A set of particles of mass moving in a gravitational field generated by the matter density distribution and described by the Newtonian potential has the following action22:23.
The first possibility, modified inertia, has been investigated by Milgrom [300, 321], who constructed modified kinetic actions24 (the first term in Eq. 13) that are functionals depending on the trajectory of the particle as well as on the acceleration constant . By construction, the gravitational potential is then still determined from the Newtonian Poisson equation, but the particle equation of motion becomes, instead of Newton’s second law,
Milgrom  investigated theories of this vein and rigorously showed that they always had to be time-nonlocal (see also Section 7.10) to be Galilean invariant25. Interestingly, he also showed that quantities such as energy and momentum had to be redefined but were then enjoying conservation laws: this even leads to a generalized virial relation for bound trajectories, and in turn to an important and robust prediction for circular orbits in an axisymmetric potential, shared by all such theories. Eq. 14 becomes for such trajectories:exactly the heuristic Milgrom’s law. Interestingly, it is this law, which is used to fit galaxy rotation curves, while in the modified gravity framework of MOND (see hereafter), one should actually calculate the exact predictions of the modified Poisson formulations, which can differ a little bit from Milgrom’s law. However, for orbits other than circular, it becomes very difficult to make predictions in modified inertia, as the time non-locality can make the anomalous acceleration at any location depend on properties of the whole orbit. For instance, if the accelerations are small on some segments of a trajectory, MOND effects can also be felt on segments where the accelerations are high, and conversely . This can give rise to different effects on bound and unbound orbits, as well as on circular and highly elliptic orbits, meaning that “predictions” of modified inertia in pressure-supported systems could differ significantly from those derived from Milgrom’s law per se. Let us finally note that testing modified inertia on Earth would require one to properly define an inertial reference frame, contrary to what has been done in [5, 179] where the laboratory itself was not an inertial frame. Proper set-ups for testing modified inertia on Earth have been described, e.g., in [201, 202]: under the circumstances described in these papers, modified inertia would inevitably predict a departure from Newtonian dynamics, even if the exact departure cannot be predicted at present, except for circular motion.
The idea of modified gravity is, on the one hand, to preserve the particle equation of motion by preserving the kinetic action, but, on the other hand, to change the gravitational action, and thus modify the Poisson equation. In that case, all the usual conservation laws will be preserved by construction.
A very general way to do so is to write : Aquadratic Lagrangian theory (AQUAL). Varying the action with respect to then leads to a non-linear generalization of the Newtonian Poisson equation26:  is an example of a flattened axisymmetric configuration for which Milgrom’s law is precisely valid, as its Newtonian potential is equivalent on both sides of the disk to that of a point mass above or below the disk respectively.
In vacuum and at very large distances from a body of mass , the isopotentials always tend to become spherical and the curl field tends to zero, while the gravitational acceleration falls well below (a regime known as the “deep-MOND” regime), so that:
An important point, demonstrated by Bekenstein & Milgrom , is that a system with a low center-of-mass acceleration, with respect to a larger (more massive) system, sees the motion of its constituents combine to give a MOND motion for the center-of-mass even if it is made up of constituents whose internal accelerations are above (for instance a compact globular cluster moving in an outer galaxy). The center-of-mass acceleration is independent of the internal structure of the system (if the mass of the system is small), namely the Weak Equivalence Principle is satisfied.
In a modified gravity theory, any time-independent system must still satisfy the virial theorem:[301, 302] showed that, in Bekenstein–Milgrom MOND, the virial is given by:  for the specific conditions for this to be valid). In this case, we can get an analytic expression for the two-body force under the approximation that the two bodies are very far apart compared to their internal sizes [301, 509, 511]. Since the kinetic energy can be separated into the orbital energy and the internal energy of the bodies , we get from the scalar virial theorem of a stationary system:
The latter equation is not valid for N-body configurations, for which the Bekenstein–Milgrom (BM) modified Poisson equation (Eq. 17) must be solved numerically (apart from highly-symmetric N-body configurations). This equation is a non-linear elliptic partial differential equation. It can be solved numerically using various methods [50, 77, 96, 147, 250, 457]. One of them [77, 457] is to use a multigrid algorithm to solve the discrete form of Eq. 17 (see also Figure 17):
The gradient component , in , is approximated in the case of by (see Figure 17).
