View source for Physics:Modified Newtonian dynamics

{{short description|Hypothesis proposing a modification of Newton's laws}} 
'''Modified Newtonian dynamics''' ('''MOND''') is a hypothesis that proposes a modification of [[Physics:Newton's laws|Newton's laws]] to account for observed properties of galaxies. It is an alternative to the hypothesis of [[Astronomy:Dark matter|dark matter]] in terms of explaining why galaxies do not appear to obey the currently understood laws of physics.

Created in 1982 and first published in 1983 by Israeli physicist [[Biography:Mordehai Milgrom|Mordehai Milgrom]],<ref name="Milgrom papers">{{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis |journal=Astrophysical Journal |volume=270 |issue= |pages=365–370 |doi=10.1086/161130 |issn= |bibcode=1983ApJ...270..365M}}. {{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics - Implications for galaxies |journal=Astrophysical Journal |volume=270 |issue= |pages=371–389 |doi=10.1086/161131 |issn= |bibcode=1983ApJ...270..371M}}. {{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics - Implications for galaxy systems |journal=Astrophysical Journal |volume=270 |issue= |pages= 384|doi= 10.1086/161132 |issn= |bibcode=1983ApJ...270..384M}}.</ref> the hypothesis' original motivation was to explain why the velocities of stars in galaxies were observed to be larger than expected based on Newtonian mechanics. Milgrom noted that this discrepancy could be resolved if the gravitational force experienced by a star in the outer regions of a galaxy was proportional to the square of its centripetal [[Physics:Acceleration|acceleration]] (as opposed to the centripetal acceleration itself, as in Newton's second law), or alternatively if gravitational force came to vary inversely with radius (as opposed to the inverse square of the radius, as in Newton's law of gravity). In MOND, violation of Newton's laws occurs at extremely small accelerations, characteristic of galaxies yet far below anything typically encountered in the [[Astronomy:Solar System|Solar System]] or on Earth.

{{unsolved|physics|
* What is the nature of [[Astronomy:Dark matter|dark matter]]? Is it a [[Physics:Elementary particle|particle]], or do the phenomena attributed to dark matter actually require a modification of the laws of gravity?
}}

MOND is an example of a class of theories known as modified gravity, and is an alternative to the hypothesis that the dynamics of galaxies are determined by massive, invisible [[Astronomy:Dark matter halo|dark matter halo]]s. Since Milgrom's original proposal, proponents of MOND have claimed to successfully predict a variety of galactic phenomena that they state are difficult to understand as consequences of [[Astronomy:Dark matter|dark matter]].<ref name="tale of two paradigms">{{Cite journal |last = McGaugh | first = S. | title = A Tale of Two Paradigms: the Mutual Incommensurability of LCDM and MOND | date=2015 | arxiv = 1404.7525|bibcode = 2015CaJPh..93..250M |doi = 10.1139/cjp-2014-0203 | volume=93 | issue = 2 | journal=Canadian Journal of Physics | pages=250–259}}</ref><ref>{{cite journal|author=Kroupa, P.|author2=Pawlowski, M.|author3=Milgrom, M.|year=2012|title=The failures of the standard model of cosmology require a new paradigm|journal=International Journal of Modern Physics D|volume=21|issue=14|page= 1230003|doi=10.1142/S0218271812300030|arxiv=1301.3907|bibcode=2012IJMPD..2130003K}}</ref> However, MOND and its generalisations do not adequately account for observed properties of galaxy clusters, and no satisfactory cosmological model has been constructed from the hypothesis.

The accurate measurement of the speed of gravitational waves compared to the speed of light in 2017 ruled out many theories which used modified gravity to rule out dark matter.<ref name="Boran">{{cite journal|last=Boran|first=Sibel|last2=Desai|first2=Shantanu|last3=Kahya|first3=Emre|last4=Woodard|first4=Richard|year=2018|title=GW170817 Falsifies Dark Matter Emulators|journal=Physical Review D|volume=97|issue=4|pages=041501|arxiv=1710.06168|doi=10.1103/PhysRevD.97.041501|bibcode=2018PhRvD..97d1501B}}</ref>
However, both Milgrom's bi-metric formulation of MOND and nonlocal MOND are not ruled out according to the same study.

== Overview ==

[[Image:M33 rotation curve HI.gif|thumb|upright 1.6|Comparison of the observed and expected rotation curves of the typical spiral galaxy [[Astronomy:Triangulum Galaxy|M33]]<ref>Data are from: {{cite journal | author=E. Corbelli |author2=P. Salucci | title=The extended rotation curve and the dark matter halo of M33|journal=Monthly Notices of the Royal Astronomical Society |date=2000|volume=311|issue=2|pages=441–447 |doi=10.1046/j.1365-8711.2000.03075.x|arxiv = astro-ph/9909252 |bibcode = 2000MNRAS.311..441C }}</ref>]]

