# MOND as manifestation of modified inertia111Bases on a talk presented at the
“MOND at 40” conference at St’ Andrews University, July 2023.

Mordehai Milgrom  Department of Particle Physics and Astrophysics, Weizmann
Institute

###### Abstract

Practically all the full-fledged MOND theories propounded to date are of the
modified-gravity (MG) type: they modify only the Newtonian Poisson action of
the gravitational potential, or the general-relativistic Einstein-Hilbert
action, leaving other terms (inertia) intact. Here, I discuss the
interpretation of MOND as modified inertia (MI). My main aim is threefold: (a)
to advocate exploring MOND theories beyond MG, and appreciating their
idiosyncracies, (b) to highlight the fact that secondary predictions of such
theories can differ materially from those of MG theories, (c) to demonstrate
some of this with specific MI models. I discuss some definitions and
generalities concerning MI. I then present instances of MI in physics, and the
lessons we can learn from them for MOND. I then concentrate on a specific
class of nonrelativistic, MOND, MI models, and contrast their predictions with
those of the two workhorse, MG theories, AQUAL and QUMOND. The MI models
predict possibly a stronger external-field effect – e.g. on low acceleration
systems in the solar neighborhood – such as very wide binary stars – and on
vertical motions in disc galaxies. More generally, the workings of the effect
are rather different, and depend in different ways on dimensionless
characteristics of the system, such as frequency ratios of the external and
internal fields, eccentricity of trajectories, etc. These models predict a
much weaker effect of the Galactic field in the inner Solar System than is
predicted by AQUAL/QUMOND. I also show how noncircular motions – such as those
perpendicular to the disc – modify the standard, algebraic mass-discrepancy-
acceleration relation (aka RAR) that is predicted by MI for exactly circular
orbits. These differences, and more that are discussed, can potentially offer
ways to distinguish between theories.

##  I Introduction

MOND [1] has been propounded as an alternative to dark matter in accounting
for the acceleration anomalies in The Universe: The gravitational
accelerations calculated in standard dynamics from the observed mass
distributions in galactic systems (and the Universe at large) fall very short
of the measured accelerations of test particles. Reviews of MOND can be found
in Refs. [2, 3, 4, 5, 6, 7]. In particular, Ref. [6] is a detailed account of
the MOND research program from the epistemological and philosophical
viewpoints.

The MOND research program starts from the following basic tenets: (a) Standard
dynamics – Newtonian and general relativity (GR, hereafter) – break down in
the limit of low accelerations, the limit being defined by a new constant,
$a_{0}$, with the dimensions of acceleration. (b) Standard dynamics is
restored when all system attributes with the dimensions of acceleration are
$\gg a_{0}$ – or when taking in a MOND theory the formal limit
$a_{0}\rightarrow 0$. (c) Gravitational, nonrelativistic dynamics become
space-time scale invariant – i.e., under $(t,{\bf r})\rightarrow(\lambda
t,\lambda{\bf r})$ – in the opposite limit, formally approached when taking,
in a MOND theory, $a_{0}\rightarrow\infty$ (and $G\rightarrow 0$, with
$A_{0}\equiv Ga_{0}$ fixed).222To these tenets one may add the small-print
requirement that no fundamental dimensionless constants of values very
different from unity appear in the theory. This means, e.g., that $a_{0}$
defines not only where the transition occurs, but also that is occurs within a
range of acceleration values of order $a_{0}$. This is a natural requirement
and is satisfied, e.g., in the contexts of the relativity and quantum
departures from classical Newtonian physics, where $c$ determines everything
in the former, and $\hbar$ in the latter.

From the basic tenets alone follow a large number of predictions; I refer to
these as “primary”, or “first-tier” predictions. Some examples of primary
prediction are as follows: (a) The salient properties of disc galaxy rotation
curves: asymptotic flatness, the mass-asymptotic-speed relation
$V_{\infty}^{4}=MGa_{0}$, and, to within small details, the full prediction of
rotation curves – tested in many works; see the above reviews. (b) The
“virial,” mass-velocity-dispersion relation for deep-MOND (low-acceleration)
systems of all types (rotation-, or pressure-supported):
$MGa_{0}=\kappa\sigma^{\scriptscriptstyle 4}$, with $\kappa\sim 1$, applied,
e.g., in Refs. [8, 9]. (c) Mass discrepancies always set in around where the
measured acceleration is $a_{0}$ (d) The presence of an external-field effect
(hereafter, EFE), whereby an external field, even if uniform, influences the
internal dynamics of a gravitating system embedded in it. (e) Definite
correlation between the mean baryonic surface density (or gravitational
acceleration) of a system (e.g., a galaxy) and that of its putative dark
matter halo [10, 11, 12]. In particular it is predicted that the latter cannot
much exceed $a_{0}$ [13]. (f) Enhanced stability of galactic discs [14, 15,
16]. These predictions, and more – with emphasis on how they follow from only
the basic tenets – are discussed in Ref. [17]. References to tests of these
predictions can be found in the above mentioned reviews.

As has been stressed repeatedly, there are, however, also phenomena on which
the basic tenets themselves do not make absolute and concrete predictions.
Different MOND theories – while embodying the basic tenets, and hence sharing
the primary predictions – may differ on what they predict for these secondary,
or lower-tier phenomena. Among these, one may count the following: (a) Minor
details of the rotation curves (e.g., Refs. [18, 19, 21, 20, 22]). (b) The
exact value of $\kappa$ in the $M~{}-~{}\sigma$ relation. (c) The exact
dependence of the effective two-body force on the masses. (d) The exact
workings (and strength) of the EFE. (e) MOND effects in high-acceleration
regions, such as the inner Solar System. (f) The workings of dynamical
friction.

Quite a few MOND theories, of varying degrees of development, have been
propose to date. Some are nonrelativistic and underlie MOND phenomenology for
self-gravitating nonrelativistic systems. Some are relativistic [23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34], and reduce to nonrelativistic MOND
theories in the nonrelativistic limit, but can vary greatly in their
predictions in the relativistic regime.

Notable among the nonrelativistic theories are the AQUAL (aquadratic
Lagrangian) [23], and the quasilinear MOND theory (QUMOND) [35]. The recently
proposed tripotential theories (TRIMOND) [36] comprise a larger class that
includes AQUAL and QUMOND as special cases. And, QUMOND itself has been
generalized in ways that go beyond the basic tenets of MOND (in that they
involve dimensioned constants other than $a_{0}$) [37]. All these are MG
theories.

AQUAL and QUMOND have become the main workhorses, and have been used in
deriving various MOND predictions, and for solving more involved problems
numerically. Just a few of the many applications of these two theories are
described in Refs. [23, 44, 18, 45, 46, 38, 35, 39, 40, 41, 42, 43, 22]). The
two theories tend to make very similar predictions, differing only in minor
details.

It seems that as the result of their ubiquitous and exclusive use, the notion
has taken root that these twin theories encompass the full scope of MOND, and
that all their predictions are universal MOND predictions.

This, however, if far from true, and one of the aims of this contribution is
to bring this fact home forcefully.

The departures of AQUAL and QUMOND from Newtonian gravity are essentially
encapsulated in a single “interpolating function” that is introduced by hand
in their Lagrangian, and which thus dictates all the predictions of these
theories. But this need not be the case even within the class of MG. Other MG
theories can make different predictions on secondary phenomena. This was
discussed and demonstrated, e.g., in Refs. [36, 37]. For example, these
theories involve interpolating functions of several variables, and show a more
complicated interpolation scheme between the Newtonian and the deep-MOND
regime.

But an even more pronounced variety in second-tier predictions appears when we
consider theories that go beyond MG – I call these modified-inertia theories,
because they depart from the Newtonian law of inertia (second law). Such was,
in fact, my initial view of MOND.

There are clear and immediately recognized differences between MG and MI. One
clear difference is that MI may entail departures from standard dynamics also
when forces other than gravity affect the motions (see e.g., Ref. [47]).
However, since all the information on the acceleration anomalies has so far
come from purely gravitating systems, we have not yet been informed
observationally on this possible aspect of MOND. But, such differences can in
principle be demonstrated with the following two examples. In the first, only
MI predicts an anomaly, and in the other only MG does.

An experiment has been proposed in Ref. [48]: For two very brief time
intervals every year, in very small regions attached to the Earth’s surface,
the acceleration due to the revolution of Earth around the Sun cancels that of
the rotation of the Earth. The latter acceleration involves also
nongravitational forces that keep the experiment attached to the surface.
While the external kinematic acceleration can be very small for this small
region of space-time, the gravitational acceleration is still very large.
Thus, an anomaly is expected only in MI.

Another experiment has been proposed, which employs the regions around the
saddle points between the Sun and a planet, where the net gravitational
acceleration of all bodies (including The Galaxy) is $\lesssim a_{0}$ (see,
e.g., Refs. [49, 50, 51, 7]). One strategy could be to have a spacecraft fly
by such a region with sensitive accelerometers on board, and make
measurements. For this small region of space-time, the spacecraft is freely
falling in a sub-$a_{0}$ gravitational field, and an anomaly is expected in
any MOND theory. But in another procedure, the spacecraft is made to follow
the saddle point, so as to spend more time in the low-gravitational-
acceleration region. This has to be done by nongravitational forces. While an
anomaly is expected in MG theories, practically none is expected in MI,
because then the kinematic acceleration of the spacecraft is much larger than
$a_{0}$.

Another immediate and phenomenologically-important difference between MG and
MI is the following: In studying the dynamics of galactic systems, the main
tool is measuring kinematic accelerations (${\bf a}=\ddot{\bf r}$) of test
particles in the system. In MG, Newton’s 2nd law, which remains intact in MG,
tells us directly the gravitational acceleration (${\bf g}={\bf F}/m$) on the
particles, which we can proceed to analyze according to various MG theories.
In MI, the directly-measured acceleration does not equal ${\bf g}$, and does
not tell us what the force is. In fact, the accelerations ${\bf a}$ of
different particles at the same position in a gravitational force field may be
different, depending on the particle’s trajectory.

And, there are many other, perhaps more subtle, differences between the
predictions of MG and MI theories. Such differences are one of the main topics
of this paper.

For many years, we had been testing only the primary predictions of MOND, and
it had been less pressing perhaps to recognize such potential differences in
secondary predictions. But in recent years we have begun to be able to test
secondary predictions. For example: details of the EFE and how it affects the
outskirts of disc galaxies [52, 53, 54, 55, 56, 57, 58], the dynamics in wide-
binary-stars [59, 60, 61, 62, 63, 64], and the appearance of tidal tails of
star clusters [65, 66]. Another example of a test of secondary predictions
that is becoming feasible is looking for the possible small effect of the
galactic acceleration on the motions of bodies in the inner Solar System [67,
68, 53]. Yet another such second-tier issue (related somewhat to the details
of the EFE) is the dynamics perpendicular to the disc in spiral galaxies [69,
70, 71, 72, 73, 74, 75].

Clearly, it is thus important that we understand which theories are
constrained by which experiment or observation, and, in particular, to assess
possible differences between MI and MG. But, besides my own preliminary
proposals on the subject [76, 77, 78, 79, 80, 81], there have been, as far as
I am aware, only very few attempts – not very promising in my judgement – to
formulate MOND as MI [82, 83, 84, 85].

There is also a more general motivation for casting one’s net wider than the
existing MOND theories. Several previous papers (e.g., the above-mentioned
reviews) discuss the shortcomings of existing MOND theories, the fact that –
very useful as they are – they can only be approximate, effective theories,
and the need to find a “FUNDAMOND” a more fundamental theory for MOND. (This
is also the state of affairs with other, more entrenched physical theories –
such as the standard model of particle physics.)

Studying and understanding even more effective theories, such as the various
MI effective theories, may eventually point the way to such a more fundamental
basis for MOND, or to the understanding of how MOND phenomenology emerges from
a microscopic picture.

In Sec. II, I describe what is meant by MI, and discuss some of the reasons
why, I think, the development of MI, MOND theories has not kept pace with that
of MG theories. Section III brings some familiar and less familiar examples of
modified, or acquired, inertia, to demonstrate and urge that this direction
may be quite promising. In Sec. IV, I raise some generalities regarding MI in
the specific context of MOND. Section V lists and details some MOND
predictions that are expected to differ among MOND theories – so-called lower-
tier predictions – because they do not follow robustly from the basic tenets
of MOND. Section VI describes a class of MI models, which I use to demonstrate
some of the possible variety in lower-tier predictions. Section VII is a
discussion.

##  II What is modified inertia?

It is usually straightforward to define MG in the context of modifying
Newtonian dynamics or GR. It would be a modification that disappears in the
absence of gravity, or in describing systems where gravity (which is always
present) can be neglected compared with other influences; this limit can be
formally effected by takeing $G\rightarrow 0$ in the theory. In such a limit,
MG leaves us with standard Newtonian dynamics and special relativity,
respectively. More specifically, in Newtonian dynamics, MG is a theory that
modifies the Poisson term in the Lagrangian (see below). And, MG of GR is a
theory that can be formulated as a metric theory that leaves intact all the
couplings of matter degrees of freedom (DoFs, hereafter) to the metric (so
called “minimal coupling”), but modifies the Einstein-Hilbert action.333There
are theories, such as TeVeS [26], that can be formulated in equivalent ways;
one as just described, and the other in terms of a metric whose action is the
Einstein-Hilbert one, but its coupling to matter is modified. These are still
MG theories.

But how to define MI? In all the MI, MOND theories proposed to date – which
are all nonrelativistic, and concern only particle dynamics (as opposed to
dynamics of fields) – the concept is well defined (see below), and involves
modifying the kinetic particle Lagrangian $mV^{2}/2$, or the “inertia term”
$m{\bf a}$ in the particle’s equations of motion. But how to define MI more
generally? One can simply view it as a generic name for any theory that is not
restricted to modifying gravity in the above sense. But, perhaps more proper
for the name, one can define MI as a theory that modifies the “free” or
“kinetic” terms in the action underlying the theory, or in the equations of
motion (or field equations). These are the terms that in the absence of
gravity depend on only one DoF (possibly tensorial). Terms that depend on more
than one degree of freedom are “interaction terms”. And gravity itself couples
to all matter degrees of freedom.

The statement that the free terms in the action encapsulate inertia can be
justified, for example, by noting that the isolated contribution of a DoF to
the energy-momentum tensor can be derived from its free action. So, these
terms describe the energy-momentum cost of the different configurations of
this DoF, and tell us how much energy-momentum have to be invested or
extracted to change configurations of the DoF – the essence of what inertia
is.444In particle dynamics, the free term tells us what the energy and
momentum are of a particle with a given speed.

In a more “physical” than formal picture, I envisage inertia – hence the free
actions – as resulting from what, in a more fundamental theory, are
interactions of the DoFs with some inertia-endowing medium, call it the
“inerton”. In an extreme, ideal picture, one would have a theory with only
interactions between the “known” DoFs, which also interact with the inerton
(in Ref. [77] I suggested that the “inerton” is the quantum vacuum).
Elimination of the inerton DoF from the theory approximately reduces these
latter interaction to approximate “free actions” for the different DoFs.