In  the Gauss–Seidel relaxation with red and black ordering is used to solve this discretized equation, with the boundary condition for the Dirichlet problem given by Eq. 20 at large radii. It is obvious that subsequently devising an evolving N-body code for this theory can only be done using particle-mesh techniques rather than the gridless multipole expansion treecode schemes widely used in standard gravity.
Finally, let us note that it could be imagined that MOND, given some of its observational problems (developed in Section 6.6), is incomplete and needs a new scale in addition to . There are several ways to implement such an idea, but for instance, Bekenstein  proposed in this vein a generalization of the AQUAL formalism by adding a velocity scale , in order to allow for effective variations of the acceleration constant as a function of the deepness of the potential, namely:) could affect the dynamics of galaxy clusters (by boosting the modification with an effectively higher value of ) compared to the previous MOND equation, while keeping the less massive systems such as galaxies typically unaffected compared to usual MOND, while other (lower) values of could allow (modulo a renormalization of ) for a stronger modification in galaxy clusters as well as milder modification in subgalactic systems such as globular clusters, which, as we shall soon see could be interesting from a phenomenological point of view (see Section 6.6). However, the possibility of too strong a modification should be carefully investigated, as well as, in a relativistic (see Section 7) version of the theory, the consequences on the dynamics of a scalar-field with a similar action.
Another way  of modifying gravity in order to reproduce Milgrom’s law is to still keep the “matter action” unchanged , thus ensuring that varying the action of a test particle with respect to the particle degrees of freedom leads to , but to invoke an auxiliary acceleration field in the gravitational action instead of invoking an aquadratic Lagrangian in . The addition of such an auxiliary field can of course be done without modifying Newtonian gravity, by writing the Newtonian gravitational action in the following way27:quasi-linear formulation of MOND (QUMOND). In order to recover the -function behavior of Milgrom’s law (Eq. 10), i.e., for and for , one needs to choose: different from the one of AQUAL. This means that, outside of high symmetry, AQUAL and QUMOND cannot be precisely equivalent. An illustration of this is given in : for a system with all its mass in an elliptical shell (in the sense of a squashed homogeneous spherical shell), the effective density of matter that would source the MOND force field in Newtonian gravity is uniformly zero in the void inside the shell for QUMOND, but nonzero for AQUAL.
The concept of the effective density of matter that would source the MOND force field in Newtonian gravity is extremely useful for an intuitive comprehension of the MOND effect, and/or for interpreting MOND in the dark matter language: indeed, subtracting from this effective density the baryonic density yields what is called the “phantom dark matter” distribution. In AQUAL, it requires deriving the Newtonian Poisson equation after having solved for the MOND one. On the other hand, in QUMOND, knowing the Newtonian potential yields direct access to the phantom dark matter distribution even before knowing the MOND potential. After choosing a -function, one defines
Numerically, for a given Newtonian potential discretized on a grid of step , the discretized phantom dark matter density is given on grid points by (see Figure 17 and cf. Eq. 25, see also ):any N-body technique (e.g., treecodes or fast multipole methods) can be adapted to QUMOND (a grid being necessary as an intermediate step). Once the Newtonian potential (or force) is locally known, the phantom dark matter density can be computed and then represented by weighted particles, whose gravitational attraction can then be computed in any traditional manner. An example is given in Figure 18, where one considers a rather typical baryonic galaxy model with a small bulge and a large disk. Applying Eq. 35 (with the -function of Eq. 43) then yields the phantom density . Interestingly, this phantom density is composed of a round “dark halo” and a flattish “dark disk” (see  for an extensive discussion of how such a dark disk component comes about; see also  and Section 6.5.2 for observational considerations). Let us note that this phantom dark matter density can be slightly separated from the baryonic density distribution in non-spherical situations , and that it can be negative [297, 490], contrary to normal dark matter. Finding the signature of such a local negative dark matter density could be a way of exhibiting a clear signature of MOND.
Finally, let us note that, as shown in [319, 509], (i) a system made of high-acceleration constituents, but with a low-acceleration center-of-mass, moves according to a low-acceleration MOND law, while (ii) the virial of a system is given byin general different in AQUAL and QUMOND, the two-body force in the deep-MOND limit is the same .
Living Rev. Relativity 15, (2012), 10
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