Several independent observations point to the fact that the visible mass in galaxies and galaxy clusters is insufficient to account for their dynamics, when analysed using Newton's laws. This discrepancy – known as the "missing mass problem" – was first identified for clusters by Swiss astronomer [[Biography:Fritz Zwicky|Fritz Zwicky]] in 1933 (who studied the Coma cluster),<ref>{{Cite journal |last=Zwicky |first=F. |date=1933 |title=Die Rotverschiebung von extragalaktischen Nebeln |journal=Helvetica Physica Acta |volume=6|pages=110–127 |bibcode=1933AcHPh...6..110Z}}</ref><ref>{{Cite journal |last=Zwicky | first=F. |title=On the Masses of Nebulae and of Clusters of Nebulae |date=1937 |journal=[[Astronomy:The Astrophysical Journal|The Astrophysical Journal]] |volume=86 |page=217 | bibcode =1937ApJ....86..217Z |doi=10.1086/143864}}</ref> and subsequently extended to include spiral galaxies by the 1939 work of Horace Babcock on [[Astronomy:Andromeda Galaxy|Andromeda]].<ref>Babcock, H, 1939, "[http://ads.nao.ac.jp/cgi-bin/nph-iarticle_query?1939LicOB..19...41B&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf The rotation of the Andromeda Nebula]", Lick Observatory bulletin ; no. 498</ref> 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 at the [[Organization:Carnegie Institution for Science|Carnegie Institute]] in Washington, 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<ref name="Rubin1970">{{Cite journal |last1=Rubin |first1=Vera C. |last2=Ford |first2=W. Kent, Jr. |date=February 1970  |title=Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions |journal=[[Astronomy:The Astrophysical Journal|The Astrophysical Journal]] |volume=159 |pages=379–403 |bibcode=1970ApJ...159..379R | doi=10.1086/150317}}</ref> – the rotation curves are said to be "flat". This observation necessitates at least one of the following: 1) There exists in galaxies large quantities of unseen matter which boosts the stars' velocities beyond what would be expected on the basis of the visible mass alone, or 2) Newton's Laws do not apply to galaxies. The former leads to the dark matter hypothesis; the latter leads to MOND.

[[Image:Milgrom Mordechai.jpg|thumb|upright|MOND was proposed by Mordehai Milgrom in 1983]]

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.<ref name="Milgrom papers"/> This law, the keystone of MOND, is chosen to reduce to the Newtonian result at high acceleration but leads to different ("deep-MOND") behaviour at low acceleration:

{{NumBlk|::|<math> F_N = m \mu \left( \frac{a}{a_0} \right) a. </math>|{{EquationRef|1}}}}

Here ''F<sub>N</sub>'' is the Newtonian force, ''m'' is the object's (gravitational) [[Physics:Mass|mass]], ''a'' is its acceleration, ''μ''(''x'') is an as-yet unspecified function (known as the "interpolating function"), and ''a''<sub>0</sub> is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires 

:<math>\begin{align} \mu(x) \to 1 && \text{for } x \gg 1, \end{align}</math>

and consistency with astronomical observations requires

:<math>\begin{align} \mu(x) \to x && \text{for } x \ll 1. \end{align}</math>

Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically.<ref>G. Gentile, B. Famaey, W.J.G. de Blok (2011). "THINGS about MOND", Astron. Astrophys. '''527''', A76. {{arxiv|1011.4148}}</ref><ref>B. Famaey, J. Binney (2005), "Modified Newtonian Dynamics in the Milky Way", MNRAS, {{arxiv|astro-ph/0506723}}</ref> Two common choices are the "simple interpolating function":

:<math> \mu \left( \frac{a}{a_0} \right) = \frac{1}{1 + \frac{a_0}{a}},</math>

and the "standard interpolating function":

:<math> \mu \left( \frac{a}{a_0} \right) = \sqrt{\frac{1}{1 + \left(\frac{a_0}{a}\right)^2}}.</math>

Thus, in the deep-MOND regime (''a'' ≪ ''a''<sub>0</sub>):

:<math> F_N = m \frac{a^2}{a_0}. </math>

Applying this to an object of mass ''m'' in [[Astronomy:Circular orbit|circular orbit]] around a point mass ''M'' (a crude approximation for a star in the outer regions of a galaxy), we find:

{{NumBlk|::|<math> \frac{G M m}{r^2} = m \frac{\left( \frac{v^2}{r} \right)^2}{a_0} \quad \Longrightarrow \quad v^4 = G M a_0 </math>|{{EquationRef|2}}}}

that is, the star's rotation velocity is independent of ''r'', its distance from the centre of the galaxy – the rotation curve is flat, as required. By fitting his law to rotation curve data, Milgrom found <math>a_0 \approx 1.2 \times 10^{-10} \mathrm{ms}^{-2}</math> to be optimal. This simple law is sufficient to make predictions for a broad range of galactic phenomena.

Milgrom's law can be interpreted in two different ways. One possibility is to treat it as a modification to the classical law of inertia (Newton's second law), so that the force on an object is not proportional to the particle's acceleration ''a'' but rather to <math>\mu\left(\frac{a}{a_0}\right)a</math>. In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example [[Physics:Electromagnetism|electromagnetism]].<ref>M. Milgrom, "MOND - Particularly as Modified Inertia", {{arxiv|1111.1611}}</ref> 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

:<math> \frac{GMm}{\mu \left (\frac{a}{a_0} \right ) r^2}.</math>

In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena.

