Those cleaving to Newtonian dynamics may take the success of MOND to reflect some very strict regularity-encompassing the whole gamut of galactic systems-relating the distribution of visible matter to that of dark matter via a simple formula. The few of us who have contributed to MOND theory and testing over the years view this success as strong indication of departure from standard dynamics in the parameter region relevant to galactic systems. Taking MOND in such a vein, one seeks to construct theories, with increasing depth and compass, that incorporate the basic tenets of MOND. Figure 3 presents the schematics of these efforts, with full-line blocks marking areas in advanced stages of development.
Figure 3.Schematics of the MOND program. |
At the nonrelativistic level, at least, MOND may be viewed as either a modification of gravity, or a modification of inertia [1]. In the former, the gravitational field produced by a given mass distribution is dictated by a new equation; in the latter the equation of motion is MONDified, while the force fields remain intact. An example of the former is the MONDification of the Poisson equation discussed in ref. [2] where the gravitational potential, , is determined by the mass distribution, , via
(1) |
Mondified inertia is discussed in refs. [3] [25]. In such theories, when derived from an action, one replaces the standard kinetic action for a particle (v^{2} / 2 dt) by a kinetic action that is a more complicated functional of the particle trajectory
(2) |
where A_{m} depends only on the body, and can be identified with its mass, and S depends only on the trajectory and on a_{0} as a parameter. Weak equivalence is thus insured. In the formal limit a_{0} 0 the action goes to the standard kinetic action. In the opposite limit, a_{0} , S a_{0}^{-1}, and inertia disappears in the very limit.
With respect to Newtonian dynamics, special relativistic dynamics is an example of modified inertia: The equation of motion of a relativistic particle moving in a force field F(r) is md (v / dt = m[a + ^{2} c^{-2}(v^{ . }a)v] = F(r), derived from the kinetic action mc^{2} d = mc^{2} ^{-1} dt. Here too there appears a parameter, c, which, like a_{0} in MOND (and in QM) both delimits the standard (classical) region, and enters the dynamics in the non-classical regime. Unlike the special-relativistic action, which is still local, the MOND action is perforce non-local if it is to be Galilei invariant [3].
Mondified gravity and mondified inertia do not differ on what we call the basic predictions of MOND: The asymptotic flatness of rotation curves (and their general shape), the M V^{4} relation, the added stability of systems in the deep MOND regime [26], etc. There are, however, important differences; some examples are: 1. In mondified gravity only systems governed by pure gravity (such as galactic systems) are affected, while in mondified inertia the modification applies for whatever combination of forces is at play. 2. in the former, the acceleration of a test particle depends only on its position in the field, while in the latter it depends strongly on other details of the trajectory (inertia is identified with acceleration only in standard Newtonian dynamics). As an example, we can see in the special-relativity case, mentioned above, that the v^{ . }a term vanishes for a circular orbit, but dominates, at high , for a linear trajectory. 3. In mondified inertia the expressions for the conserved quantities and adiabatic invariants in terms of the motion are modified [3], in contradistinction to mondified gravity.
An acceptable relativistic extension for MOND is not yet at hand. Discussions of various candidates can be found in refs. [2] [27] [28] [29], but each of these has its problems. These problems seem to be specific to the particular models (e.g. that in [2] has superluminal modes, scalar-tensor theories as discussed in [28] do not give as large a light bending as is observed, and that in [29] is based on a non-dynamical pregeometry).
Reflection over this question has convinced me that a relativistic extension will not just be a relativistic theory where a_{0} appears as a parameter, with GR restored in the limit a_{0} 0. I have always viewed MOND as an effective theory (i.e. an approximate theory that results from a deeper one in a certain limit, and/or when some of the relevant degrees of freedom are integrated out). In the present case MOND is perhaps an approximation in the limit of small sizes and short times (on the cosmological scale), and nonrelativistic motion, due to some yet-undiscovered effect connected with cosmology. An analogy will highlight the point: If we are ignorant of earth gravity as derived from the pull of the earth-such as when we are immured forever in a small laboratory near the earth's surface-dynamics is described approximately by modified inertia of the form
(3) |
where F is the applied force excluding earth gravity, and g is the free-fall acceleration on earth. This can be recast to resemble MOND inertia:
(4) |
where 1 - [e x] / x^{2}, and e g / g is a down-pointing unit vector. This is a good approximation inasmuch as this proverbial laboratory is our whole universe; i.e., for systems small compared with R_{} (analogous to the Hubble distance), and times small compared with H_{}^{-1} = t_{} R_{} / c_{}, where c_{} = (M_{}G / R_{})^{1/2} is the escape speed-analogous to the speed of light. The effective "acceleration constant", g, appearing in this modified inertia is related to the "cosmological" parameters by g = c_{} H_{}.
In a relativistic extension of MOND, or in the cosmological context, a_{0} may lose its role as a "universal constant" [3] [25] - as g does in the above analogy when dealing with, say, satellite motion for which v ~ c_{}. The peculiar situation is further highlighted by the fact that - in view of a_{0} ~ cH_{0} - the only system that is both high-field in the GR sense and in the deep-MOND regime is the universe at large. (In the quantum analogue a system in the high-field, quantum regime is of Planck scale or smaller. There, we can, at least look from outside the Planck scale, which we cannot do in MOND.) Relativistic MOND must then be understood as part and parcel of cosmology, as I elaborate more in the next section.