In the restricted treatment of MI as a modification of the particle inertia,
we leave intact the forces to which the particle is subject, including
gravity. However, it is quite conceivable that in any picture of MI, all free
actions including those of all fields, are modified. For example, if MI is to
underlie MOND, the acceleration anomalies exhibited by gravitational lensing
by galactic systems, tell us that the inertia of photons, hence the (Maxwell)
free action of the electromagnetic field, must also be modified. Also, the
Einstein-Hilbert, GR action, or the Poisson action in Newtonian dynamics, may
be viewed as the “free actions” of gravity. So it is quite conceivable that
they too are modified in an ultimate theory of MOND MI.

Thus, to justify our restricted treatment of MI, we need to assume that there
is a level of approximation which applies to our case of interest, and in
which the force field – e.g., inasmuch as they are static – are not modified;
only the action of the particle, whose DoFs are time dependent needs to be
modified. In other circumstances, such as when considering the propagation of
electromagnetic waves, or of gravitational waves, the appropriate modification
needs to be applied to their inertia as well.

Such an approximation of “selective modification”, is common in other
instances of modified physics; e.g., when we treat the quantum mechanics of
“quantum” particles in static external fields. We do so, for example, in
describing atoms by the (nonrelativistic or relativistic) quantum mechanics of
electrons in the static and non-quantum Coulomb field of a nucleus. But
ultimately we also need to treat the fields themselves quantum mechanically.
In the case of atoms such an extension was forced, for example, by the
discovery of the Lamb shift, unaccounted for by the Dirac equation, which
treats the field of the nucleus as non-quantum. But, of course we need to
treat the electromagnetic field as quantized in the context of black-body
radiation, or the photoelectric effect.

Another example of “selective modification” is the stepping stone to quantum
gravity, where one describes quantum-gravitational phenomena by quantum field
theory on an unquantized background curved space time.

More concretely now, in this review, and in my previous papers on MI in the
context of nonrelativistic MOND [76, 77, 78, 79, 80, 81], I concentrated on
the most restricted meaning of MI, in which only the nonrelativistic, kinetic,
particle action is modified into a more general functional of the particles
trajectory ${\bf r}(t)$: Start then with the Newtonian action governing the
dynamics of a nonrelativistic, self-gravitating, system of (point) masses,
$m_{\scriptscriptstyle p}$, on trajectories ${\bf r}_{\scriptscriptstyle
p}(t)$, which is $I=\int L~{}dt$, where the Lagrangian, $L$, is

$L=-\frac{1}{8\pi G}\int
d^{3}x~{}(\vec{\nabla}\phi)^{2}+\sum_{i}m_{\scriptscriptstyle
p}[\frac{1}{2}{\bf v}_{\scriptscriptstyle p}^{2}-\phi({\bf
r}_{\scriptscriptstyle p})],$ (1)

The equations of motion of the masses and the field equation for the
gravitational potential are

$m_{\scriptscriptstyle p}\ddot{\bf r}_{\scriptscriptstyle
p}=-m_{\scriptscriptstyle p}\vec{\nabla}\phi({\bf r}_{\scriptscriptstyle
p}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta\phi=4\pi
G\rho,~{}~{}~{}~{}~{}{\rm where}~{}~{}~{}~{}~{}~{}~{}\rho({\bf
r},t)=\sum_{i}m_{\scriptscriptstyle p}\delta[{\bf r}-{\bf
r}_{\scriptscriptstyle p}(t)].$ (2)

The MG, MOND theories that have been propounded to date (AQUAL [23], QUMOND
[35], TRIMOND [36], GQUMOND [37], and the more general theories treated in
Ref. [86]) modify the first term. Then, only the left-hand side of the Poisson
equation is modified to provide a modified prescription for calculating the
gravitational potential. Particles are accelerated by this potential in the
standard way.

Modified inertia in the restricted sense I adopt here modifies the particle
kinetic term in the action, $m_{\scriptscriptstyle p}\int\frac{1}{2}{\bf
v}_{\scriptscriptstyle p}^{2}~{}dt$ (or the corresponding term
$m_{\scriptscriptstyle p}\ddot{\bf r}_{\scriptscriptstyle p}$ in the equation
of motion), replacing it with some functional of the particle trajectory of
the form $m_{\scriptscriptstyle p}F[{\bf r}_{\scriptscriptstyle p}(t),a_{0}]$
(or a generalized inertia functional of the trajectory $m_{\scriptscriptstyle
p}{\bf I}[{\bf r}_{\scriptscriptstyle p}(t),a_{0}]$). The proportionality to
$m_{\scriptscriptstyle p}$ is invokes so as not to violate the universality of
free fall in a gravitational field. In a MOND theory, the action functional
depends on the MOND constant, $a_{0}$.

Note that this treatment already introduces some ambiguity on the definition
of MI. The above Newtonian action is the nonrelativistic limit of the
standard, relativistic, “free” kinetic action for a particle of mass $m$,
which is $-mc^{2}\int d\tau$. But the nonrelativistic limit of this action is
$m\int({\bf v}^{2}/2-\phi)dt$, where $\phi$ is the nonrelativistic
gravitational potential. Namely, in the presence of gravity, the relativistic
free action gives both the Newtonian particle free action and its interaction
with the potential.

It is yet to be seen whether our modification of only the nonrelativistic
particle action can result as a nonrelativistic approximation from some more
fundamental theory of MI.

###  II.1 Who is afraid of modified inertia?

The interpretation of MOND as MI was on the table from the very inception of
MOND [1]. Yet, very little work has been done in this promising vein since
then – far less than on developing MG theories, and studying their
implications analytically and numerically.

Why is it so?

It is true that MI theories seem to entail more drastic modifications of
standard dynamics, and there is a natural tendency to minimize change. To
boot, there has been much work done already on modifying general-relativistic
gravity from the very advent of GR itself. So, from what I have heard,
developers feel much more comfortable with searching and propounding MG
theories. It is also true that it has proven rather hard to incorporate the
idea of MI in full-fledged theories, based on the standard requirement –
symmetries, conservation laws, etc. All this seems to have kept developers
away. And, it is also true that the MI theories that have been put on the
table (including the ones in Ref. [34], and which I will discuss here) are
rather less amenable to application. Far fewer predictions have been extracted
from such formulations, and they are hard to simulate numerically. This seems
to have kept potential users away.

But all these should not argue against the MI approach, which is otherwise
very promising. General relativity is much harder to solve and apply than
Newtonian gravity, and quantum mechanics, to say nothing of quantum field
theory, is much harder to solve and apply than classical mechanics. The
processes of development, understanding, interpretation, and devising
approximation methods to solve and apply these more advanced theories, has
been very tortuous, and is still continuing.

Partly to allay misgivings about MI, I now point out that some well
established theories may be viewed as MI with respect to Newtonian inertia,
even though they are not usually described or thought of as such. I also
pinpoint some lessons we can learn from these about what we can expect from MI
in the context of MOND.

##  III Familiar examples of modified inertia

###  III.1 Special relativity as modified inertia

The departure of special relativity from classical dynamics, entails, among
other things, a modification of Newtonian inertia. Its kinetic (free) action
for a particle of rest mass $m$ is $-mc^{2}\int d\tau=-mc^{2}\int[1-({\bf
v}/c)^{\scriptscriptstyle 2}]^{1/2}dt$ ($\tau$ is the proper time), which
modifies the Newtonian kinetic action $m\int({\bf v}^{\scriptscriptstyle
2}/2)~{}dt$. This gives an inertia force of the form $md(\gamma{\bf v})/dt$,
instead of the Newtonian $md{\bf v}/dt$ ($\gamma$ is the Lorentz factor). The
particle equation of motion is

$\frac{dp^{\scriptscriptstyle\alpha}}{d\tau}\equiv
m\frac{d^{\scriptscriptstyle
2}x^{\scriptscriptstyle\alpha}}{d\tau^{\scriptscriptstyle
2}}=f^{\scriptscriptstyle\alpha},$ (3)

where $f^{\scriptscriptstyle\alpha}$ is the relativistic force four-vector.
Write the space part of this equation as

$\frac{dp^{\scriptscriptstyle
i}}{dt}=\gamma^{\scriptscriptstyle-1}f^{\scriptscriptstyle i}\equiv F^{i},$
(4)

where $p^{\scriptscriptstyle i}=m\gamma dx^{\scriptscriptstyle i}/dt$. For
example, for a particle of charge $e$ in an electric field ${\bf E}$, we have
${\bf F}=e{\bf E}$.

We can write this as

$\frac{md(\gamma{\bf v})}{dt}\equiv
m\overset{\text{\scriptsize{\hbox{\leftrightarrow}}}}{\mu}\left(\frac{{\bf
v}}{c}\right){\bf a}={\bf F},$ (5)

which can be inverted to read

$m{\bf
a}=\overset{\text{\scriptsize{\hbox{\leftrightarrow}}}}{\nu}\left(\frac{{\bf
v}}{c}\right){\bf F}.$ (6)

I wrote these expressions using the “interpolation matrices”
$\overset{\text{\scriptsize$\leftrightarrow$}}{\mu}$ and
$\overset{\text{\scriptsize$\leftrightarrow$}}{\nu}$, defined as

$\overset{\text{\scriptsize{\hbox{\leftrightarrow}}}}{\mu}\left(\frac{{\bf
v}}{c}\right)\equiv\gamma(1+\gamma^{2}\frac{{\bf v}\otimes{\bf
v}}{c^{2}}),~{}~{}~{}~{}~{}~{}\overset{\text{\scriptsize{\hbox{\leftrightarrow}}}}{\nu}\left(\frac{{\bf
v}}{c}\right)\equiv\overset{\text{\scriptsize{\hbox{\leftrightarrow}}}}{\mu}^{\scriptscriptstyle-1}\left(\frac{{\bf
v}}{c}\right)=\gamma^{-1}\left(1-\frac{{\bf v}\otimes{\bf v}}{c^{2}}\right),$
(7)

to resemble MOND relations – with $\mu(x)$ appearing in AQUAL, and $\nu(y)$ in
QUMOND. In the Newtonian limit, $c\rightarrow\infty$,
$\overset{\text{\scriptsize$\leftrightarrow$}}{\nu}$ and
$\overset{\text{\scriptsize$\leftrightarrow$}}{\mu}$ tend to the unit matrix.

The Lorentz factor itself is another relativistic “interpolating function”
between the Newtonian $|{\bf v}|\ll 1$ regime, and the relativistic $|{\bf
v}|\approx 1$ regime; It appears, e.g., in the expression of time dilations.

####  III.1.1 Some lessons

There are some important lessons that we can learn when comparing (special and
general) relativity with its Newtonian-dynamics ancestor.

In astronomy – e.g., when probing galaxy dynamics governed by gravity – we
measure kinematic quantities such as velocities and accelerations. In
Newtonian dynamics, the measured kinematic accelerations, $\ddot{\bf r}$,
themselves, are equal to, and thus give us directly, the gravitational forces
(per unit mass) which we are after. This is also the case in nonrelativistic,
MG MOND theories such as AQUAL, QUMOND, TRIMOND, and GQUMOND.

But, this need not be the case in other theories – as indeed is not the case
in MI MOND theories. In general, forces per unit mass are not equal
accelerations. In momentum-conserving theories, they are equal the rate of
change of some momentum, which we do not measure directly, and whose relation
to the measured kinematic quantities, such as acceleration, depends on the
specific theory.

This, as we saw, is well exemplified by special relativity.

As a result, “intuitive” conclusions that are drawn based on our familiarity
with Newtonian dynamics can be quite misleading when other theories are at
work:

Particles can have very different kinematic accelerations at the same point in
a given force field, depending on other of their orbital characteristics. In
special relativity, for example, for circular orbits:
$\overset{\text{\scriptsize$\leftrightarrow$}}{\mu}{\bf a}=\gamma{\bf a}$,
while for linear trajectories
$\overset{\text{\scriptsize$\leftrightarrow$}}{\mu}{\bf a}=\gamma^{3}{\bf a}$.
More generally, the acceleration of an electron in an electric field depends
not only on their position, but also on their velocity vector.

In particular, the kinematic acceleration, $\ddot{\bf r}$, is not necessarily
parallel to the force. And, no acceleration field is defined by a force field,
only a $\frac{d{\bf P}}{dt}$ field.

Thus, if such a theory underlays the dynamics, trying to map the force field
with different test particles assuming the ${\bf a}\propto{\bf F}$, can give
different results.

As regards interpolating functions, as they appear in relativity, some lessons
are: (a) The interpolating functions are not introduced “by hand” (in special
relativity they are dictated by Lorentz invariance). (b) Different
interpolating functions can underlie different phenomena: Besides
$\overset{\text{\scriptsize$\leftrightarrow$}}{\mu}$, and the Lorentz factor
discussed above, which are functions of ${\bf v}/c$, in GR (which is modified
Newtonian gravity), we have other interpolating functions describing, e.g.,
the transition from strong gravity near a relativistic object, to the
asymptotic Newtonian gravity, which can be thought of as a function of
$MG/Rc^{2}$. (c) Interpolating functions, when they appear in relativity,
have, as independent variables, quantities of the form $q/c$, where $q$ is a
system attribute with the dimensions of velocity, and there may be many such
variables, such as ${\bf v}/c$ and $MG/Rc^{2}$ in the examples above.

All this is to be contrasted with the situation in theories such as AQUAL and
QUMOND, where there is a single interpolating function, of a single variable,
$|\vec{\nabla}\phi|/a_{0}$, introduced by hand at the level of the action, and
thus appearing in all the predictions of this theories. In TRIMOND and GQUMOND
the interpolating functions can depend on several variables. This is also the
case for the MI models described in Se. VI below.

Finally, we learn from the history of the theory of relativity that, as
expected, one ultimately cannot stop at modifying only particle inertia, and
leave the force fields themselves untouched, although this can be a very good
approximation in some instances.

###  III.2 Quantum mechanics as modified inertia

Consider the quantum mechanics of particles in a given, unquantized force
field, such as of electrons interacting electrostatically among themselves and
with an external field. This approximation – referred to sometimes as “First
quantization” – works very well for the description of electrons in atoms, or
in solids, for example.

In quantum mechanics, the departure of particle dynamics from the Newtonian
description is even further reaching than in relativity: It is not a mere
modification of the law dictating how the particle position ${\bf r}(t)$
varies with time. The very description of the particle’s “whereabouts” and
motion is modified. For a many-body system, the Newtonian description of the
system’s history in terms of ${\bf r}_{1}(t),{\bf r}_{2}(t),...$ is replaced
by the system’s wave function $\psi(t,{\bf r}_{1},{\bf r}_{2},...)$. The
positions and velocities of the particles cannot be specified at the same
time. Only probabilities are determined by the theory. Particles have internal
degrees of freedom – their spin and its components, and spin-statistics limits
the domain of allowed many-particle wave function. Particles can tunnel
through classically impenetrable barriers. The evolution of the wave function
is dictated by the Schrödinger equation. All these have no parallels in
Newtonian dynamics.

####  III.2.1 Some lessons

Some of the message we may take from quantum mechanics as modified Newtonian
dynamics are as follows: MOND too could be much more drastic a modification
than just adding gravitational-field degrees of freedom, or changing the field
equation of the gravitational fields, or modifying the particle equations of
motion.

Regarding interpolating functions. As in MOND, the quantum mechanical
description of various phenomena is encapsulated in “interpolating functions”.
These are – as in the case of MOND – quantum mechanical expressions that are
functions of variables of the form $q/\hbar$, where $q$ has the dimensions of
$\hbar$; so it could be an angular momentum, an action variable, etc. These
expressions interpolate between the Newtonian regime – achieved in the formal
limit $\hbar\rightarrow 0$ – and the quantum regime.