By itself, Milgrom's law is not a complete and self-contained physical theory, but rather an ad-hoc empirically motivated variant of one of the several equations that constitute classical mechanics. Its status within a coherent non-relativistic hypothesis of MOND is akin to Kepler's Third Law within Newtonian mechanics; it provides a succinct description of observational facts, but must itself be explained by more fundamental concepts situated within the underlying hypothesis. Several complete classical hypotheses have been proposed (typically along "modified gravity" as opposed to "modified inertia" lines), which generally yield Milgrom's law exactly in situations of high [[Symmetry|symmetry]] and otherwise deviate from it slightly. A subset of these non-relativistic hypotheses have been further embedded within relativistic theories, which are capable of making contact with non-classical phenomena (e.g., [[Physics:Gravitational lensing|gravitational lensing]]) and [[Astronomy:Cosmology|cosmology]].<ref name = "famaey mcgaugh">B. Famaey and S. McGaugh, "Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions", {{arxiv|1112.3960}}</ref> Distinguishing both theoretically and observationally between these alternatives is a subject of current research.

The majority of astronomers, astrophysicists and cosmologists accept dark matter as the explanation for galactic rotation curves<ref>[https://www.youtube.com/watch?v=UPVGDXNSBZM Pavel Kroupa – The vast polar structures around the Milky Way and Andromeda, YouTube, Nov. 18, 2013] Pavel Kroupa claims that the majority opinion is wrong and that empirical evidence rules out the ΛCDM model.</ref> (based on general relativity, and hence Newtonian mechanics), and are committed to a dark matter solution of the missing-mass problem. MOND, by contrast, is actively studied by only a handful of researchers. The primary difference between supporters of ΛCDM and MOND is in the observations for which they demand a robust, quantitative explanation and those for which they are satisfied with a qualitative account, or are prepared to leave for future work. Proponents of MOND emphasize predictions made on galaxy scales (where MOND enjoys its most notable successes) and believe that a cosmological model consistent with galaxy dynamics has yet to be discovered; proponents of ΛCDM require high levels of cosmological accuracy (which concordance cosmology provides) and argue that a resolution of galaxy-scale issues will follow from a better understanding of the complicated baryonic astrophysics underlying galaxy formation.<ref name="tale of two paradigms"/><ref>{{cite journal|author=Sanders, R. H.|title=A historical perspective on modified Newtonian dynamics|journal=Canadian Journal of Physics|volume=93|issue=2|year=2014|pages=126–138|doi=10.1139/cjp-2014-0206|arxiv=1404.0531|bibcode=2015CaJPh..93..126S}} [https://arxiv.org/abs/1404.0531 arXiv.org preprint]</ref>

== Observational evidence for MOND ==

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.<ref name="famaey mcgaugh"/><ref>M. Milgrom (2013), "MOND Laws of Galaxy Dynamics", {{arxiv|1212.2568}}</ref> 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. Observationally, this is known as the baryonic [[Astronomy:Tully–Fisher relation|Tully–Fisher relation]] (BTFR),<ref>S. S. McGaugh, J. M. Schombert, G. D. Bothun,2 and W. J. G. de Blok (2000), "The Baryonic Tully-Fisher Relation", {{arxiv|astro-ph/0003001}}</ref> and is found to conform quite closely to the MOND prediction.<ref>S. McGaugh (2011), "The Baryonic Tully-Fisher Relation of Gas-Rich Galaxies as a Test of ΛCDM and MOND", ApJ, {{arxiv|1107.2934}}</ref>
* 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".<ref name = "famaey mcgaugh"/>
* 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.<ref>R. Sanders, "Mass discrepancies in galaxies: dark matter and alternatives", The Astronomy and Astrophysics Review 1990, Volume 2, Issue 1, pp 1-28</ref><ref>S. McGaugh (2004), "The Mass Discrepancy-Acceleration Relation: Disk Mass and the Dark Matter Distribution", ApJ, {{arxiv|astro-ph/0403610}}</ref> One aspect of the MOND prediction is that the mass of the inferred dark matter go to zero when the stellar centripetal acceleration becomes greater than ''a''<sub>0</sub>, where MOND reverts to Newtonian mechanics. In 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.<ref name = "tale of two paradigms"/>
* Both MOND and dark matter halos stabilise disk galaxies, helping them retain their rotation-supported structure and preventing their transformation into [[Astronomy:Elliptical galaxy|elliptical galaxies]]. In MOND, this added stability is only available for regions of galaxies within the deep-MOND regime (i.e., with ''a'' < ''a''<sub>0</sub>), suggesting that spirals with ''a'' > ''a''<sub>0</sub> in their central regions should be prone to instabilities and hence less likely to survive to the present day.<ref>M. A. Jimenez and X. Hernandez (2014), "Local galactic disk stability under MONDian gravity", {{arxiv|1406.0537}}</ref> This may explain the "Freeman limit" to the observed central surface mass density of spiral galaxies, which is roughly ''a''<sub>0</sub>/''G''.<ref>{{cite journal | last1 = McGaugh | first1 = S. | date = 1998 | title = Testing the Hypothesis of Modified Dynamics with Low Surface Brightness Galaxies and Other Evidence | url = | journal = Astrophys J | volume = 499 | issue = 1| pages = 66–81 | doi=10.1086/305629 | bibcode=1998ApJ...499...66M|arxiv = astro-ph/9801102 }}</ref> This scale must be put in by hand in dark matter-based galaxy formation models.<ref>{{cite journal | last1 = McGaugh | first1 = S. | date = 2005 | title = Balance of Dark and Luminous Mass in Rotating Galaxies | url = | journal = Phys. Rev. Lett. | volume = 95 | issue = 17| page = 171302 | arxiv = astro-ph/0509305 | doi=10.1103/physrevlett.95.171302|bibcode = 2005PhRvL..95q1302M | pmid=16383816}}</ref>
* Particularly massive galaxies are within the Newtonian regime (''a'' > ''a''<sub>0</sub>) 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.<ref>{{cite journal | last1 = Romanowsky | first1 = A.J. | last2 = Douglas | first2 = N.G. | last3 = Arnaboldi | first3 = M. | last4 = Kuijken | first4 = K. | last5 = Merrifield | first5 = M.R. | last6 = Napolitano | first6 = N.R. | last7 = Capaccioli | first7 = M. | last8 = Freeman | first8 = K.C. | date = 2003 | title = A Dearth of Dark Matter in Ordinary Elliptical Galaxies | url = | journal = Science | volume = 301 | issue = 5640| pages = 1696–1698 |arxiv=astro-ph/0308518 | doi=10.1126/science.1087441 | pmid=12947033|bibcode = 2003Sci...301.1696R }}</</ref><ref>{{cite journal | last1 = Milgrom | first1 = M. | last2 = Sanders | first2 = R.H. | date = 2003 | title = Modified Newtonian Dynamics and the 'Dearth of Dark Matter in Ordinary Elliptical Galaxies' | url = | journal = Astrophys J | volume = 599 | issue = 1| pages = 25–28 | arxiv = astro-ph/0309617 |bibcode = 2003ApJ...599L..25M |doi = 10.1086/381138 }}</ref>
* In MOND, all gravitationally bound objects with ''a'' < ''a''<sub>0</sub> – regardless of their origin – should exhibit a mass discrepancy when analysed 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 ("[[Astronomy:Dwarf galaxy|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.<ref>F. Bournaud, P.-A. Duc, E. Brinks, M. Boquien, P. Amram, U. Lisenfeld, B. S. Koribalski, F. Walter, V. Charmandaris (2007), "Missing Mass in Collisional Debris from Galaxies", {{arxiv|0705.1356}}</ref><ref>G. Gentile, B. Famaey, F. Combes, P. Kroupa, H. S. Zhao, O. Tiret (2007), "Tidal dwarf galaxies as a test of fundamental physics" {{arxiv|0706.1976}}</ref><ref>P. Kroupa (2012), "The dark matter crisis: falsification of the current standard model of cosmology", {{arxiv|1204.2546}}</ref>
* Recent work has shown that many of the dwarf galaxies around the [[Astronomy:Milky Way|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 further 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.<ref>P. Kroupa (2014), "Lessons from the Local Group (and beyond) on dark matter", {{arxiv|1409.6302}}</ref>

