An implementation of MOND as a non-relativistic modified gravity
was discussed by
[Bekenstein & Milgrom
(1984)],
who replaced the standard Poisson action
*S*_{}
in eq. (3) by an action of the form

(4) |

This gives, upon variation on , the equation

(5) |

where *µ*(*x*)
*dF*(*y*) /
*dy*|_{y=x2}. This theory, since it
is derived from an action that has all the usual symmetries,
satisfies all the standard conservation laws. Its various
implications have been discussed in
[Bekenstein & Milgrom
(1984)],
[Milgrom (1986)],
[Milgrom
(1997)],
and others.

One important point to note is that this theory gives the desired center-of-mass motion of composite systems: Stars, star clusters, etc. moving in a galaxy with a low center-of-mass acceleration are made of constituents whose internal accelerations are much higher than 0. If we look at individual constituents we see bodies whose total accelerations are high and so whose overall motion is very nearly Newtonian. Yet, their motion should somehow combine to give a MOND motion for the center of mass. This is satisfied in the above theory as shown in [Bekenstein & Milgrom (1984)]. (A similar situation exists in GR: imagine a system made of very tightly bound black holes moving in the weak field of a galaxy, say. While the motions of the individual components is highly relativistic, governed by a non-linear theory, we know that these motions combine to give a simple Newtonian motion for the center of mass.)

This field equation, generically, requires numerical solution, but
it is straightforward to solve in cases of high symmetry
(spherical, cylindrical, or planar symmetry), where the
application of the Gauss law to eq. (5) gives the
exact algebraic relation between the MOND (
= -
) and
Newtonian ( = -
_{N})
acceleration fields:

(6) |

which is identical to the heuristic MOND relation we started with. Note that in general, for configurations of lower symmetry, this algebraic relation does not hold (and, in general, and are not even parallel).

It is worth pointing out that in such a modified-gravity theory, the deep-MOND limit corresponds to a theory that is conformally invariant, as discussed in [Milgrom (1997)]. Whether this has some fundamental bearings is not clear, but it does make MOND unique, and enables one to derive useful analytic results, such as an expression for the two-body force, and a virial relation, despite the obstacle of nonlinearity.

There is a large number of
physical phenomena that are governed by an equation like
eq. (5), each with its own form of the function
*µ*(*x*), as detailed in
[Milgrom
(1997)],
or [Milgrom
(2001)].
I would like to concentrate here on one, in particular. It is well
known that a stationary, potential flow is described by the
Poisson equation: If the velocity field
() is derived from a potential,
=
, then the
continuity equation, which here determines the flow, reads
^{ . }
= *s*() /
_{0},
where *s*() is the source density, and
_{0} is the
(constant) density of the fluid. When the fluid is compressible,
but still irrotational, and barotropic [i.e. has an equation of
state of the form *p* =
*p*()] the
stationary flow is described by
the nonlinear Poisson equation. The Euler equation reduces to
Bernoulli's law

(7) |

where *dh* /
*d*
^{-1}
*dp* /
*d*.
This tell us that
is
a function of *u* = |
|. Substituting
this in the continuity equation gives

(8) |

which has the same form as eq. (5) if we identify
as
*µ*, and the source density *s* with the normalized
gravitational mass density 4
*G*
. Note, however, that from the Bernoulli law,
*d*
/ *d*| *u*| =
- |
*u*|/*c*^{2}, where *c*^{2} = *dp* /
*d*
is the formal squared speed of sound. Thus, in
the case of MOND, where we have that *µ* is an increasing
function of its argument, the model fluid has to have a negative
compressibility *c*^{2} < 0. A cosmological-constant
equation of state, *p* = - *c*^{2}
, with
*c* the speed of light gives
(*u*) =
_{0}
*exp*(*u*^{2} / 2*c*^{2}), which is not
what we need for MOND. The deep-MOND limit, *µ*(*u*)
*u* /
*a*_{0}, corresponds to
*p* = - (*a*_{0}^{2}/3)
^{3}. To get
the Newtonian limit at large values
of *u* the equation of state has to become incompressible
at some finite density
_{0}, so that
eq. (8) goes to the Poisson equation.

The gravitational force is then the pressure+drage force on
sources. For a small (test) static source *s*, at a position where
the fluid speed is , the source imparts momentum to the flow
at a rate *s*, and so is subject to a force - *s*. The force
between sources of the same sign is attractive, as befits gravity.

Note that in such a picture the fluid density itself
does not contribute to the sources of the potential equation, so
it does not, itself, gravitate. Also note that, because
= *p* = 0
for = 0, the fluid behaves as if it has no
existence without the sources (masses) that induce velocities in it.
Obviously, this picture is anything but directly applicable as
an explanation of Newtonian gravity. For example, it is not clear
how to obtain the barotropic equation of state that is needed to
reproduce MOND. In particular, how does the infinite
compressibility appear at a finite critical density, and what is the
meaning of this density? Is this due to some phase transition?
What happens at densities higher than this critical density? are
they accessible at all? Also, there seem to be a drag force on
moving sources. In the context of a time-dependent configuration
the above equation of state is problematic.