An example is the black-body function, where the variable $q=kT/\omega$, which
interpolates between the classical Rayleigh–Jeans law, for $q\ll\hbar$, and
the Wien law for $q\gg\hbar$. Another example is the quantum mechanical
expression for the specific heat of solids – modelled first by Einstein and
then improved on by Debye – interpolating between the classical Dulong-Petit
expression at high temperature and the quantum behavior at low
temperatures.555Einstein’s model invoked some material-characteristic
frequency, $\omega_{\scriptscriptstyle 0}$, and the crucial variable became
$q=kT/\omega_{\scriptscriptstyle 0}$. Debye’s model involved a continuum of
characteristic frequencies. Yet another examples is the expression for the
barrier penetration probability, where the crucial variable is
$q=d[m(V_{0}-E)]^{1/2}$ (d is the characteristic width of the barrier and
$V_{0}$ its height, while $m$ is the mass of the tunneling particle and $E$
its energy.)

In contradistinction with the interpolating functions that appear in theories
like AQUAL and QUMOND, those appearing in quantum mechanics are not introduced
by hand at the fundamental level, they are derived expressions; are not of a
single, universal form that applies to all the phenomena; and, are not
necessarily functions of a single variable. A single variable appears in the
above examples, but it may well happen that more than one crucial variable
appears, for example, in systems with more than one characteristic frequency –
as, e.g. in the description of molecules characterized by rotational-
vibrational frequencies describing roughly the motion of the nuclei, and the
electronic frequencies.

Theories such as GQUMOND and TRIMOND were designed to demonstrate that even in
MG it is possible to have more than one variable appearing. We shall see that
this is also the case in the MI models discussed below. It may even happen
that some of the variables are in one limit (e.g. Newtonian, $q/a_{0}\gg 1$)
and others are in the deep-MOND limit ($q/a_{0}\ll 1$), just as this can
happen in the quantum case.

Another lesson comes from our knowledge that we cannot stop at quantum
mechanics (“first quantization”). The “force fields” must also be consistently
quantized leading to quantum field theory in flat space-time, through the
approximation of describing quantum fields in a classical curved space-time,
and to the yet-to-be-achieved quantization of space-time itself.

In MOND, if we go beyond the restricted MG, most probably we will have to
modify all parts of the action as well.

Lastly, a lesson regarding the degree to which theories lend themselves to the
penetration, manipulation, and solution by us, humans. Some MOND theories are
rather less wieldy than the simple and linear Newtonian dynamics. They are
more difficult to solve, and in some cases it is even hard to see how they can
be solved for general N-body systems. Contrary to what I have heard
occasionally, our difficulties, or technical inability to solve such problems
do not, in any way, argue against the validity of such theories in describing
nature. We know that the similar difficulties arise even in quantum mechanics,
the first approximation in the quantum development, to say nothing of the
higher stages, such as quantum-field theory (e.g., the standard model of
particle physics). And the same is true of the theory of GR. It has taken many
years to develop even crude approximation methods to many of the problems that
arise in such theories.

###  III.3 Other instances of modified or acquired inertia in physics

The idea that inertia is an acquired property is old, generally going under
the name of “Mach’s principle”. According to it, inertia is due to the
interaction of bodies with some omnipresent medium. Ideas along this line
differ on what the medium is and on the nature of the interaction. But if
indeed such is the origin of inertia, then inertia is not some property of
ultimate fundamentality, but can take different forms depending on where we
are in parameter space of the body with respect to signposts and boundary
stones defined by the characteristics of the medium.

For example, I suggested in Ref. [77] that the quantum vacuum is the
omnipresent medium, the interaction with which endows bodies with inertia.
This was shown to be appealing in the context of MOND, because the quantum
vacuum is imprinted with some characteristics of the global state of the
Universe. For example, in a de Sitter cosmological background, the de Sitter
radius, $\ell_{\scriptscriptstyle\Lambda}$, enters the Gibbons-Hawking
temperature, and the Unruh temperature of accelerated systems. This radius is
also known to be related numerically to $a_{0}\sim
c^{2}/\ell_{\scriptscriptstyle\Lambda}$.

Examples of acquired or modified inertia are very common in physics. In the
standard model of particle physics, the masses of the leptons and quarks
(“fundamentally massless”) are acquired by their interaction with the
omnipresent Higgs field, each according to the strength of its interaction.
The effective inertia of the electroweak mediators as they are measured in the
laboratory also appear in “low-energy” circumstances. The inertia of all these
would look very different under other circumstances (with the same physics).

Mass renormalization in quantum-field theory is another example, where the
interaction with the vacuum changes dramatically the effective masses of
particles.

Electrons in solids attain a dispersion relation – a relation between energy
and momentum – that can differ greatly from the relation
$E_{\scriptscriptstyle K}=P^{2}/2m$, which they obey when free, $m$ being the
electron (inertial) mass. In some solids and in some regions of phase space
they still behave like free electrons but obey this relation with a very
different mass.

It has been observed that in some regions of phase space, electrons in certain
systems/materials, such as graphene, behave as massless, dirac particle, with
a dispersion relation $E_{\scriptscriptstyle K}\propto|{\bf P}|$ (see Fig. 1).
We can also see in Fig. 1 that the general “inertial law is more complicated
and takes a different form in other regions of phase space.

Figure 1: Dirac cones in graphene: The kinetic-energy-vs-momentum surfaces of
electrons in graphene. It is characterized by cones that are analogue to light
cones – only with an effective “light speed$\ll c$, around which electrons
behave like massless particles.

Ab initio calculations of the inertia of the electron (due essentially to its
electrostatic energy) were popular around the turn of the 20th century, before
special relativity took a firm hold in the mind. There were, e.g., the results
of Abraham, of Bucherer, and of Lorentz, which Kaufmann set to test
experimentally [88]

In Ref. [87] I showed how certain objects, weakly interacting with a
compressible, irrotational, nonviscous fluid acquire effective inertia, as
well as interacting “gravitationally” among themselves. They have no free
Lagrangian to star with, but the interaction with the fluid endows them with a
“free” Lagrangian of the form $Mc^{2}\ell(v^{2}/c^{2})$, where $M$, the
acquired “rest mass” is a constant that depends on the structure of the body,
$c$ is the fluid’s speed of sound, playing the role of the speed of light, and
$v$ is the velocity of the body with respect to the fluid. As in relativity,
$\ell$ diverges as its argument approaches 1. Interestingly, in three
dimensions this acoustic analog of inertia gives the same inertia as derived
by Abraham and tested by Kaufmann a hundred years earlier, and in a different
setting. In four dimensions, the Lagrangian has a form nearly that of special
relativity.

There are many other instances of acquired inertia within well accepted
physical theories.

##  IV Restricted modified inertia in the context of MOND

Let us then confine our subsequent discussion to nonrelativistic MOND theories
that can be described at MI in the restricted sense defined in Sec. II. Such
theories are required to retain the standard conservation laws of momentum,
energy, and angular momentum. One should also retain some kind boost
invariance. Without it, there would be a meaning to absolute overall velocity
of a system, which will affect the system’s internal dynamics: two similar
systems could have different internal dynamics if they move with respect to
each other, which we wish to avoid.

These requirements are automatically ensured if the modification involves
modifying the Newtonian particle action $I_{\scriptscriptstyle
K}=\int\frac{1}{2}m{\bf v}^{2}~{}dt$ in a way that the standard symmetries
(space and time translations, rotation, and boost) are retained. One can also
modify directly the Newtonian particle-inertia term in the equation of motion,
$m{\bf a}$, in a way that retains the above requirements.

In Ref. [76] I showed that a nonrelativistic, MOND, kinetic action for a
particle, with the above requirements, and the boost symmetry being Galilean,
has to be time nonlocal. Namely, it cannot be of the form $\int L~{}dt$, where
the Lagrangian $L$ is a function of a finite number of time derivatives of the
particle trajectory ${\bf r}(t)$, not even a limit of such Lagrangian with
infinite order of the highest derivative.

An example of a simplistic attempt that fails the above criteria was the
pristine MOND equation of motion [1] of a particle subject to a Newtonian
gravitational acceleration, ${\bf g}_{\scriptscriptstyle N}$,which replaced
the Newtonian equation, ${\bf a}={\bf g}_{\scriptscriptstyle N}$, by
$\mu(a/a_{0}){\bf a}={\bf g}_{\scriptscriptstyle N}$. The inertia term
$\mu(a/a_{0}){\bf a}$ is translation and rotation invariant, and is also
Galilei invariant. However, it leads to nonconservation of momentum in a many-
body system. The symmetries do not ensure conservation laws if the equation of
motion is not derivable from an action – as is the case here. This is why in
Ref. [1], I had to limit its applicability to test particles.

In another simplistic, but rejectable, example, the kinetic action

$I_{\scriptscriptstyle K}=\int\frac{1}{2}m\mathcal{Q}(|\dot{\bf v}|/a_{0}){\bf
v}^{2}~{}dt,$ (8)

is local, obeys the standard symmetries and hence defines conserved momentum,
energy, and angular momentum. Its (fourth-order) equation of motion also
predicts the standard, algebraic MOND relation
$\mu(a/a_{0})a=g_{\scriptscriptstyle N}$, for rotation curves as measured in a
frame where the velocity of the center of a galaxy vanishes. But this action
has no boost invariance, and would predict dependence of the internal dynamics
on some center-of-mass velocity of a system – not clear with respect to what.

It may be that a MOND MI theory can be found that satisfies some boost
invariance other than Galilei – some MOND analogue of Lorentz invariance.
Then, such a theory may perhaps be local, without having unwanted properties.
Such a theory has, however, proven difficult to find for me. For this reason,
all the MI models I discussed in Refs. [76, 77, 78, 79, 80, 81] are time
nonlocal.

##  V Some lower-tier phenomena where predictions of MI and MG may differ
materially

One of the purposes of discussing the specific MI MOND models below in Sec.
VI, is to demonstrate, with concrete examples, some of the potential variety
among MOND theories in making secondary predictions. In contradistinction, the
primary MOND predictions are those that follow from only the basic tenets of
MOND and hence are common to all MOND theories that embody these tenets. Many
of these primary predictions and the ways they follow from the basic tenets
are discussed in Ref. [17].

But before discussing the specific MI models, and a sample of their
predictions, I now described in more general terms some such secondary
phenomena and effects, and explain why they might emerge and appear
differently in different MOND theories.

###  V.1 External-field effect

An important effect, idiosyncratic to MOND, is the external-field effect. It
acts on the internal dynamics of a small, self-gravitating system that is
embedded (e.g., falling freely) in the field of a mother system. The external
field, even if it is uniform across the subsystem (so no tidal effects are
present) makes itself felt in the internal dynamics; the general effect being
to suppress MOND effects and bring the MOND predictions closer to those of
Newtonian dynamics.

The centrality of the effect was pointed out and exemplified with some
specific applications in Ref. [1] based on the pristine formulation of MOND,
and studied theoretically in detail in the AQUAL theory in Ref. [23].

The EFE is not allowed by the strong equivalence principle, which is obeyed by
GR, and there is no explanation of it in standard dynamics with dark matter.
So, a robust identification of its presence at low accelerations would
constitute a “smoking gun” in favour of MOND.

There is a large number of theoretical, phenomenological, and observational
studies of this MOND effect in the context of various systems and phenomena.
Some examples of applications are as follows: Reference [45] showed that the
EFE by a satellite galaxy can cause a warp in the outer parts of a mother disc
galaxy, demonstrating this with the EFE of the Magellanic clouds on the Milky
Way. Reference [46] described the EFE on dwarf satellites as they move in the
field of a mother galaxy, and Ref. [8] finds signs for an EFE in the internal
dynamics of some satellites of the Andromeda galaxy. Reference [89] discussed
the EFE on satellites of elliptical galaxies. References [90, 91] discuss the
importance of the EFE in ultradiffuse galaxies. References [54, 53, 52, 55,
56, 57, 58] showed that the MOND predictions of rotation curved perform
somewhat better if one includes the EFE from surrounding structure. Moreover,
Refs. [55, 56] find evidence that the small departure of the rotation curves
from flatness in the outer parts correlates with their estimates of the
external accelerations produced by large-scale cluster in the way predicted by
AQUAL/QUMOND. Reference [92] demonstrates that the Fornax cluster EFE on
dwarf-galaxy members makes them vulnerable to tidal distortion and disruption,
in accordance with observations. The EFE of the Galactic field on star
clusters may cause asymmetries in their tidal tails (otherwise hard to explain
in the DM paradigm). References [65, 66] discuss and describe observations of
such asymmetries in globular and open clusters.

The EFE of the Galactic acceleration near the Sun – whose strength is $\approx
2a_{0}$ – greatly quenches MOND effects in the internal dynamics of wide
binaries in the solar neighborhood [59, 60, 61, 62, 63, 64]. Because we are
near the MOND-Newtonian transition, the degree of such quenching is sensitive
to the exact predicted workings of the effect.

The EFE is also responsible for quenching, practically completely, all MOND
effects in experiments on Earth, where the (external) gravitational and
kinematical accelerations are many orders of magnitude larger than $a_{0}$.

This extensive body of work attests to the ubiquity and importance of the EFE,
making it doubly important that we appreciate the possible variety in its
predicted workings in different MOND theories, and that we recognize which
aspects of it are primary and which can differ among MOND theories.

The presence of the effect is quite generic in MOND, in the sense that it is
difficult to avoid its appearance in one form or another in any MOND theory
[17], but its exact strength and detailed workings can differ substantially
among MOND theories.

I will try to make some general and schematic statements concentrating on
purely gravitational systems. Suppose the characteristic MOND acceleration of
the external field is $g_{\scriptscriptstyle ex}$ (it can have other
characteristics, such as the presence of many variable accelerations and their
variability time scales, etc.) and that internal to the system is
$g_{\scriptscriptstyle in}$. There is no primary prediction of MOND for the
case where $g_{\scriptscriptstyle ex}\ll g_{\scriptscriptstyle in}$ – such as
when considering the effect of the galactic field on the high-acceleration
dynamics in the inner Solar System – or even when $g_{\scriptscriptstyle
ex}\sim g_{\scriptscriptstyle in}$.

