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
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 (in the Solar System and on Earth), 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 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 \frac{G M m}{\nu\left( \frac{a_0}{a} \right) r^2}. 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 (typically along "modified gravity" as opposed to "modified inertia" lines). 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 (e.g.,
gravitational lensing) 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. 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" (EFE), The external field effect is best described by classifying physical systems according to their relative values of
ain (the characteristic acceleration of one object within a subsystem due to the influence of another),
aex (the acceleration of the entire subsystem due to forces exerted by objects outside of it), and
a0: • a_{\mathrm{in}} > a_0 : Newtonian regime • a_{\mathrm{ex}} : Deep-MOND regime • a_{\mathrm{in}} : The external field is dominant and the behavior of the system is Newtonian. • a_{\mathrm{in}} : 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 (but not the
weak equivalence principle which is required by the
Lagrangian 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.
Complete MOND theories Milgrom's law requires incorporation into a complete hypothesis 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, but produce different behavior in detail. 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. 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 F. This function F reduces to the \mu-version of the interpolation function after varying the over \phi using the
principle of least action. In Newtonian gravity and AQUAL the Lagrangians are: :\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 ), \qquad \text{with } \quad \mu(x) = \frac{dF(x^2)}{dx}. \end{align} where \phi 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 generalization of the
Newton–Poisson equation: : \nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\phi \right\|}{a_0} \right) \nabla\phi\right] = 4\pi G \rho 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). AQUAL uses the \mu-version of the chosen interpolation function.
QUMOND 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
aN. The Lagrangian may be constructed so that
aN 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, 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. The QUMOND Lagrangian is: :\begin{align} \mathcal{L}_\text{QUMOND} = \frac{1}{2} \rho v^2 -\rho\phi - \frac{1}{8\pi G} \left (2\nabla \phi \cdot \nabla \phi_N - a_0^2 Q\left ( (a_0/\nabla \phi_N)^2\right ) \right ) \end{align} Since this Lagrangian does not explicitly depend on time and is invariant under spatial translations this means energy and momentum are conserved according to
Noether's theorem. Varying over r yields ma=mg showing that the
weak equivalence principle always applies in QUMOND. However, since \phi and \phi_N are not identical and are non-linearly related this means that the
strong equivalence principle must be violated. This can be observed by measuring the external field effect. Furthermore, by varying over \phi we get the following
Newton-Poisson equation familiar from Newtonian gravity but now with a subscript to denote that in QUMOND this equation determines the auxiliary gravitational field \phi_N: == Observational evidence for MOND ==