AQUAL is a theory of gravity based on

Modified Newtonian Dynamics Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of universal gravitation to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining ...

(MOND), but using a

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

. It was developed by

Jacob Bekenstein Jacob David Bekenstein ( he, יעקב בקנשטיין; May 1, 1947 – August 16, 2015) was an American and Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of ...

and

Mordehai Milgrom Mordehai "Moti" Milgrom is an Israeli physicist and professor in the department of Particle Physics and Astrophysics at the Weizmann Institute in Rehovot, Israel. Biography He received his B.Sc. degree from the Hebrew University of Jerusalem in 1 ...

in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian gravity?". "AQUAL" stands for "A QUAdratic Lagrangian". The gravitational force law obtained from MOND, :

m \mu \left (\frac \right) a = \frac,

has a serious defect: it violates

Newton's third law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...

, and therefore fails to

conserve Conserve may refer to: * Conserve (condiment), a preserve made from a mixture of fruits or vegetables * Conserve (NGO), an Indian environmental organization * Conserve (publisher), a Dutch publisher * Conserved sequence, a protein or nucleic acid ...

momentum and energy. To see this, consider two objects with

m \neq M

; then we have: :

\mu \left (\frac \right) m a_m = \frac = \frac = \mu \left (\frac \right) M a_M

but the third law gives

m a_m = M a_M,

so we would get :

\mu \left (\frac \right)=\mu \left (\frac \right)

even though

a_m \neq a_M,

and

\mu

would therefore be constant, contrary to the MOND assumption that it is linear for small arguments. This problem can be rectified by deriving the force law from a Lagrangian, at the cost of possibly modifying the general form of the force law. Then conservation laws could then be derived from the Lagrangian by the usual means. The AQUAL Lagrangian is: :

\rho \Phi + \frac a_0^2 F \left (\frac \right);

this leads to a modified Poisson equation: :

\nabla \cdot \left  (\mu \left (\frac \right ) \nabla\Phi \right ) = 4 \pi G \rho, \qquad \text \quad \mu(x) = \frac.

where the predicted acceleration is

-\nabla\Phi = a.

These equations reduce to the MOND equations in the spherically symmetric case, although they differ somewhat in the disc case needed for modelling spiral or lenticular galaxies. However, the difference is only 10–15%, so does not seriously impact the results. According to Sanders and McGaugh, one problem with AQUAL (or any

scalar–tensor theory In theoretical physics, a scalar–tensor theory is a field theory that includes both a scalar field and a tensor field to represent a certain interaction. For example, the Brans–Dicke theory of gravitation uses both a scalar field and a t ...

in which the scalar field enters as a conformal factor multiplying Einstein's metric) is AQUAL's failure to predict the amount of

gravitational lens A gravitational lens is a distribution of matter (such as a cluster of galaxies) between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer. This effect is known ...

ing actually observed in rich clusters of galaxies.

References

* * * {{theories of gravitation Astrophysics Theories of gravity Lagrangian mechanics