In situation where $g_{\scriptscriptstyle ex}\gg g_{\scriptscriptstyle in}$,
the total acceleration everywhere in the system is $\approx
g_{\scriptscriptstyle ex}$, and inasmuch as this is constant, we can expect
the MOND correction over Newtonian behavior to be the same everywhere in the
subsystem; namely the Newtonian accelerations are multiplied by the same
“discrepancy” factor. This would imply roughly that the internal dynamics
should be quasi Newtonian with an overall correction factor that can be viewed
as a renormalization of G and which, on dimensional grounds can depend only on
$g_{\scriptscriptstyle ex}/a_{0}$ and on additional dimensionless attributes
of the system. So, schematically, we expect an overall correction,
encapsulated in renormalizing $G$ to an effective value

$G_{e}=G\eta(g_{\scriptscriptstyle ex}/a_{0},\alpha_{\scriptscriptstyle
1},\alpha_{\scriptscriptstyle 2},...),$ (9)

and additional departures from Newtonian dynamics that depend on the
dimensionless attributes $\alpha_{\scriptscriptstyle k}$. When
$g_{\scriptscriptstyle ex}/a_{0}\rightarrow\infty$, the MOND tenets require
that $\eta\rightarrow 1$. In the opposite limit, $g_{\scriptscriptstyle
ex}/a_{0}\rightarrow 0$ (but still $g_{\scriptscriptstyle ex}\gg
g_{\scriptscriptstyle in}$), scale invariance of the deep-MOND limit dictates
that666The reason for this expectation is that in the Newtonian-like internal
dynamics that we have here, internal accelerations scale as $a\sim
G_{e}M/r^{2}$. Under space-time scaling, $g_{\scriptscriptstyle ex}$, which is
a MOND acceleration due to external bodies, transforms as
$g_{\scriptscriptstyle ex}\rightarrow\lambda^{-1}g_{\scriptscriptstyle ex}$.
We thus have to have $G_{e}\propto a_{0}/g_{\scriptscriptstyle ex}$, for
$a\rightarrow\lambda^{\scriptscriptstyle-1}a$, as it should ($a_{0}$ appears
for dimensional reasons). Another way to see this is to note that scale
invariance dictates that only the product $A_{0}=Ga_{0}$ can appear in any
deep-MOND-limit prediction, not $G$ and/or $a_{0}$ separately. Thus, we must
have $\eta(x,...)\propto 1/x$, for $x\rightarrow 0$.

$\eta=\frac{a_{0}}{g_{\scriptscriptstyle ex}}s(\alpha_{\scriptscriptstyle
1},\alpha_{\scriptscriptstyle 2},...).$ (10)

AQUAL and QUMOND are underlain by a single interpolating function of a single
variable, $\mu(x)$. So, inevitably, it is this one function that dictates both
rotation curves via $g_{\scriptscriptstyle N}(r)\approx a\mu(a/a_{0})$, and
the above function $\eta$ that enters the prediction of the EFE for dominant
external fields $\eta(x,...)=1/\mu(x)$.777In these theories the direction of
the external field (encapsulated in some of the $\alpha_{\scriptscriptstyle
k}$ parameters) is also felt in the internal dynamics – in a somewhat
different way for the two theories [23, 35, 93, 94].

But this simplification is not at all a general MOND prediction. The phenomena
of rotation curves and the EFE are quite unrelated in MOND, and the rotational
acceleration in a disc galaxy, is not the same quantity as the external
acceleration that appears in the EFE. There is no reason why the two phenomena
should be tightly related in MOND, and underlain by the same formula. So, the
function $\eta(x,..)$ despite the similarity with $1/\mu$ that underlies
rotation curves, may have nothing to do with it in more general MOND theories,
as we shall see in the examples below.

####  V.1.1 Implications of a stronger EFE

One of the predictions of the specific MI models I discuss below, are that the
EFE may be stronger than is predicted by AQUAL/QUMOND. The way this happens in
the models might happen more generally: Even in the restricted case whereby
the EFE is controlled by the same interpolating function as appears in the
predictions of rotation curves, some $\mu(x)$, its argument that enters the
strength of the EFE need not be $x=g_{\scriptscriptstyle ex}/a_{0}$ itself,
but $\alpha g_{\scriptscriptstyle ex}/a_{0}$, where $\alpha$ is one of the
dimensionless system parameters that can enter the EFE in theories that are
underlain by more general interpolating schemes than AQUAL/QUMOND. In the MI
models below, $\alpha>1$ is a function of the ratio of frequencies of the
external field variations to those of the internal ones.

The basic tenets do tell us robustly – making it a primary prediction – that
in the low-acceleration, asymptotic region of rotation curves, the rotational
speed is $V(r)\approx V_{\scriptscriptstyle\infty}\equiv(MGa_{0})^{1/4}$ ($M$
is the mass of the galaxy). So the ratio of the MOND acceleration,
$a(r)=V^{2}(r)/r$, and the Newtonian gravitational acceleration there,
$g_{\scriptscriptstyle N}=MG/r^{2}$, is $\eta\approx a_{0}/a$ (with the
standard normalization of $a_{0}$). However, there is nothing in the basic
tenets that tells us that in the prediction of the mass anomaly in subsystems
in a dominant, deep-MOND external field, $g_{\scriptscriptstyle ex}$ we also
have $\eta\approx a_{0}/g_{\scriptscriptstyle ex}$ – namely that in Eq. (10)
we have $s=1$.

This important point is worth recapping: asymptotic rotation curves, and a
dominant, deep-MOND EFE are unrelated phenomena, and there is nothing in MOND
that predicts that they should be described by the same expression.

A stronger EFE, as predicted by some MOND theories, stronger even by a factor
of a few, may have important implications in cases where
$g_{\scriptscriptstyle ex}\lesssim a_{0}$ (when $g_{\scriptscriptstyle ex}\gg
a_{0}$ the internal dynamics is approximately Newtonian anyhow). The following
are some examples:

In the outskirts of disc galaxies, a stronger EFE may show itself more clearly
in causing a more pronounced decline in the outer parts of rotation curve, in
departure from the asymptotic flatness predicted for isolated systems,
affecting such analyses as in Refs. [54, 53, 52, 55, 56, 57, 58] . It can also
increase the effectiveness of satellites inducing warps in a mother disc,
beyond what is predicted by AQUAL (or QUMOND), as described in Ref. [45].

The field of the Galaxy at the position of the solar system is $\approx
2a_{0}$. The EFE is then expected to quench MOND effects in nearby systems of
intrinsic accelerations $\lesssim a_{0}$, such as wide binary stars. Indeed,
analysis of Gaia data of such binaries [59, 60, 61, 62, 63, 64] compare with
the predictions of AQUAL/QUMOND that give $\eta\approx 1/\mu(2)\approx 1.5$.
This leads to a quasi-Newtonian behavior, but with velocities elevated by a
factor $\eta^{\scriptscriptstyle 1/2}\approx 1.2$. However, if the correct
argument value for $\mu$ is several times larger, the expected velocity
anomaly is even smaller, only 8% (instead of 20%) if $\alpha=3$, for example.

In MOND, vertical dynamics in disc galaxies cannot be separated from the
radial dynamics, because total accelerations are important. The way the radial
acceleration – largely the one that causes the circular rotation in the disc –
enters the vertical dynamics may be viewed as a special case of the EFE. All
analyses of vertical disc dynamics in MOND to date, e.g., in Refs. [69, 70,
71, 73, 72, 75], have adopted the simplistic, single-interpolating-function
scheme using either the AQUAL/QUMOND predictions, or even, unjustifiably,
assumed the algebraic relation between the Newtonian and MOND accelerations
outside the disc. (There is no MOND theory that predicts this; MI predict such
an algebraic relation only for exactly circular orbits in the disc’s
midplane.) Such analyses would require major modifications in the framework of
other MOND theories.

###  V.2 Effects of The Galactic field in the Solar System

It has been shown that AQUAL and QUMOND predict an anomaly (departure from
standard dynamics) due to the presence of the gravitational field of the
Galaxy in which the Solar System is embedded [67, 68]. The anomaly is an added
contribution to the solar field that is very small, but may be detectable,
even in the inner Solar System where the accelerations due to the Sun,
$g_{\scriptscriptstyle\odot}({\bf r})$, are many orders of magnitude larger
than $a_{0}$. This anomaly does not decrease like
$1-\mu(g_{\scriptscriptstyle\odot}/a_{0})$, as some other MOND anomalies, and
so it does not become negligible even if $\mu(x)$ approach 1 very fast at
large $x$.

To understand why this is not a primary MOND prediction, and why it can
differs greatly from the predictions of MI, I next explain the origin of this
anomaly. It arises as follows: one solves the AQUAL or QUMOND gravitational
field of the Sun with the Galactic field as boundary condition at infinity,
and then subtract the latter to get the acceleration relative to the Sun. The
MOND gravitational potential $\phi({\bf r})$, in such MG theories can be
thought of as the Newtonian potential (solution of the Poisson equation), but
with a modified source density, $\hat{\rho}=\Delta\phi/4\pi G$, not the actual
density, $\rho$, that sources the MOND equation. The excess,
$\rho_{\scriptscriptstyle p}\equiv\hat{\rho}-\rho$ is called the “phantom mass
density,” and its Newtonian field constitutes the anomaly.

There is a region around the point – somewhat different in AQUAL and QUMOND,
but several thousand astronomical units away from the Sun – where the Galactic
field cancels that of the Sun, where the gravitational accelerations are
$\lesssim a_{0}$, and where AQUAL and QUMOND predict some distribution of
phantom matter. This is then felt everywhere in the Solar System as an
additive anomaly – the Newtonian field of the phantom matter, which is
independent of the local value of the acceleration. In the inner solar system,
much nearer the Sun than the balance point, this anomaly can be approximated
by a quadrupole field, where at a distance $r$ from the Sun, this added
acceleration is $\approx qa_{0}(r/r_{\scriptscriptstyle M})$ (with some
quadrupolar angular dependence), where $r_{\scriptscriptstyle
M}\equiv(M_{\odot}G/a_{0})^{1/2}\approx 7000$ astronomical units is the “MOND
radius” of the Sun, and q is a dimensionless prefactor, which is roughly
proportional to $1-\mu(g_{\scriptscriptstyle g}/a_{0})$, with
$g_{\scriptscriptstyle g}\approx 2a_{0}$ the Galactic field at the Sun’s
position [67, 68]. So, as expected, the value of the anomalous acceleration,
at position ${\bf r}$, does not depend on the local value of
$g_{\scriptscriptstyle\odot}$. The relative correction to the Sun’s Newtonian
acceleration is $\approx q(g_{\scriptscriptstyle\odot}/a_{0})^{-3/2}$, which
decreases only as a power $-3/2$ of the ratio
$g_{\scriptscriptstyle\odot}/a_{0}$, no matter how fast MOND approaches
Newtonian dynamics at high acceleration, as described by $1-\mu(x)\rightarrow
0$. The MOND to Newtonian transition occurs when $g_{\scriptscriptstyle
g}/a_{0}$ becomes large, not when $g_{\scriptscriptstyle\odot}/a_{0}$ becomes
large.

Phenomenological consequences and constraints that derive from this anomaly
were discussed in Refs. [67, 68, 53]. Some studies have, in fact, attributed
observed Solar-System anomalies to this effect [95, 96, 97, 98].

In a sense, this anomaly is the special case of the EFE when the external
acceleration is much weaker that those in the region of the subsystem we are
interested in. But there is an important difference: The main features of the
EFE when the external field dominates over all internal accelerations seem
generic in MOND, despite some important differences in detail. But the present
anomaly is not generic; in some MI formulations it is predicted to essentially
not be present.

It has to be realized that the anomaly, predicted by AQUAL and QUMOND, is the
result of idiosyncracies of these theories, specifically, the Poisson-like
nature of their field equations. These, like the Poisson equation itself, are
elliptic equations, which introduces an element of space-nonlocality. The
solution of these equation in a certain region depends on what happens in
other regions, in the form of sources, or of dictated boundary
conditions.888For example, in electrostatics the solution depends on the
charge density everywhere, and on the presence of conductors that dictate a
constant potential on their surface. Thus, it is enough that there is some
region in space where the MOND gravitational accelerations are small – such as
here, the region around Galaxy-Sun balance point – to induce an anomalous
acceleration everywhere, independent of whether the local total acceleration
is very high. In contradistinction, as has been explained already in Ref.
[67], MI theory might make very different predictions on this phenomenon, as
indeed will be demonstrated by the specific MI models I describe below.999A
quote from Ref. [67]: “Small, nonrelativistic MOND anomalies in high-
acceleration motions may possibly have several origins. In non-local, modified
inertia formulations of MOND, anomalies may appear in the inner solar system
in the motion of bodies that are on highly eccentric trajectories;
trajectories that take them to large distances, where accelerations are low
(Milgrom, in preparation). Cases in point are the long-period comets, and the
Pioneer spacecraft. Because of the non-locality of such theories, the motions
in high-acceleration portions of the orbit may be affected by the low
acceleration ones, giving rise to small anomalies. Such MOND effects have been
proposed as a possible mechanism for generating the Pioneer anomaly, without
affecting the motions of planets, whose orbits are wholly in the high
acceleration regime (Milgrom 2002, 2005, in preparation).”  In MI theories –
such as our MI models below – the MOND modification depends only on what
happens along the trajectory itself. If all accelerations there, $a$, are
high, as happens with bodies that live only in the inner Solar System – MOND
approaches Newtonian dynamics as fast as $1-\mu(a/a_{0})$. It does not matter
that there are some other regions in space where the accelerations would be
small.

###  V.3 Additional secondary phenomena

One important effect, which I mention here only in passing because I cannot
offer any concrete statements as to how it works in MI – but which might be
described very differently in MG and MI – is dynamical friction. The MOND
predictions for this effect in AQUAL and QUMOND have been analyzed
analytically and numerically, and applied, e.g., in Refs. [99, 100, 101, 102,
103, 104, 105, 106].

Gravitational waves are a relativistic phenomena and so goes beyond the basic
tenets of MOND, and cannot be thought of as a primary MOND phenomenon. Indeed,
existing MG relativistic extensions of MOND show a large variety in what they
predict for gravitational waves: the type of modes there are, and their
propagation.

Without having any specific result, my only observation at this point is that
we can expect very different predictions for gravitational wave in purely MG,
relativistic theories, and MI theories. Sine MG theories modify gravity
itself, they necessarily modify the properties of gravitational waves in major
ways, generically adding modes to the two general-relativistic propagation
modes, modifying their speeds, etc. This may not be necessary in MI theories.

There is at least one class of MG, relativistic extensions of MOND where the
tensor mode of gravitational waves propagate in the same way as photons [30].
But this is by no means a generic result; for example, it is not so the
original form of TeVeS. In MI it is more natural for the path followed by
gravitational waves (gravitons) to be the same as that for photons.

##  VI A class of Modified-inertia MOND models

Here I describe, somewhat briefly, the class of MI, MOND models propounded
recently in Ref. [81]. I call them “models” because, as discussed in Ref.
[81], there are some matter-of-principle issues regarding them still to be
investigated, for example, the issue of causality. But these models are quite
useful as heuristic tools for demonstrating how MI formulations of MOND can
differ substantially in their secondary predictions, from the two MG
formulations we extensively use today. These new models are also rather more
amenable to analysis than MI models propounded before, such as in Ref. [76].
Beyond the summary description of the models, and the discussion of some of
their predictions discussed in Ref. [81], I discuss here some new
ramifications.

In constructing the models, I do not start from an action principle (it
remains to be seen if they can be derived from an action), but directly from
the equations of motion.