== Complete MOND hypotheses ==

Milgrom's law requires incorporation into a complete hypotheses if it is to satisfy conservation laws and provide a unique solution for the time evolution of any physical system. Each of the theories described here reduce to Milgrom's law in situations of high symmetry (and thus enjoy the successes described above), but produce different behaviour in detail.

=== Nonrelativistic ===

The first hypothesis of MOND (dubbed [[Physics:AQUAL|AQUAL]]) was constructed in 1984 by Milgrom and [[Biography:Jacob Bekenstein|Jacob Bekenstein]].<ref>{{cite journal|author=Jacob Bekenstein |author2=M. Milgrom |last-author-amp=yes |title=Does the missing mass problem signal the breakdown of Newtonian gravity?|journal=Astrophys. J.|volume=286|pages=7–14|date=1984|bibcode = 1984ApJ...286....7B |doi = 10.1086/162570 }}</ref> AQUAL generates MONDian behaviour by modifying the gravitational term in the classical [[Physics:Lagrangian mechanics|Lagrangian]] from being quadratic in the gradient of the Newtonian potential to a more general function. (AQUAL is an acronym for AQUAdratic Lagrangian.) In formulae:

:<math>\begin{align}
\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{align}</math>

where  <math>\phi</math> is the standard Newtonian gravitational potential and ''F'' is a new dimensionless function. Applying the Euler–Lagrange equations in the standard way then leads to a non-linear generalisation of the [[Physics:Poisson's equation|Newton–Poisson equation]]:

:<math> \nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\phi \right\|}{a_0} \right) \nabla\phi\right] = 4\pi G \rho </math>

This can be solved given suitable boundary conditions and choice of F to yield Milgrom's law (up to a [[Curl (mathematics)|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<sub>N</sub>'''. The Lagrangian may be constructed so that '''a<sub>N</sub>''' 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,<ref>M. Milgrom (2010), "Quasi-linear formulation of MOND", {{arxiv|0911.5464}}</ref> 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.<ref name = "famaey mcgaugh"/>

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 (mathematics)|functional]] depending on the trajectory of the particle. Such "modified inertia" theories, however, are difficult to use because they are time-nonlocal, require [[Physics:Energy|energy]] and [[Physics:Momentum|momentum]] to be non-trivially redefined to be conserved, and have predictions that depend on the entirety of a particle's orbit.<ref name = "famaey mcgaugh"/>

