AQUAL

{{short description|Theory of gravity}}

{{about|the theory of gravity|other uses|aqual (disambiguation)}}

AQUAL is a theory of gravity based on Modified Newtonian Dynamics (MOND), but using a Lagrangian. It was developed by Jacob Bekenstein and Mordehai Milgrom in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian gravity?",{{cite journal|author=Jacob Bekenstein|author2=M. Milgrom|name-list-style=amp|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 }} as well as a 1986 paper by Milgrom.{{cite journal|author=Milgrom, M|title=Solutions for the modified Newtonian dynamics field equation|journal=Astrophys. J.|volume=302|pages=617–625|date=1986|bibcode = 1986ApJ...302..617M |doi = 10.1086/164021 |doi-access=free}} "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 $|\backslash nabla\backslash Phi|$.

The gravitational force law obtained from MOND,

:$m\; \backslash mu\; \backslash left\; (\backslash frac\{a\}\{a\_0\}\; \backslash right)\; a\; =\; \backslash frac\{GMm\}\{r^2\},$

has a serious defect: it violates Newton's third law of motion, and therefore fails to conserve momentum and energy.{{cite journal | last=Felten | first=J. E. | title=Milgrom's revision of Newton's laws - Dynamical and cosmological consequences | journal=The Astrophysical Journal | volume=286 | date=1984 | issn=0004-637X | doi=10.1086/162569 | doi-access=free | page=3}} To see this, consider two objects with $m\; \backslash neq\; M$; then we have:

:$\backslash mu\; \backslash left\; (\backslash frac\{a\_m\}\{a\_0\}\; \backslash right)\; m\; a\_m\; =\; \backslash frac\{GMm\}\{r^2\}\; =\; \backslash frac\{GMm\}\{r^2\}\; =\; \backslash mu\; \backslash left\; (\backslash frac\{a\_M\}\{a\_0\}\; \backslash right)\; M\; a\_M$

but the third law gives $m\; a\_m\; =\; M\; a\_M,$ so we would get

:$\backslash mu\; \backslash left\; (\backslash frac\{a\_m\}\{a\_0\}\; \backslash right)=\backslash mu\; \backslash left\; (\backslash frac\{a\_M\}\{a\_0\}\; \backslash right)$

even though $a\_m\; \backslash neq\; a\_M,$ and $\backslash mu$ 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:

:$\backslash rho\; \backslash Phi\; +\; \backslash frac\{1\}\{8\; \backslash pi\; G\}\; a\_0^2\; F\; \backslash left\; (\backslash frac$

\nabla\Phi|^2}{a_0^2} \right); this leads to a modified Poisson equation: :$\backslash nabla\; \backslash cdot\; \backslash left\; (\backslash mu\; \backslash left\; (\backslash frac\{|\backslash nabla\backslash Phi$

{a_0} \right ) \nabla\Phi \right ) = 4 \pi G \rho, \qquad \text{with } \quad \mu(x) = \frac{dF(x^2)}{dx}.

where the predicted acceleration is $-\backslash nabla\backslash Phi\; =\; a.$ 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 which the scalar field enters as a conformal factor multiplying Einstein's metric) is AQUAL's failure to predict the amount of gravitational lensing actually observed in rich clusters of galaxies.{{cite journal|author=Sanders, Robert H.|author2=McGaugh, Stacy S.|title=Modified Newtonian dynamics as an alternative to dark matter|journal=Annual Review of Astronomy and Astrophysics|volume=40|issue=1|year=2002|pages=263–317|arxiv=astro-ph/0204521|bibcode=2002ARA&A..40..263S|doi=10.1146/annurev.astro.40.060401.093923}} AQUAL has been claimed to match observations of wide binary star orbits.{{cite journal | last=Chae | first=Kyu-Hyun | title=Breakdown of the Newton–Einstein Standard Gravity at Low Acceleration in Internal Dynamics of Wide Binary Stars | journal=The Astrophysical Journal | volume=952 | issue=2 | date=2023-08-01 | issn=0004-637X | doi=10.3847/1538-4357/ace101 | doi-access=free | page=128| arxiv=2305.04613 }}