The models aim at describing the dynamics of a system of (pointlike) bodies of
masses $m_{\scriptscriptstyle p}$, whose trajectories, ${\bf
r}_{\scriptscriptstyle p}(t)$, the models should dictate. Each body $p$ is
subject to (Newtonian, since we only modify inertia) forces ${\bf
f}_{\scriptscriptstyle pq}({\bf r}_{\scriptscriptstyle p},{\bf
r}_{\scriptscriptstyle q})$ from other bodies $q$ (${\bf
f}_{\scriptscriptstyle qp}=-{\bf f}_{\scriptscriptstyle pq}$). So, body $p$ is
subject to the total, time-dependent force

${\bf F}_{p}(t)=\sum_{\scriptscriptstyle q\not=p}{\bf f}_{\scriptscriptstyle
pq}[{\bf r}_{\scriptscriptstyle p}(t),{\bf r}_{\scriptscriptstyle q}(t)]$ (11)

along its trajectory. Concentrate on one of the bodies; its Newtonian law of
motion (omitting its index $p$) is $m{\bf a}_{\scriptscriptstyle N}(t)={\bf
F}(t)$ reads in Fourier space

$m\hat{\bf a}_{N}(\omega)=\hat{\bf F}(\omega).$ (12)

We modify the left-hand, inertia term writing, for the actual (MOND) Fourier
component of the particle’s acceleration, $\hat{\bf a}(\omega)$,

$m\hat{\bf a}(\omega)\mathcal{I}[\\{\hat{\bf r}\\},\omega,a_{0}]=\hat{\bf
F}(\omega),~{}~{}~{}~{}~{}{\rm or}~{}~{}~{}~{}~{}\hat{\bf
a}(\omega)\mathcal{I}[\\{\hat{\bf r}\\},\omega,a_{0}]=\hat{\bf
a}_{\scriptscriptstyle N}(\omega).$ (13)

The “inertia functional”, $\mathcal{I}[\\{\hat{\bf r}\\},\omega,a_{0}]$, is a
dimensionless functional of the whole trajectory [designated $\\{\hat{\bf
r}\\}$ to distinguish it from the number $\hat{\bf r}(\omega)$], and a
function of $\omega$, and of $a_{0}$, which is the only dimensioned constant
appearing in $\mathcal{I}$.

$\mathcal{I}$ has to conform to the two limits defined in the basic tenets of
MOND.

$\mathcal{I}~{}~{}\overset{a_{0}\rightarrow 0}{\longrightarrow}~{}~{}1,$ (14)

for the theory to tend to the Newtonian relation (12) at high accelerations.
And, Ref. [81] explains why in the opposite limit, the scale invariance of the
deep-MOND limit dictates that101010This follows when we consider a system
where the forces are purely gravitational – what we know of MOND so far all
comes from considering gravitational systems. In the context of more general
forces, the exact definition of the deep-MOND limit requires further
deliberation.

$\mathcal{I}~{}~{}\overset{a_{0}\rightarrow\infty}{\longrightarrow}~{}~{}\mathcal{A}[\\{\hat{\bf
r}\\},\omega]/a_{0},$ (15)

where $\mathcal{A}[\\{\hat{\bf r}\\},\omega]$ is a functional of the
trajectory, and a function of $\omega$, having the dimensions of acceleration.

###  VI.1 Some general properties

Note that the modified equations of motion (13) lead to a time nonlocal
theory, since $\\{\hat{\bf r}\\}$ appearing as an argument in $\mathcal{I}$
require knowledge of the whole trajectory. Note also that $\hat{\bf
F}(\omega)$, or $\hat{\bf a}_{\scriptscriptstyle N}(\omega)$ appearing on the
right-hand sides in Eq. (13), while they are the Newtonian expressions, are
calculated for the MOND trajectories (not the Newtonian ones), which are not
known a priori. So the problem has to be solved self consistently.

####  VI.1.1 Conservation laws

The requirements for the models to enjoy the standard symmetries; time and
space translations, rotations, and Galilei are discuss at length in Ref. [81].
I shall not repeat them here, only to say that the specific examples I
concentrate on below satisfy these requirements. However, remember that the
symmetries do not ensure the standard conservation laws without there being an
action basis for the theory. Nevertheless, while we still lack such an action
for the present models, they do satisfy conservation laws for some modified
definitions of the momentum, energy, and angular momentum – modifying inertia
always necessitates redefinition of these quantities as exemplified by special
relativity. These conserved quantities are defined here through their Fourier
transform, and are time nonlocal in nature. We define the momentum, for
example, such that it reduces to the standard momentum $m{\bf v}$ in the limit
$a_{0}\rightarrow 0$, and that it satisfies $\dot{\bf P}(t)={\bf F}(t)$, which
in Fourier space reads $i\omega\hat{\bf P}(\omega)=\hat{\bf F}(\omega)$. This
leads, from Eq. (13), to the definition

$\hat{\bf P}(\omega)=m\hat{\bf v}(\omega)\mathcal{I}[\\{\hat{\bf
r}\\},\omega,a_{0}].$ (16)

In isolated systems $\sum_{p}{\bf F}_{p}=0$; so we have $d(\sum_{p}{\bf
P}_{p})/dt=0$.

Similarly, a kinetic energy that satisfies $dE_{k}/dt={\bf v}(t)\cdot{\bf
F}(t)$ will give a conserved total energy when the interbody forces are
derivable from potentials, $\phi_{pq}({\bf r}_{pq})$, since then

$\frac{d}{dt}(\sum_{p}E^{k}_{p}+\sum_{\scriptscriptstyle q<p}\phi_{pq})=0.$
(17)

This leads to the definition [81]

$\hat{E}_{k}(\omega)=\frac{m}{2\pi}\int\frac{\omega^{\prime}}{\omega}\hat{\bf
v}(\omega-\omega^{\prime})\cdot\hat{\bf
v}(\omega^{\prime})\mathcal{I}[\\{\hat{\bf
r}\\},\omega^{\prime},a_{0}]d\omega^{\prime},$ (18)

which reduces to the standard $E_{k}=mv^{2}/2$ for $a_{0}\rightarrow 0$.
Similarly, we define the angular momentum as

$\hat{\bf J}(\omega)=\frac{m}{2\pi}\int\frac{\omega^{\prime}}{\omega}\hat{\bf
r}(\omega-\omega^{\prime})\times\hat{\bf
v}(\omega^{\prime})\mathcal{I}[\\{\hat{\bf
r}\\},\omega^{\prime},a_{0}]d\omega^{\prime},$ (19)

whose inverse Fourier transform is conserved in systems not subject to an
external moment.

####  VI.1.2 “Center of mass”

A natural (nonlocal) definition of the center of mass of the many-body system
is

$\hat{\bf R}(\omega)=\frac{\sum_{p}M_{p}\hat{\bf
r}_{p}(\omega)}{\sum_{p}M_{p}},~{}~{}~{}~{}~{}~{}M_{p}(\omega)\equiv
m_{p}\mathcal{I}[\\{\hat{\bf r}_{p}\\},\omega,a_{0}],$ (20)

which by the conservation of momentum for isolated systems, satisfies
$\ddot{\bf R}=0$, and reduces to the standard definition for $a_{0}\rightarrow
0$.

####  VI.1.3 The two-body problem in the deep-MOND regime

As in Newtonian dynamics, the two-body problem in the deep-MOND limit of the
present models can be reduced to the problem of a single body in a force field
[81], for which the equation of motion in Fourier space is

$\bar{m}\hat{\bf
a}_{12}(\omega)\frac{\mathcal{A}_{12}(\omega)}{a_{0}}=\hat{\bf F}(\omega),$
(21)

where the reduced mass is given by

$\bar{m}=\frac{m_{1}m_{2}}{(m_{1}^{1/2}+m_{2}^{1/2})^{2}}$ (22)

to be compared with the Newtonian reduced mass
$\bar{m}=m_{1}m_{2}/(m_{1}+m_{2})$. The quantities with subscript $12$ refer
to the relative distances and accelerations, and $\mathcal{A}$ is what defines
the deep-MOND limit as in Eq. (15).

The relative velocity, $V_{12}$, between two masses, $m_{\scriptscriptstyle
1}$ and $m_{\scriptscriptstyle 2}$, on deep-MOND limit circular orbits around
each other is given by111111For circular orbits, there is only one frequency
appearing, and $\mathcal{A}_{12}$, which has dimensions of acceleration, can
only be proportional to $V^{2}_{12}/R_{12}$. This is why such a general result
can emerge without specifying details of the model. Arbitrary constants that
appear are then absorbed into the definition of $a_{0}$, which we always take
so that for a test mass around a larger mass we have $V_{12}^{4}=MGa_{0}$.

$\Delta V^{4}_{12}=(q_{1}^{1/2}+q_{2}^{1/2})^{2}MGa_{0},$ (23)

with $M=m_{\scriptscriptstyle 1}+m_{\scriptscriptstyle 2}$, and
$q_{\scriptscriptstyle p}=m_{\scriptscriptstyle p}/M$. This is to be
contrasted with the prediction of AQUAL, QUMOND, and the general class of MG
theories described in Ref. [86]:

$V^{4}_{12}=\frac{4}{9}\left(\frac{1-q_{\scriptscriptstyle
1}^{\scriptscriptstyle 3/2}-q_{\scriptscriptstyle 2}^{\scriptscriptstyle
3/2}}{q_{\scriptscriptstyle 1}q_{\scriptscriptstyle 2}}\right)^{2}MGa_{0}.$
(24)

When $q_{\scriptscriptstyle 1}\ll q_{\scriptscriptstyle 2}$, both expressions
give the MOND mass-asymptotic-speed relation (“baryonic Tully-Fisher
relation”), $V_{12}^{4}=MGa_{0}$. This is indeed a primary MOND prediction.
For equal masses the coefficient in Eq, (23) is 2, and in Eq. (24) it is
$\approx 0.61$ predicting a relative speed $\approx 25$ percent lower.

The result that in the deep-MOND limit $V^{4}_{12}/MGa_{0}$ has to be of order
unity is a primary prediction that follows from the basic tenets of MOND; but,
as exemplified here, the exact value of the ratio is theory dependent, and
constitutes a secondary prediction.

###  VI.2 Subclass of models

So far, not much was specified about the inertia functional, $\mathcal{I}$,
besides its asymptotic forms. To enable us to discuss possible predictions
more concretely, I first limit the discussion to a subclass of models, and
then I will further concentrate on specific examples within this subclass.

In the subclass of model, the inertia functional is expressed as a function of
a finite number of variables

$\mathcal{I}[\\{\hat{\bf
r}\\},\omega,a_{0}]=\bar{\mu}\left[\frac{\mathcal{A}_{1}(\omega)}{a_{0}},\frac{\mathcal{A}_{2}(\omega)}{a_{0}},...\right],$
(25)

where $\mathcal{A}_{i}(\omega)$ are themselves functionals of the trajectory,
and explicit functions of the frequency, with the dimensions of acceleration.
In the limit where all $\mathcal{A}_{i}\gg a_{0}$ (formally $a_{0}\rightarrow
0$) we should have $\bar{\mu}\approx 1$ to restore Newtonian dynamics.

In the deep-MOND limit, where $\mathcal{A}_{i}\ll a_{0}$, or, formally, when
$a_{0}\rightarrow\infty$, scale invariance of gravitational dynamics – for
which the right-hand side of the equation of motion (13) has a scaling
dimension $-2$ – dictates that $\bar{\mu}$ becomes homogeneous of degree 1 in
its variables,121212Since the variables $\mathcal{A}_{i}$ are constructed only
from the particle-trajectory degrees of freedom, and since they have
dimensions of acceleration, their scaling dimension must be that of kinematic
acceleration, namely -1: under space-time scaling $(t,{\bf
r})\rightarrow\lambda(t,{\bf r})$ we have
$\mathcal{A}_{i}\rightarrow\lambda^{\scriptscriptstyle-1}\mathcal{A}_{i}$.
namely that in this limit $\bar{\mu}$ has the property that
$\bar{\mu}(\lambda^{\scriptscriptstyle-1}x_{\scriptscriptstyle
1},\lambda^{\scriptscriptstyle-1}x_{\scriptscriptstyle
2},...)=\lambda^{\scriptscriptstyle-1}\bar{\mu}(x_{\scriptscriptstyle
1},x_{\scriptscriptstyle 2},...)$. This means (seen by taking
$\lambda=x_{\scriptscriptstyle 1}$) that the deep-MOND limit of $\bar{\mu}$
can be written most generally as

$\bar{\mu}\overset{a_{0}\rightarrow\infty}{\longrightarrow}\frac{\mathcal{A}_{1}}{a_{0}}K\left(\frac{\mathcal{A}_{2}}{\mathcal{A}_{1}},\frac{\mathcal{A}_{3}}{\mathcal{A}_{1}},...\right).$
(26)

Because different acceleration functionals $\mathcal{A}_{i}$ can be defined,
we see that in this subclass of models the transition from the Newtonian to
the deep-MOND limit can be rather more involved than in AQUAL/QUMOND. For
example, in the latter the deep-MOND limit behavior is totally fixed, in the
former we still have to dictate a function $K$ of several (dimensionless)
variables. This would lead, in effect, to different interpolating functions
governing different phenomena, or even different types of trajectories (e.g.,
circular ones and radial ones).

####  VI.2.1 Example

In Ref. [81], I discussed requirements on the way $\mathcal{A}_{i}$ have to
depend on $\\{\hat{\bf r}\\}$ and $\omega$, so that the models satisfy various
desiderata – for example, the invariance requirements under time and space
translation, and under rotations, and the requirement that they predict a
correct center-of-mass motion. I will not repeat this discussion, but only
give an example of models that employ two choices of $\mathcal{A}_{i}$ that
satisfy these requirements:

$\mathcal{A}_{1}(\omega)=\frac{1}{2^{\scriptscriptstyle
3/2}\pi}\int_{\scriptscriptstyle-\infty}^{\scriptscriptstyle\infty}\theta_{1}\left(\frac{\omega^{\prime}}{\omega}\right)[\hat{\bf
r}(\omega^{\prime})\cdot\hat{\bf r}^{*}(\omega^{\prime})]^{1/2}\omega^{\prime
2}d\omega^{\prime},$ (27)
$\mathcal{A}_{2}(\omega)=\frac{1}{2^{\scriptscriptstyle
3/2}\pi}\int_{\scriptscriptstyle-\infty}^{\scriptscriptstyle\infty}\theta_{2}\left(\frac{\omega^{\prime}}{\omega}\right)|\hat{\bf
r}(\omega^{\prime})\cdot\hat{\bf r}(\omega^{\prime})|^{1/2}\omega^{\prime
2}d\omega^{\prime}.$ (28)

The equation of motion is thus

$\hat{\bf
a}(\omega)\bar{\mu}\left[\frac{\mathcal{A}_{1}(\omega)}{a_{0}},\frac{\mathcal{A}_{2}(\omega)}{a_{0}}\right]=\hat{\bf
a}_{\scriptscriptstyle N}(\omega),$ (29)

and in the deep-MOND limit

$\bar{\mu}\approx\frac{\mathcal{A}_{1}}{a_{0}}K(\mathcal{A}_{2}/\mathcal{A}_{1}),$
(30)

with $K(X)$ not dictated by the MOND tenets, since it is scale invariant, and
the $\mathcal{A}_{1}$ prefactor gives $\bar{\mu}$ the correct scaling
behavior.

