy clusters known as the "

A significant piece of evidence in favor of standard dark matter is the observed anisotropies in the cosmic microwave background.^[67] While ΛCDM is able to explain the observed angular power spectrum, MOND has a much harder time, although it is possible to construct relativistic generalizations of MOND that can fit the observations too.^[68] MOND also encounters difficulties explaining structure formation, with density perturbations in MOND perhaps growing so rapidly that too much structure is formed by the present epoch.^[69] However, forming galaxies more rapidly than in ΛCDM can be a good thing to some extent.^[70]

Several other studies have noted observational difficulties with MOND. For example, it has been claimed that MOND offers a poor fit to the velocity dispersion profile of

Besides these observational issues, MOND and its relativistic generalizations are plagued by theoretical difficulties.[74]^[77] Several ad hoc and inelegant additions to general relativity are required to create a theory compatible with a non-Newtonian non-relativistic limit, though the predictions in this limit are rather clear. This is the case for the more commonly used modified gravity versions of MOND, but some formulations (most prominently those based on modified inertia) have long suffered from poor compatibility with cherished physical principles such as conservation laws. Researchers working on MOND generally do not interpret it as a modification of inertia, with only very limited work done on this area.

Proposals for testing MOND

Several observational and experimental tests have been proposed to help distinguish^[78] between MOND and dark matter-based models:

• The detection of particles suitable for constituting cosmological dark matter would strongly suggest that ΛCDM is correct and no modification to Newton's laws is required.
• If MOND is taken as a theory of modified inertia, it predicts the existence of anomalous accelerations on the Earth at particular places and times of the year. These could be detected in a precision experiment. This prediction would not hold if MOND is taken as a theory of modified gravity, as the external field effect produced by the Earth would cancel MONDian effects at the Earth's surface.^[79][80]
• It has been suggested that MOND could be tested in the Solar System using the
  perihelion precession of the planets in the Solar System,^[82] or a purpose-built spacecraft.^[83]
• One potential astrophysical test of MOND is to investigate whether isolated galaxies behave differently from otherwise-identical galaxies that are under the influence of a strong external field. Another is to search for non-Newtonian behaviour in the motion of binary star systems where the stars are sufficiently separated for their accelerations to be below a₀.^[84]
• Testing MOND using the redshift-dependence of radial acceleration – Sabine Hossenfelder and Tobias Mistele propose a parameter-free MOND model they call Covariant Emergent Gravity and suggest that as measurements of radial acceleration improve, various MOND models and particle dark matter might be distinguishable because MOND predicts a much smaller redshift-dependence.^[85]

See also

Notes

References

Further reading

Technical:

• ISBN 9781108492690
• Milgrom, Mordehai (2014). "The MOND paradigm of modified dynamics". Scholarpedia. 9 (6): 31410.
  doi:10.4249/scholarpedia.31410
  .
• Famaey, Benoît; McGaugh, Stacy S. (2012). "Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions". Living Reviews in Relativity. 15 (1): 10.
  PMID 28163623
  .
• McGaugh, Stacy S. (2015). "A tale of two paradigms: The mutual incommensurability of ΛCDM and MOND". Canadian Journal of Physics. 93 (2): 250–259.
  S2CID 51822163
  .
• Milgrom, Mordehai (2015). "MOND theory". Canadian Journal of Physics. 93 (2): 107–118.
  S2CID 119183394
  .
• Kroupa, Pavel (2015). "Galaxies as simple dynamical systems: Observational data disfavor dark matter and stochastic star formation". Canadian Journal of Physics. 93 (2): 169–202.
  S2CID 118479184
  .
• Clifton, Timothy; Ferreira, Pedro G.; Padilla, Antonio; Skordis, Constantinos (2012). "Modified gravity and cosmology". Physics Reports. 513 (1–3): 1–189.
  S2CID 119258154
  .
• Mannheim, P. (2006). "Alternatives to dark matter and dark energy". Progress in Particle and Nuclear Physics. 56 (2): 340–445.
  S2CID 14024934
  .

Popular:

• A non-Standard model, David Merritt, Aeon Magazine, July 2021
• Dark matter critics focus on details, ignore big picture, Lee, 14 Nov 2012
• Milgrom, Mordehai (2009). "MOND: Time for a change of mind?".
  arXiv:0908.3842 [astro-ph.CO
  ].
• "Dark matter" doubters not silenced yet Archived 2016-05-20 at the Wayback Machine, World Science, 2 Aug 2007
• Does Dark Matter Really Exist?, Milgrom, Scientific American, Aug 2002

External links

• Media related to Modified Newtonian Dynamic at Wikimedia Commons

Wikiquote has quotations related to Modified Newtonian dynamics.

• The MOND pages, Stacy McGaugh
• Mordehai Milgrom's website
• "The Dark Matter Crisis" blog, Pavel Kroupa, Marcel Pawlowski
• Pavel Kroupa's website
• Hossenfelder, Sabine (1 Feb 2016). "The superfluid Universe". Retrieved 2 Feb 2016. Superfluid dark matter may provide a more natural way to arrive at the MOND equation.