=== Relativistic ===

In 2004, Jacob Bekenstein formulated TeVeS, the first complete relativistic hypothesis using MONDian behaviour.<ref name=Bekenstein>{{cite journal |arxiv=astro-ph/0403694 |title=Relativistic gravitation theory for the MOND paradigm |author=Jacob D. Bekenstein |journal =Phys. Rev. |date=2004 |volume=D70 |issue=8 |pages=83509 |doi=10.1103/PhysRevD.70.083509|bibcode = 2004PhRvD..70h3509B }}</ref> TeVeS is constructed from a local Lagrangian (and hence respects conservation laws), and employs a unit [[Vector field|vector field]], a dynamical and non-dynamical [[Physics:Scalar field|scalar field]], a free function and a non-Einsteinian [[Physics:Metric tensor (general relativity)|metric]] in order to yield AQUAL in the non-relativistic limit (low speeds and weak gravity). TeVeS has enjoyed some success in making contact with gravitational lensing and [[Astronomy:Structure formation|structure formation]] observations,<ref>T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis (2011), "Modified Gravity and Cosmology", {{arxiv|1106.2476}}</ref> but faces problems when confronted with data on the [[Anisotropy|anisotropy]] of the [[Astronomy:Cosmic microwave background|cosmic microwave background]],<ref>See Slosar, Melchiorri and Silk {{arxiv|astro-ph/0508048}}</ref> the lifetime of compact objects,<ref name=Seifert2007>{{Citation |last1=Seifert |first1=M. D. |date=2007 |title=Stability of spherically symmetric solutions in modified theories of gravity |journal=[[Physics:Physical Review D|Physical Review D]] |volume=76 |issue=6 |pages=064002 |arxiv=gr-qc/0703060 |bibcode=2007PhRvD..76f4002S |doi=10.1103/PhysRevD.76.064002}}</ref> and the relationship between the lensing and matter overdensity potentials.<ref>{{Citation |last1=Zhang |first1=P. |last2=Liguori |first2=M. |last3=Bean |first3=R. |last4=Dodelson |first4=S. |date=2007 |title=Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=99 |issue=14 |pages=141302 |arxiv=0704.1932 |bibcode=2007PhRvL..99n1302Z |doi=10.1103/PhysRevLett.99.141302 |pmid=17930657}}</ref>

Several alternative relativistic generalisations of MOND exist, including BIMOND and generalised Einstein-Aether theories.<ref name = "famaey mcgaugh"/> There is also a relativistic generalisation of MOND that assumes a Lorentz-type invariance as the physical basis of MOND phenomenology.<ref name = "alzain">M. Alzain, "MOND as a modification of Newtonian inertia", {{arxiv|1708.05385}}</ref>

== The 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" (EFE).<ref name = "Milgrom papers"/>

The external field effect is best described by classifying physical systems according to their relative values of ''a''<sub>in</sub> (the characteristic acceleration of one object within a subsystem due to the influence of another), ''a''<sub>ex</sub> (the acceleration of the entire subsystem due to forces exerted by objects outside of it), and ''a''<sub>0</sub>:

* <math> a_{\mathrm{in}} > a_0 </math> :  Newtonian regime

* <math> a_{\mathrm{ex}} < a_{\mathrm{in}} < a_0 </math> :  Deep-MOND regime

* <math> a_{\mathrm{in}} < a_0 < a_{\mathrm{ex}} </math> : The external field is dominant and the behaviour of the system is Newtonian.

* <math> a_{\mathrm{in}} < a_{\mathrm{ex}} < a_0 </math> : 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 ''a''<sub>0</sub>/''a''<sub>ex</sub>.<ref>S. McGaugh, [http://www.astro.umd.edu/~ssm/mond/EFE.html The EFE in MOND]</ref>

The external field effect implies a fundamental break with the strong equivalence principle (but not necessarily the weak 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 a<sub>0</sub>. It has since come to be recognised as a crucial element of the MOND paradigm.

The dependence in MOND of the internal dynamics of a system on its external environment (in principle, the rest of the [[Astronomy:Universe|universe]]) is strongly reminiscent of [[Physics:Mach's principle|Mach's principle]], and may hint towards a more fundamental structure underlying Milgrom's law. In this regard, Milgrom has commented:<ref>M. Milgrom (2008), "The MOND paradigm", {{arxiv|0801.3133}}</ref>

<blockquote>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.</blockquote>

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 ''a''<sub>0</sub> (determined by fits to internal properties of galaxies) is within an order of magnitude of ''cH''<sub>0</sub>, where ''c'' is the [[Physics:Speed of light|speed of light]] and ''H''<sub>0</sub> is the [[Physics:Hubble constant|Hubble constant]] (a measure of the present-day expansion rate of the universe).<ref name = "Milgrom papers"/> It is also close to the acceleration rate of the universe, and hence the [[Physics:Cosmological constant|cosmological constant]]. However, as yet no full hypothesis  has been constructed which manifests these connections in a natural way.