To understand the meaning of the two acceleration functionals, we note that
for a single Fourier component

${\bf r}(t)=\frac{1}{\sqrt{2}}({\bf r}_{\scriptscriptstyle
0}e^{i\omega_{\scriptscriptstyle 0}t}+{\bf r}_{\scriptscriptstyle
0}^{*}e^{-i\omega_{\scriptscriptstyle 0}t});~{}~{}~{}~{}~{}~{}~{}{\bf
r}^{2}(t)={\bf r}_{\scriptscriptstyle 0}\cdot{\bf r}_{\scriptscriptstyle
0}^{*}+|{\bf r}_{\scriptscriptstyle 0}\cdot{\bf r}_{\scriptscriptstyle
0}|\cos{(2\omega_{\scriptscriptstyle 0}t+\varphi)}.$ (31)

The amplitude vector, ${\bf r}_{\scriptscriptstyle 0}={\bf
r}_{\scriptscriptstyle R}+i{\bf r}_{\scriptscriptstyle I}$ is a complex-valued
vector, and

${\bf r}_{\scriptscriptstyle 0}\cdot{\bf r}_{\scriptscriptstyle 0}^{*}={\bf
r}_{\scriptscriptstyle R}^{2}+{\bf r}_{\scriptscriptstyle
I}^{2},~{}~{}~{}~{}~{}~{}~{}{\bf r}_{\scriptscriptstyle 0}\cdot{\bf
r}_{\scriptscriptstyle 0}={\bf r}_{\scriptscriptstyle R}^{2}-{\bf
r}_{\scriptscriptstyle I}^{2}+2i{\bf r}_{\scriptscriptstyle R}\cdot{\bf
r}_{\scriptscriptstyle I}.$ (32)

The Fourier component of ${\bf r}(t)$ is

$\hat{\bf r}(\omega)=[{\bf r}_{\scriptscriptstyle
0}\delta(\omega-\omega_{\scriptscriptstyle 0})+{\bf r}_{\scriptscriptstyle
0}^{*}\delta(\omega+\omega_{\scriptscriptstyle 0})].$ (33)

We see from the second of Eqs. (31) that $(\hat{\bf r}\cdot\hat{\bf
r}^{*})^{1/2}$, which appears in the integrand of eq. (27), which defines
$\mathcal{A}_{1}$, is the root-mean-square radius for the component, and the
integrand is the root-mean-square acceleration weighted by $\theta_{1}$. We
also see that $|\hat{\bf r}\cdot\hat{\bf r}|^{1/2}$, which appears in the
integrand of eq. (28), which defines $\mathcal{A}_{2}$, measures the departure
from circularity of the trajectory: it vanishes for the circular case (for
which ${\bf r}_{\scriptscriptstyle R}^{2}={\bf r}_{\scriptscriptstyle I}^{2}$
and ${\bf r}_{\scriptscriptstyle R}\cdot{\bf r}_{\scriptscriptstyle I}=0$),
and equals $(\hat{\bf r}\cdot\hat{\bf r}^{*})^{1/2}$ for the radial case. So,
circular Fourier components do not contribute to $\mathcal{A}_{2}$.

The forms and normalizations of $\theta_{i}(\zeta)$ are still free. With the
convenient normalization $\theta_{1}(1)=1$, which we adopt, $\mathcal{A}_{1}$
equals the acceleration $V^{2}/R$ for an exactly circular trajectory. It can
be seen that for circular, deep-MOND trajectories around an attracting mass
$M$, for which $\mathcal{A}_{2}=0$, and only $\omega^{\prime}=\omega$
contributes in Eq. (27), the theory is easily solved and one gets for the
orbital speed

$V^{4}=MGa_{0}/\theta_{1}(1)K(0).$ (34)

The standard normalization of $a_{0}$ in MOND is then achieved by taking
$K(0)=1$.

The normalization of $\theta_{2}$ is degenerate with the dependence of
$\bar{\mu}$ on $\mathcal{A}_{2}$, and we can fix it at $\theta_{2}(1)=1$, as
well. This dependence on $\mathcal{A}_{2}$, and, in particular, the behaviour
of the deep-MOND limit $K(X)$, hardly affects the study of rotation curves
(see below), and I cannot offer, at present, constraints on them (if we take
these models seriously). To constrain them we need information from test
particles on noncircular trajectories; e.g., the comparison of acceleration
discrepancies as deduced in the same system from circular trajectories (e.g.,
rotation curves) and noncircular ones (e.g., motions of satellites, such as
dwarf-galaxies or globular clusters).

It was also argued in Ref. [81], and explained below in Sec. VI.4.1, that we
have to have

$\theta_{i}(\zeta)\overset{\zeta\rightarrow\infty}{\longrightarrow}0$ (35)

fast enough. This ensures that the center-of-mass motion of bodies comprising
components at high accelerations – such as stars in a galaxy – is still
MONDian. This is because then the internal, high-acceleration, but also high-
frequency, motions, do not affect the low-frequency, lower-acceleration
components of the center-of-mass motion. It makes sense to take
$\theta_{i}(\zeta)$ to be monotonic, in which case they also decrease between
$0$ and $1$. So, for instance, $\theta_{i}(0)>1$, which has important
consequences for the EFE, which depends on $\theta_{i}(0)$, since typically
the characteristic frequencies of the external acceleration are much smaller
than those of the internal ones (see below).

###  VI.3 Phenomenology

I now discuss some predictions of the restricted class described in Sec.
VI.2.1. In Ref. [81], I concentrated, for concreteness’s sake, on the even
simpler models that involve only $\mathcal{A}_{1}$ as the acceleration
functional appearing as the variable in $\bar{\mu}$ (alluding only in passing
to the possibility of defining $\mathcal{A}_{2}$). Even this was enough to
demonstrate that different interpolating functions can appear in different
phenomena, because the relevant values of $\theta_{1}(\zeta)$ may differ when
there is more than one underlying frequency.

Here, I emphasize that having more than one acceleration variable enriches the
possibilities even further, demonstrating even more forcefully the potential
diversity of secondary predictions among MOND formulations.

Below, I discuss some phenomenological consequences of the above model
examples, mentioning only in passing those already discussed in Ref. [81], and
some new results in more detail.

####  VI.3.1 Generalities

To be able to present some analytic results of the models, we try to identify
situations where the relevant trajectories have only a finite number of
discrete Fourier components – ideally only two major ones. In general, this is
not possible. The equation of motion (13) is to be solved selfconsistently,
with the expression of the solution trajectory, ${\bf r}(t)$, substituted in
the expression for the force on the right-hand side. Consider even the simpler
case of a single particle moving in a given force field (as contrasted with a
many-body problem). If one tries to substitute in ${\bf F}({\bf r})$ some
trial ${\bf r}(t)$ that has a finite number of Fourier components at
frequencies $\omega_{\scriptscriptstyle k}$, in general, the resulting
$\hat{\bf F}(\omega)$ will have all the combined frequencies
$p\omega_{\scriptscriptstyle k}\pm q\omega_{\scriptscriptstyle n}$, which
shows that this ${\bf r}(t)$ cannot be a consistent solution.

But, there are situations, where the physical trajectory can be written,
approximately, as a sum of “separable” components, each characterized by a
single frequency – or a bunch of neighboring frequencies, which we lump into
one for heuristic purposes,

${\bf r}(t)=\sum_{k}{\bf r}_{\scriptscriptstyle
k}(t),~{}~{}~{}~{}~{}~{}~{}~{}{\bf r}_{\scriptscriptstyle
k}(t)=\frac{1}{\sqrt{2}}(\bar{\bf r}_{\scriptscriptstyle
k}e^{\scriptscriptstyle i\omega_{\scriptscriptstyle k}t}+\bar{\bf
r}^{\scriptscriptstyle*}_{\scriptscriptstyle
k}e^{\scriptscriptstyle-i\omega_{\scriptscriptstyle k}t}).$ (36)

Furthermore, in such instances, the (Newtonian) force itself separates, to a
good approximation, into forces that depend on the separate component
trajectories:

${\bf F}[{\bf r}(t)]\approx\sum_{k}{\bf F}_{k}[{\bf r}_{\scriptscriptstyle
k}(t)].$ (37)

An example is the approximate separation between vertical and radial dynamics
in a thin galactic disk. For a particle that does not wonder much in $R$
(ideally, that is on a circular orbit), the $R$ dependence of the force in the
perpendicular, $z$, direction becomes irrelevant, and vice versa if the
particle does not wonder much in $z$. We can then write

${\bf F}[{\bf r}(t)]\approx F_{R}(R){\bf e}_{\scriptscriptstyle
R}+F_{z}(z){\bf e}_{\scriptscriptstyle z}$ (38)

(${\bf e}_{\scriptscriptstyle R}$ and ${\bf e}_{\scriptscriptstyle z}$ are
unit vectors in the $R$ and $z$ directions, respectively). Other examples
discussed in Ref. [81] are the motion in a triaxial harmonic field where the
three principal components are exactly separable, and the motion along each
axis is still harmonic (single frequency) but the motion frequencies differ
from those appearing in the expression for the potential field:
$\phi=(1/2)\sum\omega_{\scriptscriptstyle k}^{\scriptscriptstyle
2}x_{\scriptscriptstyle k}^{\scriptscriptstyle 2}$.

If such an approximation is justified, we can write separate equations of
motion for the different components

$m\hat{\bf a}_{\scriptscriptstyle k}(\omega_{\scriptscriptstyle
k})\bar{\mu}\left[\frac{\mathcal{A}_{1}(\omega_{\scriptscriptstyle
k})}{a_{0}},\frac{\mathcal{A}_{2}(\omega_{\scriptscriptstyle
k})}{a_{0}}\right]=\hat{\bf F}_{k}(\omega_{\scriptscriptstyle k}).$ (39)

Note, importantly, that while the different components are separated on the
right-hand side, they are coupled through $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$, which are algebraic functions of all the $\bar{\bf
r}_{\scriptscriptstyle k}$ and $\omega_{\scriptscriptstyle k}$. In Eq. (39) we
need the values of $\mathcal{A}_{1},~{}\mathcal{A}_{2}$ at the spectral
frequencies themselves. For the $n$th component we have, with our
normalizations $\theta_{1}(1)=\theta_{2}(1)=1$,

$\mathcal{A}_{1}(\omega_{\scriptscriptstyle n})=\omega_{\scriptscriptstyle
n}^{\scriptscriptstyle 2}(\bar{\bf r}_{\scriptscriptstyle n}\cdot\bar{\bf
r}_{\scriptscriptstyle n}^{*})^{1/2}+\sum_{k\not=n}\omega_{\scriptscriptstyle
k}^{\scriptscriptstyle 2}(\bar{\bf r}_{\scriptscriptstyle k}\cdot\bar{\bf
r}_{\scriptscriptstyle
k}^{*})^{1/2}\theta_{1}\left(\frac{\omega_{\scriptscriptstyle
k}}{\omega_{\scriptscriptstyle
n}}\right),~{}~{}~{}~{}~{}~{}~{}\mathcal{A}_{2}(\omega_{\scriptscriptstyle
n})=\omega_{\scriptscriptstyle n}^{\scriptscriptstyle 2}|\bar{\bf
r}_{\scriptscriptstyle n}\cdot\bar{\bf r}_{\scriptscriptstyle
n}|^{1/2}+\sum_{k\not=n}\omega_{\scriptscriptstyle k}^{\scriptscriptstyle
2}|\bar{\bf r}_{\scriptscriptstyle k}\cdot\bar{\bf r}_{\scriptscriptstyle
k}|^{1/2}\theta_{2}\left(\frac{\omega_{\scriptscriptstyle
k}}{\omega_{\scriptscriptstyle n}}\right).$ (40)

As I stated above, the functions $\theta_{1},~{}\theta_{2}$ have to vanish
fast enough for high values of their arguments (‘fast enough’ – in the sense
discussed in Sec. VI.4.1 below). This implies that the dynamics at a given
frequency $\omega_{\scriptscriptstyle n}$ is affected only little by much
higher frequencies, $\omega_{\scriptscriptstyle k}$.

But lower-frequency component do affect the dynamics at
$\omega_{\scriptscriptstyle n}$. For example, if $\omega_{\scriptscriptstyle
k}\ll\omega_{\scriptscriptstyle n}$, the contribution of its acceleration in
$\mathcal{A}_{1}$ is $\approx\omega_{\scriptscriptstyle k}^{\scriptscriptstyle
2}|\bar{\bf r}_{\scriptscriptstyle k}|\theta(0)\approx a_{\scriptscriptstyle
k}\theta(0)$ compared with the acceleration at $\omega_{\scriptscriptstyle n}$
itself $a_{\scriptscriptstyle n}=\omega_{\scriptscriptstyle
n}^{\scriptscriptstyle 2}|\bar{\bf r}_{\scriptscriptstyle n}|$, which is not
enhanced by the factor $\theta_{1}(0)>1$.

####  VI.3.2 Rotation curves – exact circular trajectories

For Exact circular motion in an axisymmetric field, $\mathcal{A}_{2}=0$. The
trajectory has only one frequency; so only $\theta(1)=1$ enters in expression
(40), which gives $\mathcal{A}_{1}=\omega^{\scriptscriptstyle 2}|\bar{\bf
r}|=V^{2}/R$, and we get the standard “algebraic relation” between the
Newtonian gravitational acceleration, $g_{\scriptscriptstyle N}$, and MOND
kinematic accelerations, $a=V^{2}/R$:

$a\mu(a/a_{0})=g_{\scriptscriptstyle N},$ (41)

where $\mu(x)=\bar{\mu}(x,0)$. With the adopted normalization $K(0)=1$ in Eq.
(30), we have in deep-MOND limit $\mu(x)=x$, and so $a_{0}$ has the standard
normalization corresponding to $V^{4}_{\scriptscriptstyle\infty}=MGa_{0}$, for
the asymptotic rotational speed.

This is the original, basic MOND relation put forth in Ref. [1], known as the
mass-discrepancy-acceleration relation, or the radial-acceleration relation.
Almost without exception, this is the relation that has been used to analyze
rotation curves in MOND, in the form (41), or its inverted form
$a=g_{\scriptscriptstyle N}\nu(g_{\scriptscriptstyle N}/a_{0})$.

This algebraic relation is not only a prediction of the simplistic subclass of
heuristic models of Sec. VI.2.1. It was shown in Ref. [76] that this is a most
general result for any MOND MI theory defined in the restricted sense of Sec.
IV; but it applies only to exactly circular, constant-speed trajectories. We
shall see below, in Sec. VI.3.3, that it takes corrections when there are
other motion components, such as motions perpendicular to the galactic disc.

In theories like AQUAL and QUMOND, observations of rotation curves essentially
determine the interpolating function, and with it the full theory, with all
its predictions. An important lesson, with more general ramifications, learnt
from the present models, is that this need not be the case in other MOND
theories. Here, observations of rotation curves for exact circular orbits, do
not inform us on the value of $\theta_{1}$ for arguments different from 1 –
which enter, for example the EFE, where the frequencies of the internal and
external motions are different (see Sec. VI.4.2 below). They also tell us
practically nothing about the dependence of the inertia functional $\bar{\mu}$
on $\mathcal{A}_{2}$, which may enter materially in all phenomena other than
rotation curves for exact circular motions.

####  VI.3.3 Rotation curves – effects of departure from exact circular
motions

The test particles used to measure rotation curves of disc galaxies – such as
stars or neutral gas – do not generally move on exact circular trajectories.
They have also components of radial and vertical motions, which modify the
prediction of the radial acceleration, and hence of the rotational speed.
Furthermore, different types of test bodies probe different parts of the
gravitational field; so the departure from exact circular motion depends, in
each galaxy, on the particle type. For example, the neutral gas is generally
more concentrated near the midplane of the disc, with smaller acceleration
components perpendicular to the disc than stars. And, among the latter, older
stellar populations form thicker discs than the younger ones and the cold gas;
so they have higher perpendicular velocity dispersions, and they sense, on
average, larger perpendicular accelerations. Also, the $z$ motion of the
bodies is not harmonic and each type has its own spectrum. A full treatment of
the problem, even within our simple models, would require much work beyond our
scope here.

