AQUAL is a

theory of gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force be ...

based on

Modified Newtonian Dynamics Modified Newtonian dynamics (MOND) 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 resul ...

(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 (; May 1, 1947 – August 16, 2015) was a Mexican-born American-Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of the connections betwee ...

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 their 1984 paper, ''"Does the missing mass problem signal the breakdown of Newtonian gravity?"'', as well as a 1986 paper by Milgrom. "AQUAL" stands for "AQUAdratic Lagrangian", stemming from the fact that, in contrast to Newtonian gravity, the proposed Lagrangian is non-quadratic in the potential gradient

, \nabla\Phi,

. 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 physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body r ...

, and therefore fails to conserve 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 non-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 te ...

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 matter, such as a galaxy cluster, cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's Ge ...

ing actually observed in rich clusters of galaxies. AQUAL has been claimed to match observations of wide binary star orbits.

References

{{theories of gravitation Astrophysics Theories of gravity Lagrangian mechanics