== Responses and criticism ==

=== Attempts to explain MOND phenomenology using dark matter ===

While acknowledging that Milgrom's law provides a succinct and accurate description of a range of galactic phenomena, many physicists reject the idea that classical dynamics itself needs to be modified and attempt instead to explain the law's success by reference to the behaviour of dark matter. Some effort has gone towards establishing the presence of a characteristic acceleration scale as a natural consequence of the behaviour of cold dark matter halos,<ref>M. Kaplinghat and M. Turner (2002), "How Cold Dark Matter Theory Explains Milgrom's Law", {{arxiv|astro-ph/0107284}}</ref><ref>M. H. Chan (2013), "Reconciliation of MOND and Dark Matter theory", {{arxiv|1310.6810}}</ref> although Milgrom has argued that such arguments explain only a small subset of MOND phenomena.<ref>M. Milgrom (2002), "Do Modified Newtonian Dynamics Follow from the Cold Dark Matter Paradigm?", {{arxiv|astro-ph/0110362}}</ref> An alternative proposal is to modify the properties of dark matter (e.g., to make it [[Astronomy:Strongly interacting massive particle|interact strongly]] with itself or baryons) in order to induce the tight coupling between the baryonic and dark matter mass that the observations point to.<ref>J. Bullock (2014), [http://www.cfa.harvard.edu/events/2014/sackler/index/talks/Harvard2014_Bullock.pdf Self-Interacting Dark Matter]</ref> Finally, some researchers suggest that explaining the empirical success of Milgrom's law requires a more radical break with conventional assumptions about the nature of dark matter. One idea (dubbed "dipolar dark matter") is to make dark matter gravitationally polarisable by ordinary matter and have this polarisation enhance the gravitational attraction between baryons.<ref>L. Blanchet, "Gravitational polarization and the phenomenology of MOND", Class. Quantum Grav., 24, 3529–3539, (2007), {{arXiv|astro-ph/0605637}}</ref>

=== Outstanding problems for MOND ===

The most serious problem facing Milgrom's law is that it cannot completely eliminate the need for dark matter in all astrophysical systems: galaxy clusters show a residual mass discrepancy even when analysed using MOND.<ref name = "tale of two paradigms"/> The fact that some form of unseen mass must exist in these systems detracts from the elegance 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&nbsp;eV neutrinos could account for the cluster observations in MOND while preserving the hypothesis's successes at the galaxy scale.<ref>{{Cite journal |last=Angus |first=Garry W. |last2=Shan |first2=Huan Yuan |last3=Zhao |first3=Hong Sheng |last4=Famaey |first4=Benoit |lastauthoramp=yes |date=2007 |title=On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos |journal=The Astrophysical Journal Letters |volume=654 |issue=1 |pages=L13–L16 |doi=10.1086/510738 |arxiv=astro-ph/0609125 |bibcode=2007ApJ...654L..13A}}</ref><ref>{{Cite journal|author=R.H. Sanders|date=2007|title=Neutrinos as cluster dark matter|journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]]|arxiv=astro-ph/0703590|bibcode = 2007MNRAS.380..331S |doi = 10.1111/j.1365-2966.2007.12073.x|volume=380|issue=1|pages=331–338}}</ref> 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  <ref>{{Cite journal|title=How Zwicky already ruled out modified gravity theories without dark matter|first=Theodorus M.|last=Nieuwenhuizen |journal=Fortschritte der Physik|volume=65|issue=6–8|page=1600050|doi=10.1002/prop.201600050|arxiv=1610.01543|date=2016}}</ref>, and 1.8 eV neutrinos are needed in clusters. <ref>{{Cite journal|title=Dirac neutrino mass from a neutrino dark matter model for the galaxy cluster Abell 1689|first=Theodorus M.|last=Nieuwenhuizen |journal=Journal of Physics: Conference Series|volume=701|issue=1|page=012022(13pp)|date=2015|doi=10.1088/1742-6596/701/1/012022|arxiv=1510.06958|bibcode=2016JPhCS.701a2022N}}</ref>

The 2006 observation of a pair of colliding galaxy clusters known as the "[[Astronomy:Bullet Cluster|Bullet Cluster]]",<ref>{{Cite journal |last=Clowe |first=Douglas |last2=Bradač |first2=Maruša |last3=Gonzalez |first3=Anthony H. |last4=Markevitch |first4=Maxim |last5=Randall |first5=Scott W. |last6=Jones |first6=Christine |last7=Zaritsky |first7=Dennis |lastauthoramp=yes |date=2006 |title=A Direct Empirical Proof of the Existence of Dark Matter |journal=The Astrophysical Journal Letters |volume=648 |issue=2 |pages=L109–L113 |doi=10.1086/508162 |arxiv=astro-ph/0608407 |bibcode=2006ApJ...648L.109C}}</ref> poses a significant challenge for all theories proposing a modified gravity solution to the missing mass problem, including MOND. Astronomers measured the distribution of stellar and gas mass in the clusters using [[Physics:Visible light|visible]] and [[Physics:X-ray|X-ray]] light, respectively, and in addition mapped the inferred dark matter density using gravitational lensing. In MOND, one would expect the missing mass (which is only apparent since it results from using Newtonian as opposed to MONDian dynamics) to be centred on the visible mass. In ΛCDM, on the other hand, one would expect the dark matter to be significantly offset from the visible mass because the halos of the two colliding clusters would pass through each other (assuming, as is conventional, that dark matter is collisionless), whilst the cluster gas would interact and end up at the centre. An offset is clearly seen in the observations. It has been suggested, however, that MOND-based models may be able to generate such an offset in strongly non-spherically-symmetric systems, such as the Bullet Cluster.<ref>{{cite journal| author=G.W. Angus |author2=B. Famaey |author3=H. Zhao  |last-author-amp=yes | title=Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry| journal=Mon. Not. R. Astron. Soc.| volume=371| issue=1| pages=138–146 |date=September 2006| doi= 10.1111/j.1365-2966.2006.10668.x| arxiv=astro-ph/0606216v1|bibcode = 2006MNRAS.371..138A }}</ref>