Here I describe an idealized treatment of the problem to get an idea of the
possible effects and of how different attributes enter into it. Assume that
the test particles have just two components of motion: an exact circular one –
neglecting radial motions – with frequency $\omega=V/R$, and a component of
motion in the perpendicular, $z$, direction, which I approximate by a harmonic
linear motion with a single frequency, $\omega_{z}$, with an amplitude
$\bar{\bf r}_{\scriptscriptstyle k}=z_{\scriptscriptstyle 0}{\bf
e}_{\scriptscriptstyle z}$, as defined in Eq. (36). We then ask how the
presence of the $z$ motion affects the rotational speed of that particle.

We employ Eq. (40) with the circular component being $n$ and the $z$ component
being $k$ in the sums. The former, being circular, does not contribute to
$\mathcal{A}_{2}$, and the second, being linear, enters with the same
acceleration in both terms. Thus we have

$\mathcal{A}_{1}(\omega)=\frac{V^{2}}{R}+\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle
0}\theta_{1}\left(\frac{\omega_{z}}{\omega}\right),~{}~{}~{}~{}~{}~{}~{}\mathcal{A}_{2}(\omega)=\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle 0}\theta_{2}\left(\frac{\omega_{z}}{\omega}\right).$
(42)

The effect of this correction, is to replace Eq. (41) with

$\frac{V^{2}}{R}\bar{\mu}(\mathcal{A}_{1}/a_{0},\mathcal{A}_{2}/a_{0})=g_{\scriptscriptstyle
N}.$ (43)

In most of the galactic disc, the correction is expected to be small, because,
except near the center, $\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle 0}\ll V^{2}/R$, and because $\omega_{z}>>\omega$, and
$\theta_{i}(\zeta)$ has to decrease from its value of $1$ at $\zeta=1$. But as
we approach the center, the ratio of the mean $z$ acceleration to the radial
one increases, and the ratio of frequencies may decrease; so the predicted
correction to the rotation curve may become more important. Inasmuch as the
correction is small and we can expand relation (43) in it, we can write

$\frac{V^{2}}{R}\mu\left(\frac{V^{2}}{Ra_{0}}\right)\left\\{1+\frac{a_{\scriptscriptstyle
z}}{a_{\scriptscriptstyle
R}}\hat{\mu}\left[\theta_{1}\left(\frac{\omega_{z}}{\omega}\right)+\theta_{2}\left(\frac{\omega_{z}}{\omega}\right)\tilde{\mu}\right]\right\\}=g_{\scriptscriptstyle
N},$ (44)

where $a_{\scriptscriptstyle z}=\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle 0},~{}~{}~{}~{}a_{\scriptscriptstyle
R}=\omega^{\scriptscriptstyle 2}R,$ and

$\hat{\mu}\equiv\frac{d\ln\mu(x)}{d\ln
x}=\frac{\partial\ln\bar{\mu}(x,0)}{\partial\ln
x},~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\mu}\equiv\frac{\bar{\mu}_{x_{\scriptscriptstyle
2}}(a_{\scriptscriptstyle R}/a_{0},0)}{\bar{\mu}_{x_{\scriptscriptstyle
1}}(a_{\scriptscriptstyle R}/a_{0},0)},$ (45)

($\bar{\mu}_{x_{\scriptscriptstyle i}}$ is the partial derivative of
$\bar{\mu}$ with respect to $x_{\scriptscriptstyle i}$.)

We see that even in the simplified, heuristic models we employ here, the
departure of the MOND prediction from that for exact circular motion Eq. (41)
can depend in complicated ways on the local acceleration components and the
frequency ratio.

As already noted, we are not able to specify $\tilde{\mu}$. But we know from
the successes of MOND phenomenology that $\hat{\mu}$ is positive (and equals 1
in the deep-MOND regime). Thus, inasmuch as $\tilde{\mu}$ is not negative and
large enough to give a negative value to the square bracket in Eq. (44), the
curly bracket in this equation is larger than 1. And since $x\mu(x)$ is
increasing, the correction due to $z$ motion reduces the value of the
predicted rotational speed relative to the algebraic relation.

To get some idea of the possible corrections, we can hark back to the analysis
of Ref. [18]. As us here, they tried to see how a correction to the algebraic
relation that accounts in some way for the contribution from $z$ accelerations
affects the MOND prediction, and how the corrected prediction compares with
the prediction of AQUAL (the touchstone for MOND at the time). They did this
by applying the algebraic relation between the MOND and Newtonian
accelerations just outside the disc, instead of in the midplane, calculated
from

${\bf a}^{\scriptscriptstyle+}\mu(|{\bf a}^{\scriptscriptstyle+}|/a_{0})={\bf
g}^{\scriptscriptstyle+}_{\scriptscriptstyle N}.$ (46)

We need the radial component of this equations for the rotation curves.
(Unlike the vertical component, the radial accelerations do not vary across
the thickness of a thin disc.) This gives

$V^{2}/R=g_{\scriptscriptstyle N}/\mu(a^{\scriptscriptstyle+}/a_{0}),$ (47)

where $g_{\scriptscriptstyle N}$ is the radial Newtonian, gravitational
acceleration, and $a^{\scriptscriptstyle+}\equiv|{\bf
a}|^{\scriptscriptstyle+}$ represents the total MOND acceleration just outside
the disc, calculated from the Newtonian total acceleration just outside the
disc, $g_{\scriptscriptstyle N}^{\scriptscriptstyle+}=[g_{\scriptscriptstyle
N}^{\scriptscriptstyle 2}+(2\pi G\Sigma)^{2}]^{1/2}$, from
$\mu(a^{\scriptscriptstyle+}/a_{0})a^{\scriptscriptstyle+}=g_{\scriptscriptstyle
N}^{\scriptscriptstyle+}$. ($\Sigma$ is the disc surface density at position
$R$.) So the correction to the algebraic relation is in modifying the argument
of the interpolating function to the total acceleration outside the disc
instead of the one in the midplane.

This procedure does not account for the effect of the $z$-accelerations on the
predicted rotation curves in exactly the way dictated by our MI models, but it
does show similarly how the correction is small at large radii, becomes
important at small radii, and brings the predicted MI velocities closer to
those predicted by AQUAL. All these are shown in Fig. 2 for three disc models.

Figure 2: Comparison of the rotation curves predicted by AQUAL (solid line)
with the prediction of MI for exactly circular trajectories in the midplane of
the disc (the algebraic relations, triangles), and with and algebraic relation
given in Eq. (47) (squares) for three pure-disc galaxy models, as indicated.
From Ref. [18].

Another implication of the necessity of the above corrections it to reduce the
predicted value of the $Q$ parameter defined in Ref. [19] (and studied in Ref.
[20]) for MI relative to the value for MG, perhaps reducing the difference
between them and diminishing the value of this parameter as a discriminant
between theories.

Note also, that the corrections of the type we discuss above are trajectory
dependent, and so may lead to somewhat different predicted rotational speeds
for different test particles in a galaxy. For example older discs in galaxies
are, by and large, thicker and have higher velocity dispersions than the
younger stellar discs, and even more so than the gas disc. They are thus
characterized by larger $z$ accelerations, and the above considerations may
lead to lower rotational speeds for them than for the thinner, younger
components. Indeed, Ref. [107] find that, near the Sun, the Milky Way’s thick
disc rotates somewhat slower (by about 5 percent, but with high significance)
than the thin disc. But this could be due to reasons other than those we
discuss here (e.g., a higher asymmetric-drift contribution).

The correction discussed here it not a function of only the (Newtonian)
acceleration, it depends also on other parameters that vary from point to
point, and may even depend on the test particles studied. It is thus predicted
to produce some scatter in the observed mass-discrepancy-acceleration
($a-g_{\scriptscriptstyle N}$) relation, on top of the systematic reduction in
predicted $a$ values. This may account for the small “intrinsic” scatter
deduced in Ref. [58].

We note in this connection that the analysis of Ref. [22], which compared the
predictions of MI with those of AQUAL, and claims to have found preference to
AQUAL over MI, used for the MI MOND prediction the algebraic relation (41),
which applies only to exact circular orbits, without the type of corrections
that must be introduced in realistic analysis, and which do bring the
predictions closer to those of AQUAL.

####  VI.3.4 Vertical dynamics in a (galactic) disc

To get an idea of how the radial acceleration in a disc galaxy affects the
perpendicular dynamics, I employ the same idealized treatment as in the
previous section. Use Eq. (40) with the $z$ component labeled $n$, and the
circular component labeled $k$ in the sums. Since the latter is idealized as
exactly circular, it does not contribute to $\mathcal{A}_{2}$. We then have

$\mathcal{A}_{1}(\omega_{z})=\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle 0}+\omega^{\scriptscriptstyle
2}R\theta_{1}\left(\frac{\omega}{\omega_{z}}\right),~{}~{}~{}~{}~{}~{}~{}\mathcal{A}_{2}(\omega_{z})=\omega_{z}^{\scriptscriptstyle
2}z_{\scriptscriptstyle 0}.$ (48)

By Eq. (39), the ratio of the Newtonian to the MONDian Fourier components of
the accelerations is then

$\frac{\hat{g}_{\scriptscriptstyle N}^{\scriptscriptstyle
z}(\omega_{z})}{\hat{a}_{\scriptscriptstyle
z}(\omega_{z})}\approx\bar{\mu}\left[\frac{\mathcal{A}_{1}(\omega_{z})}{a_{0}},\frac{\mathcal{A}_{2}(\omega_{z})}{a_{0}}\right].$
(49)

For example, in cases where we can neglect the $z$ acceleration compared with
the circular one, $\omega_{z}^{\scriptscriptstyle 2}z_{\scriptscriptstyle
0}\ll\theta_{1}(\omega/\omega_{z})\omega^{\scriptscriptstyle 2}R$, we have

$\frac{\hat{g}_{\scriptscriptstyle N}^{\scriptscriptstyle
z}(\omega_{z})}{\hat{a}_{\scriptscriptstyle
z}(\omega_{z})}\approx\mu\left[\theta_{1}\left(\frac{\omega}{\omega_{z}}\right)\frac{\omega^{\scriptscriptstyle
2}R}{a_{0}}\right]$ (50)

[$\mu(x)\equiv\bar{\mu}(x,0)$]. Note, importantly, that this is not the same
factor that gives the rotation-curve anomaly at $R$, which is
$\mu(\omega^{\scriptscriptstyle 2}R/a_{0})$, without the factor $\theta_{1}$
that appears in the argument of expression (50). If, e.g.
$\omega\ll\omega_{z}$, as is many times the case – for example in the solar
neighborhood for most disc components – then this factor can be approximated
by $\theta_{1}(0)$. Since $\theta_{1}$ is assumed to be decreasing and
$\theta_{1}(1)=1$, $\theta_{1}(0)$ could be of the order of a few.

The presence of this factor in the argument could greatly change the MOND
predictions of $z$ dynamics compared with those of AQUAL/QUMOND.

###  VI.4 A system falling in an external field

Consider the motion of a small composite body, made of gravitating
constituents, in the field of a larger body. For example, a star, a gas cloud,
or a star cluster falling in the field of a galaxy, or a galaxy falling in the
field of a galaxy cluster. Two crucial questions that arise are: (a) How do
the internal accelerations of the constituents affect the center-of-mass
motion of the whole system? (b) How does the external field affect the
internal dynamics? Both questions have clear-cut and satisfactory answers
within existing MG theories such as AQUAL, QUMOND, and TRIMOND. For MI, I
already discussed in Sec. V.1 generalities of the answer to the second
question, which go under the term EFE, and I’ll discuss the specific
predictions of our model examples below in Sec. VI.4.2. But first I discuss
the first question within these models.

####  VI.4.1 Center-of-mass motion

Again, to be able to derive concrete predictions, and treat the body in
question as a well defined entity, with a well defined overall motion, I
assume that the body is of characteristic size $r$ much smaller than the scale
$R$ that characterizes its “external” trajectory. So the motions of each
constituent can be clearly separated to “external” Fourier components, of
frequencies collectively designated $\Omega$, and amplitudes, collectively
designated $R$, and “intrinsic” Fourier components with analogous
characteristics $\omega$ and $r$. The characteristic external accelerations
are then $\sim\Omega^{2}R$, and the inner ones are
$\sim\omega^{\scriptscriptstyle 2}r$.

Observations tell us that the overall motion of such bodies obeys MOND in a
way that is oblivious to the internal dynamics; e.g., stars of all types, gas
clouds, etc. give roughly the same rotation curve, even though they have very
different internal structures. Such a behavior is also in line with the weak
equivalence principle (universality of free fall) as applied to such composite
bodies – which we wish to retain. Indeed, observations tell us that it holds,
at least approximately, also in the MOND regime.

To satisfy this requirement, we need that all constituents share, to a good
enough approximation, the same “external” components of motion, and that these
external components obey the MOND equation appropriate for their own
acceleration. A sufficient condition for this is that we can neglect in Eq.
(40) all the contributions of the internal motions to the “inertia functional”
$\bar{\mu}$ of the “external” motions.

Consider then expressions (40) for the “external” components. For these, all
the contributions of the internal motions appear as terms in the sum over $k$;
so they all appear with a factor $\theta_{i}$. Thus their contribution to
$\mathcal{A}_{i}$ relative to the main term, which is of order $\Omega^{2}R$,
is

$\sim\frac{\omega^{\scriptscriptstyle
2}r}{\Omega^{2}R}\theta_{i}\left(\frac{\omega}{\Omega}\right).$ (51)

If $\omega^{\scriptscriptstyle 2}r\ll\Omega^{2}R$ the internal accelerations
are immaterial for the external ones. If this is not so, then
$\omega\gg\Omega$, because $R\gg r$. Thus, even if $\omega^{\scriptscriptstyle
2}r\gg\Omega^{2}R$, the relative contributions are very small if
$\theta_{i}(\zeta)$ decrease fast enough with $\zeta$. For example, if
$\theta_{i}(\zeta)\sim\zeta^{\scriptscriptstyle-2}$ for $\zeta\gg 1$,
expression (51) is $\sim r/R\ll 1$; in fact, it then becomes of the order of
the tidal effects, which we neglect anyhow.

Of course, $\theta_{i}$ could decrease even faster (such as exponentially)
which would suppress even more the effect of the internal on the external
motions.

####  VI.4.2 The external-field effect

Consider again the simplified picture where the subsystem falls in the
external field on a trajectory whose frequencies are $\Omega_{m}$ and the
corresponding (complex) amplitude vectors are $\bar{\bf R}_{m}$. The internal
motions have frequencies $\omega_{\scriptscriptstyle k}$ with amplitudes
$\bar{\bf r}_{\scriptscriptstyle k}$. In most past and foreseeable
applications, we have $\Omega_{m}/\omega_{\scriptscriptstyle k}\ll 1$ for all
$m$ and $k$, which I assume, and so put these ratios to be zero, without
losing much accuracy. I also assume that $|\bar{\bf R}_{m}|\gg|\bar{\bf
r}_{\scriptscriptstyle k}|$, so the finite size of the subsystem can be
neglected, and all constituents can be assumed to have the same “external”
motion.

If either of these strong inequalities do not hold, we cannot justify the
decoupling of motions that underlie the equations of motion (39), and a rather
more complicated treatment would be necessary.