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 globular clusters and the temperature profile of galaxy clusters,<ref name=Seife>{{cite book |title=Alpha and Omega |pages = [https://archive.org/details/alphaomegasearch0000seif/page/100 100]–101 |url=https://archive.org/details/alphaomegasearch0000seif |url-access=registration |quote=Modified Newtonian dynamics. |author=Charles Seife |isbn=0-14-200446-4 |publisher=Penguin Books |date=2004}}</ref><ref name=Aguirre>{{cite journal |title=Problems for Modified Newtonian Dynamics in Clusters and the Lyα Forest? |author=Anthony Aguirre |author2=Joop Schaye |author3=Eliot Quataert  |last-author-amp=yes |date=2001 |journal=The Astrophysical Journal |volume=561 |issue=2 |pages=550–558 |doi=10.1086/323376 |bibcode=2001ApJ...561..550A|arxiv = astro-ph/0105184 }}</ref> that different values of a<sub>0</sub> are required for agreement with different galaxies' rotation curves,<ref>S. M. Kent, "Dark matter in spiral galaxies. II - Galaxies with H I rotation curves", 1987, AJ, 93, 816</ref> and that MOND is naturally unsuited to forming the basis of a hypothesis  of cosmology.<ref name = "fin du mond">{{Cite journal |last1=Scott |first1 = D. | last2 = White | first2 = M. | last3 = Cohn | first3 = J. D. | last4 = Pierpaoli | first4 = E. | date = 2001 | title = Cosmological Difficulties with Modified Newtonian Dynamics (or: La Fin du MOND?) |pages = arXiv:astro–ph/0104435 | arxiv = astro-ph/0104435|bibcode = 2001astro.ph..4435S }}</ref> Furthermore, many versions of MOND predict that the speed of light is different from the speed of gravity, but in 2017 the speed of gravitational waves was measured to be equal to the speed of light.<ref name="Boran"/>

Besides these observational issues, MOND and its generalisations are plagued by theoretical difficulties.<ref name = "fin du mond"/><ref>C. R. Contaldi, T. Wiseman, B. Withers (2008), "TeVeS gets caught on caustics", {{arxiv|0802.1215}}</ref> Several ad-hoc and inelegant additions to general relativity are required to create a hypothesis with a non-Newtonian non-relativistic limit, the plethora of different versions of the hypothesis offer diverging predictions in simple physical situations and thus make it difficult to test the framework conclusively, and some formulations (most prominently those based on modified inertia) have long suffered from poor compatibility with cherished physical principles such as conservation laws.

== Proposals for testing MOND ==

Several observational and experimental tests have been proposed to help distinguish<ref>{{cite journal|arxiv=0705.3408|title=Testing Gravity in the Outer Solar System: Results from Trans-Neptunian Objects|author=Wallin, John F.|author2=Dixon, David S.|author3=Page, Gary L.|date=23 May 2007|doi=10.1086/520528|volume=666|issue=2|journal=The Astrophysical Journal|pages=1296–1302|bibcode=2007ApJ...666.1296W}}</ref> between MOND and dark matter-based models:

* The [[Astronomy:Dark matter#Detection|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.
* MOND 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.<ref>A. Ignatiev (2014), "Testing MOND on Earth", {{arxiv|1408.3059}}</ref><ref>V. A. De Lorenci, M. Faundez-Abans, J. P. Pereira (2010). "Testing the Newton second law in the regime of small accelerations" {{arxiv|1002.2766}}</ref>
* It has been suggested that MOND could be tested in the Solar System using the [[Astronomy:LISA Pathfinder|LISA Pathfinder]] mission (launched in 2015). In particular, it may be possible to detect the anomalous tidal stresses predicted by MOND to exist at the Earth-Sun saddlepoint of the Newtonian gravitational potential.<ref>Christian Trenkel, Steve Kemble, Neil Bevis, Joao Magueijo (2010). "Testing MOND/TEVES with LISA Pathfinder" {{arxiv|1001.1303}}</ref> It may also be possible to measure MOND corrections to the perihelion precession of the planets in the Solar System,<ref>L. Blanchet (2011), "Testing MOND in the Solar System", {{arxiv|1105.5815}}</ref> or a purpose-built spacecraft.<ref>V. Sahni, Y. Shtanov (2006), "APSIS - an Artificial Planetary System in Space to probe extra-dimensional gravity and MOND", {{arxiv|gr-qc/0606063}}</ref>
* 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 [[Astronomy:Binary star|binary star systems]] where the stars are sufficiently separated for their accelerations to be below a<sub>0</sub>.<ref>X. Hernandez, M. A. Jimenez, and C. Allen (2012), "Wide binaries as a critical test of Classical Gravity", {{arxiv|1105.1873}}</ref>
* Testing modified gravity with black hole shadows - [[Biography:John Moffat (physicist)|John Moffat]] calculates that the black hole shadow should appear larger by a factor of about ten in [[Biography:John Moffat (physicist)#Modified Gravity Theory|MOG]] as compared to general relativity.<ref>{{cite web|url=http://backreaction.blogspot.in/2015/05/testing-modified-gravity-with-black.html|title=Testing modified gravity with black hole shadows|date=2015-05-06}}</ref>
*Testing MOND using the redshift-dependence of radial acceleration - [[Biography:Sabine Hossenfelder|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.<ref>S. Hossenfelder and T. Mistele (2018), "The Redshift-Dependence of Radial Acceleration:
Modified Gravity versus Particle Dark Matter", {{arxiv|1803.08683v1}}</ref>