We are now interested in the internal dynamics, and the way they are affected
by the external motions. One first solves for the external motion, which is
oblivious to the internal dynamics under our assumptions, as shown above in
Sec. VI.4.1. Then, the equations of motions are applied to the
$\omega_{\scriptscriptstyle n}$ components, with the relevant arguments of
$\bar{\mu}$ calculated from Eq. (40)

$\mathcal{A}_{1}(\omega_{\scriptscriptstyle n})=\omega_{\scriptscriptstyle
n}^{\scriptscriptstyle 2}(\bar{\bf r}_{\scriptscriptstyle n}\cdot\bar{\bf
r}_{\scriptscriptstyle
n}^{*})^{1/2}+\sum_{k\not=n}\theta_{1}\left(\frac{\omega_{\scriptscriptstyle
k}}{\omega_{\scriptscriptstyle n}}\right)\omega_{\scriptscriptstyle
k}^{\scriptscriptstyle 2}(\bar{\bf r}_{\scriptscriptstyle k}\cdot\bar{\bf
r}_{\scriptscriptstyle
k}^{*})^{1/2}+\theta_{1}(0)\sum_{m}\Omega_{m}^{2}(\bar{\bf R}_{m}\cdot\bar{\bf
R}_{m}^{*})^{1/2},$ $\mathcal{A}_{2}(\omega_{\scriptscriptstyle
n})=\omega_{\scriptscriptstyle n}^{\scriptscriptstyle 2}|\bar{\bf
r}_{\scriptscriptstyle n}\cdot\bar{\bf r}_{\scriptscriptstyle
n}|^{1/2}+\sum_{k\not=n}\theta_{2}\left(\frac{\omega_{\scriptscriptstyle
k}}{\omega_{\scriptscriptstyle n}}\right)\omega_{\scriptscriptstyle
k}^{\scriptscriptstyle 2}|\bar{\bf r}_{\scriptscriptstyle k}\cdot\bar{\bf
r}_{\scriptscriptstyle k}|^{1/2}+\theta_{2}(0)\sum_{m}\Omega_{m}^{2}|\bar{\bf
R}_{m}\cdot\bar{\bf R}_{m}|^{1/2}.$ (52)

Thus, the external motion enters as a constant (independent of
$\omega_{\scriptscriptstyle n}$), additive contribution to the
$\mathcal{A}_{i}$. Note that I have not assumed that the intrinsic frequencies
decouple among themselves – only from the external motions – in the sense of
Eq. (37). So the values of $\mathcal{A}_{i}$ from Eq. (52) are to be used in
the general equation of motion, Eq. (29).

The result is rather simple when the external accelerations are much dominant
over the internal ones, $\Omega_{m}^{2}|\bar{\bf
R}_{m}|\gg\omega_{\scriptscriptstyle k}^{2}|\bar{r}_{\scriptscriptstyle k}|$
(in addition to the other strong inequalities).131313$\theta_{i}$ are
decreasing so are maximal at $0$. Then

$\mathcal{A}_{1}(\omega)\approx\mathcal{A}_{1}^{\scriptscriptstyle
ex}\equiv\theta_{1}(0)\sum_{m}\Omega_{m}^{2}(\bar{\bf R}_{m}\cdot\bar{\bf
R}_{m}^{*})^{1/2},~{}~{}~{}~{}~{}~{}~{}\mathcal{A}_{2}(\omega)\approx\mathcal{A}_{2}^{\scriptscriptstyle
ex}\equiv\theta_{2}(0)\sum_{m}\Omega_{m}^{2}|\bar{\bf R}_{m}\cdot\bar{\bf
R}_{m}|^{1/2}$ (53)

are the same for all intrinsic frequencies, and the equations of motion (29)
for the internal dynamics are

$\hat{\bf a}(\omega)\bar{\mu}_{\scriptscriptstyle ex}=\hat{\bf
a}_{\scriptscriptstyle N}(\omega),$ (54)

where

$\bar{\mu}_{\scriptscriptstyle
ex}\equiv\bar{\mu}\left[\frac{\mathcal{A}_{1}^{ex}}{a_{0}},\frac{\mathcal{A}_{2}^{ex}}{a_{0}}\right]$
(55)

is a constant.

We see that in this case the system obeys, intrinsically, the equations of
motion of Newtonian dynamics, $m\bar{\mu}_{\scriptscriptstyle ex}\hat{\bf
a}(\omega)=\hat{\bf F}(\omega)$ ($\hat{\bf F}$ is the Newtonian force fourier
component), but with all the inertial masses, $m$, of its constituents, but
not the gravitational masses, renormalized, by a constant factor, to smaller
values $m\bar{\mu}_{\scriptscriptstyle ex}$. Specifically, in a gravitating
system, this has the same effect as replacing $G$ with the larger, effective
$G_{e}=G/\bar{\mu}_{\scriptscriptstyle ex}$.

In more detail, it can be seen that starting from a Newtonian solution for the
trajectories of the constituents ${\bf r}^{\scriptscriptstyle
N}_{\scriptscriptstyle p}(t)$ gotten by dictating the initial conditions ${\bf
r}^{\scriptscriptstyle N}_{\scriptscriptstyle p}(0)$, ${\bf
v}^{\scriptscriptstyle N}_{\scriptscriptstyle p}(0)$, the MOND solution, for
the initial conditions ${\bf r}_{\scriptscriptstyle p}(0)={\bf
r}^{\scriptscriptstyle N}_{\scriptscriptstyle p}(0)$, ${\bf
v}_{\scriptscriptstyle p}(0)=\lambda{\bf v}^{\scriptscriptstyle
N}_{\scriptscriptstyle p}(0)$, is ${\bf r}_{\scriptscriptstyle p}(t)={\bf
r}^{\scriptscriptstyle N}_{\scriptscriptstyle p}(\lambda t)$, where
$\lambda=1/\sqrt{\bar{\mu}_{\scriptscriptstyle ex}}$. Thus the orbits are the
same as the Newtonian ones, but the particles run along them at speeds
$1/\sqrt{\bar{\mu}_{\scriptscriptstyle ex}}$ times higher, ${\bf
v}_{\scriptscriptstyle p}(t)=\lambda{\bf v}^{\scriptscriptstyle
N}_{\scriptscriptstyle p}(\lambda t)$, and accelerations
$1/\bar{\mu}_{\scriptscriptstyle ex}$ times higher, ${\bf
a}_{\scriptscriptstyle p}(t)=\lambda^{\scriptscriptstyle 2}{\bf
a}^{\scriptscriptstyle N}_{\scriptscriptstyle p}(\lambda t)$.

Compare this result with the analog one (for a purely gravitational system) in
AQUAL and QUMOND: A subsystem of characteristic size, frequency, and
accelerations, $r$, $\omega$, and $g_{\scriptscriptstyle
in}\sim\omega^{\scriptscriptstyle 2}r$, respectively, in an external
acceleration field, ${\bf g}_{\scriptscriptstyle ex}$, that varies on typical
length scales $R$, and on time scales $1/\Omega$, and whose momentary value
much dominates over the internal ones, $|{\bf g}_{\scriptscriptstyle ex}|\gg
g_{\scriptscriptstyle in}$. As above, we also assume here that $r\ll R$, so
the external field is approximately constant across the subsystem, and that
$\Omega\ll\omega$, so the external field can be assumed time independent (over
the intrinsic dynamical times), and if it does change, this causes adiabatic
changes in the internal dynamics.141414Reference [46] showed that the
conditions of nonadiabaticity and substantial tidal effects occur roughly
simultaneously for a dwarf system on an elongated trajectory in the field of a
mother system. Then, AQUAL and QUMOND both predict that the internal dynamics
is “Newtonian like”, but with an effective $G_{e}=G/\mu(g_{\scriptscriptstyle
ex}/a_{0})$, where $\mu$ is the interpolating function of these
theories.151515In QUMOND $\mu(x)$ is uniquely defined by the ‘inverse’
interpolating function, $\nu(y)$. In addition, there appear some distortion
effects of order unity, that depend also on the direction of ${\bf
g}_{\scriptscriptstyle ex}$ (and differ somewhat between QUMOND and AQUAL)
[23, 35, 93, 94].

These predictions look similar in some sense to the ones above – encapsulated
in Eqs. (54-55) – that follow from our model MI theories: The intrinsic
dynamics is “Newtonian like”, with a renormalized value of $G_{e}$ that is
Larger than $G$ and is given by $G/\mu^{*}$ where $\mu^{*}$ is some constant
that is the value of some interpolation function with arguments that depend
only on characteristics of the external acceleration.

But there are also important differences between the predictions as follows:
(a) In the MG theories, the effect depends only on the momentary value of
${\bf g}_{\scriptscriptstyle ex}$, and so it changes along the external
trajectory. In the MI models, the effect depends on global characteristics of
the external orbit, and is the same everywhere along it. (b) The MG theories
predict also some (order unity) dependence on the direction of
$g_{\scriptscriptstyle ex}$; not so in the MI predictions. (c) The
renormalizing factor $\mu^{*}=\mu(g_{\scriptscriptstyle ex}/a_{0})$ in the MG
predictions depends only on one number, $g_{\scriptscriptstyle ex}$; this
$\mu^{*}$ is fully determined by what rotation-curve analysis informs us,
since it determines the interpolating function. In the MI prediction there
appears $\mu^{*}$ that a function of two variables
$\mu^{*}=\bar{\mu}(x_{1},x_{2})$. But rotation curves tell us nothing about
the dependence on $x_{2}$; and the variable $x_{1}$ involves the factor
$\theta_{1}(0)$, which does not appear in predictions of rotation curves.

####  VI.4.3 Galaxy effect on the inner Solar System

I discussed in Sec. V.2 a secondary prediction of AQUAL/QUMOND, of a possible
small MOND anomaly due to the galactic acceleration, which might show up in
the motion of bodies – such as planets – in the inner Solar System, even
though their accelerations due to the Sun are many orders of magnitude larger
than $a_{0}$ [67, 68, 53]. I explained why this prediction is idiosyncratic to
AQUAL/QUMOND (and possibly to similar MG theories, such as TRIMOND). I also
explained why MI theories might predict very differently, which I now
demonstrate in the context of our heuristic MI models.

Consider a test particle (e.g., a planet) that moves around the Sun, and stays
always at distances from the Sun where its acceleration is
$g_{\scriptscriptstyle\odot}\gg a_{0}$ (unlike, e.g., very-long-period comets
whose orbit may reach beyond the MOND radius of the Sun; see footnote 9
above). The trajectory of the Sun itself in the Galaxy, ${\bf R}(t)$, is
assumed oblivious to the effects of the planet. It is thus solved for first
and then it is a given of our problem. As in the MG case, we take the limit
$|\bar{\bf r}|/|\bar{\bf R}|\rightarrow 0$; so the Sun and the planet are
subject to the same galactic field. The real problem, including the effects of
other planets, etc. is rather more complicated. But in the MG theories the
anomaly appears even in this simplified case, and we want to see what the case
is in the MI models.

We can then describe the trajectory of the planet as ${\bf R}(t)+{\bf
r}_{\scriptscriptstyle N}(t)+{\bf r}_{\scriptscriptstyle a}(t)$, where ${\bf
r}_{\scriptscriptstyle N}(t)$ is the Newtonian trajectory – calculated for
some initial conditions – and ${\bf r}_{\scriptscriptstyle a}(t)$ is a
possible small correction (the anomaly) that might appear in a MOND
calculation that includes the galactic field, and which we are after. In
principle, there may be a small MOND anomaly even without the presence of the
Galactic field, and even at the high accelerations,
$g_{\scriptscriptstyle\odot}$, in the inner solar system, because the
Newtonian limit is not yet fully reached even at such high acceleration. In
theories, such as AQUAL/QUMOND this can be expressed as the possibility that
$1-\mu(g_{\scriptscriptstyle\odot}/a_{0})\not=0$. This has been considered and
constrained , e.g., in Refs. [1, 67, 108]. This is not the anomaly we are
after, and it can be easily put aside and ignored if MOND approaches standard
dynamics fast enough at high accelerations. For example, with the $\nu$
interpolating function, $\nu(y)=1/(1-e^{-y^{1/2}})$ that is now often used for
rotation-curve analysis, we have for the Earth $\nu-1\approx e^{-10^{4}}$,
totally negligible.

As explained in Sec. V.2, the anomaly in AQUAL/QUMOND does not disappear if we
put $1-\mu(g_{\scriptscriptstyle\odot}/a_{0})=0$. The question we ask is
whether it does disappear in our MI models. We can use Eq. (52) here as well
to see that the exact Newtonian solution without an anomaly applies here since
the values of $\mathcal{A}_{i}$ calculated for it are many orders of magnitude
larger than $a_{0}$ in the inner solar system. (For the Newtonian solution,
the first terms in Eq. (52) are very large.) The galactic field enters only as
a small addition to the Newtonian contribution in the arguments of
$\bar{\mu}$. So, unlike the AQUAL/QUMOND anomaly, here the anomaly does
disappear if the theory tends to Newtonian dynamics fast enough.

##  VII Discussion

I have strived to bring home the following main points:

(a) It is quite worthwhile, and highly desirable, to explore MOND theories
beyond the restricted class of modified-gravity theories, which have been
studied almost exclusively, so far.

(b) Roughly speaking, predictions made by MOND theories can be divided into
primary ones, those that follow, within a margin of minor latitude, from only
the basic tenets of MOND, and secondary ones, which do not. The former are
then shared by all MOND theories (defined as those that embody the basic
tenets), while the latter can differ greatly among MOND theories, and can thus
are much better suited for distinguishing observationally between theories. I
thus also warn against viewing, indiscriminately, predictions of the two
workhorse MOND MG theories, AQUAL and QUMOND, as absolute predictions of MOND.

(c) In particular, it is advisable to explore “modified-inertia”
interpretation of MOND, despite the facts that they seem to be more drastic
modifications of standard dynamics, and perhaps more difficult to devise and
explore. Effective appearances of modified inertia, or emergent inertia, are
rife in standard physics. And, this avenue seems to me very promising for MOND
as well, which, as far as it now appears, is also an effective theory – an
approximation of a more fundamental theory – a FUNDAMOND. This assessment of
MOND as we now know it being an effective theory is based on the fact that all
present MOND theories still introduce by hand an interpolation scheme between
the standard dynamics (Newtonian and general relativistic) and deep-MOND
dynamics. In a FUNDAMOND, we expect that various results would have a
standard-dynamics limit at high accelerations, and a deep-MOND limit at low
ones, but not that this should be described by some universal interpolating
function that is introduced by hand at the fundamental level of the theory.
The proximity of the MOND acceleration constant, $a_{0}$, to accelerations of
cosmological significance may also hint at a FUNDAMOND in which such a
“coincidence” would appear naturally [77, 109].

(d) Modified-inertia, MOND theories are not only worthwhile exploring because
they may turn out to be the correct way to FUNDAMOND. In being distant in
concept from MG theories, they also demonstrate clearly how MOND theories can
differ greatly in predicting the dynamics in secondary phenomena. This can be
understood on general grounds, and demonstrated with specific examples.

To demonstrate and exemplify such possible variety in secondary predictions, I
describe briefly the class of MI models propounded in Ref. [81], and derive
some of their predictions of primary and secondary phenomena. I stress that I
do not propound these models as full-fledged, MI, MOND theories. There are
still unresolved matter-of principle aspects of these models, and it is also
not clear how to apply them in all circumstances, as discussed in more detail
in Ref. [81].

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