== See also ==

{{div col|colwidth=22em}}
*MOND researchers:
** {{annotated link|Biography:Mordehai Milgrom}}
** {{annotated link|Biography:Jacob Bekenstein}}
* {{annotated link|Physics:AQUAL}}
** {{annotated link|Physics:Entropic gravity}}
* {{annotated link|Astronomy:Cold dark matter}}
* {{annotated link|Astronomy:Dark matter}}
* {{annotated link|Astronomy:Tully–Fisher relation}}
{{div col end}}

==References==
{{Reflist}}

==Further reading==

Technical:
* {{cite journal |doi=10.4249/scholarpedia.31410|bibcode=2014SchpJ...931410M|title=The MOND paradigm of modified dynamics|year=2014|last1=Milgrom|first1=Mordehai|journal=Scholarpedia|volume=9|issue=6|pages=31410|doi-access=free}}
* {{cite journal |arxiv=1112.3960|doi=10.12942/lrr-2012-10|pmid=28163623|pmc=5255531|bibcode=2012LRR....15...10F|title=Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions|year=2012|last1=Famaey|first1=Benoît|last2=McGaugh|first2=Stacy S.|journal=Living Reviews in Relativity|volume=15|issue=1|pages=10}}
* {{cite journal |arxiv=1404.7525|doi=10.1139/cjp-2014-0203|bibcode=2015CaJPh..93..250M|title=A tale of two paradigms: The mutual incommensurability of ΛCDM and MOND|year=2015|last1=McGaugh|first1=Stacy S.|journal=Canadian Journal of Physics|volume=93|issue=2|pages=250–259}}
* {{cite journal |arxiv=1404.7661|doi=10.1139/cjp-2014-0211|bibcode=2015CaJPh..93..107M|title=MOND theory|year=2015|last1=Milgrom|first1=Mordehai|journal=Canadian Journal of Physics|volume=93|issue=2|pages=107–118}}
* {{cite journal |arxiv=1406.4860|doi=10.1139/cjp-2014-0179|bibcode=2015CaJPh..93..169K|title=Galaxies as simple dynamical systems: Observational data disfavor dark matter and stochastic star formation|year=2015|last1=Kroupa|first1=Pavel|journal=Canadian Journal of Physics|volume=93|issue=2|pages=169–202}}
* {{cite journal |arxiv=1106.2476|doi=10.1016/j.physrep.2012.01.001|bibcode=2012PhR...513....1C|title=Modified gravity and cosmology|year=2012|last1=Clifton|first1=Timothy|last2=Ferreira|first2=Pedro G.|last3=Padilla|first3=Antonio|last4=Skordis|first4=Constantinos|journal=Physics Reports|volume=513|issue=1–3|pages=1–189}}
* {{cite journal |arxiv=astro-ph/0505266|doi=10.1016/j.ppnp.2005.08.001|bibcode=2006PrPNP..56..340M|title=Alternatives to dark matter and dark energy|year=2006|last1=Mannheim|first1=P.|journal=Progress in Particle and Nuclear Physics|volume=56|issue=2|pages=340–445}}

Popular:
* {{cite arxiv |eprint=0908.3842|last1=Milgrom|first1=Mordehai|title=MOND: Time for a change of mind?|class=astro-ph.CO|year=2009}}
* [http://www.world-science.net/othernews/070802_darkmatter.htm "Dark matter" doubters not silenced yet], World Science, 2 Aug 2007
* [http://www.astro.umd.edu/~ssm/mond/sad0802Milg6p.pdf Does Dark Matter Really Exist?], Milgrom, Scientific American, Aug 2002
* [https://arstechnica.com/science/2012/11/dark-matter-critics-focus-on-details-ignore-big-picture Dark matter critics focus on details, ignore big picture], Lee, 14 Nov 2012

==External links==
* [http://astroweb.case.edu/ssm/mond/ The MOND pages], Stacy McGaugh
* [http://www.weizmann.ac.il/weizsites/milgrom/ Mordehai Milgrom's website]
* [http://www.scilogs.com/the-dark-matter-crisis/ "The Dark Matter Crisis" blog], Pavel Kroupa, Marcel Pawlowski
* [http://www.astro.uni-bonn.de/~pavel/kroupa_cosmology.html Pavel Kroupa's website]
* {{cite web
 | url=https://aeon.co/essays/is-dark-matter-subatomic-particles-a-superfluid-or-both
 | title=The superfluid Universe
 | last=Hossenfelder
 | first=Sabine
 | date=1 Feb 2016
 | access-date=2 Feb 2016 }} Superfluid dark matter may provide a more natural way to arrive at the MOND equation.

{{theories of gravitation}}

{{DEFAULTSORT:Modified Newtonian Dynamics (MoND)}}
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[[Category:Theories of gravitation]]
[[Category:Unsolved problems in physics]]
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