The External Field Effect

One of the curious aspects of MOND is the External Field Efect (EFE). Because the theory violates the strong equivalence principle (but not necessarily the Einstein or weak equivalence principle), one can not ignore the influence of external masses that impose an acceleration gext on systems with small inernal accelerations gin.
[The EFE often seems to get confused with a putative Internal Field Effect, in which high internal accelerations of a sub-system (like a solar system) are imagined to preclude that sub-system from obeying MOND within a larger system (like a galaxy). This is a misconception. The external field of the galaxy might affect what goes on in a sub-system like a solar system; the internal motions of the sub-system are irrelavant to how it moves in the larger system. Only the center of mass of the sub-system matters to that (see below).]

There are four broad regimes, each with an applicable mass estimator:


Note that for this illustration, the mass estimator assumes that one is comfortably in an extremal regime (g << or >> a0). There can be small corrections due to the interpolation function μ(g/a0) when g is within a factor of a few of a0.


In words, these regimes are:

  1. Newtonian Regime: gin > a0
    A system with accelerations well in excess of a0 behaves in a purely Newtonian manner: g = gN.
    (For a point mass M, gN = GM/r2.)

    Examples of systems where this occurs:

    • The Earth
    • The solar system
    • The inner regions of elliptical galaxies

  2. MOND Regime: gext < gin < a0
    A system with accelerations well below a0 but above those imposed by any external source is in the MOND regime: g = √(gN a0).

    Examples of systems where this occurs:

    • Low Surface Brightness Galaxies
      Any diffuse, isolated extragalactic system with surface density < a0/G (≈ 860 M pc-2 ≈ a sheet of paper).
    • The outer regions of high surface brightness galaxies

  3. Externally Imposed Newtonian Regime: gin < a0 < gext
    In this case, we imagine a system that has very small internal accelerations but which is influenced by a larger system that imposes accelerations in the Newtonian regime. In this case, the large external acceleration "wins" and the internal behavior is purely Newtonian irrespective of how small the system's internal accelerations may be.

    Examples of systems where this occurs:

    • Eotvos type experiments
      Experimentalists have devised systems with internal accelerations several orders of magnitude smaller than a0. These do not detect MONDian effects, nor should they. The experiments sit on the surface of the earth, where gext = 1011a0, so the internal dynamics behave in a purely Newtonian fashion.
    • Star clusters or binary stars in the solar neighborhood
      For current (2011) estimates of the Galactic constants, the solar neighborhood has a centripetal acceleration about the Galactic center of approximately 1.8a0. Therefore binary stars and local star clusters are still marginally in the Newtonian regime and should not exhibit pronounced MONDian behavior, even when their internal accelerations fall below a0. (I say "pronounced" because 1.8a0 is close enough to the MOND regime that there can be small deviations from purely Newtonian behavior as the interpolation function μ(x = g/a0) starts to deviate from the Newtonian limit (μ(x) → 1), which strictly applies only for x >> 1 [i.e., g >> a0].)
    • Galaxies in the cores of rich clusters of galaxies
      Some very large clusters obtain accelerations close to or even in excess of a0 in their central regions. This might cause morphological transformations as spirals that venture through this region lose the stability imparted by MOND. NGC 4438 comes to mind. Exactly what happens would depend on how long the spiral was subject to how large an external acceleration, but this might go some way to explain why ellipticals dominate the cores of rich clusters.

    Note that this is where the violation of the strong equivalence principle is most obvious - a system of interacting particles cares about the external universe in a disturbingly Machian sort of way.

  4. Quasi-Newtonian Regime: gin < gext < a0
    A system in which all accelerations are in the MOND regime but where the external acceleration exceeds the interal acceleration is in the Quasi-Newtonian Regime. Here the behavior is Newtonian in the sense that the effective force still follows the inverse square law, but the effective value of G is enhanced by the factor a0/gext. (Strictly speaking, by the inverse of the interpolation function 1/μ(x) where x = gext/a0.)

    Systems in this regime will show a mass discrepancy (M will be interpreted to be enhanced instead of G) but may also show a Keplerian decline in their rotation curve. Ultimately, all extragalactic systems must reach this regime at some point, as the acceleration due to the rest of the universe exceeds that of the central object. So rotation curves should not stay flat forever, even in MOND, just because there are other masses in the universe.

    Examples of systems where this occurs:

    • Dwarf Spheroidals
      Many of the dwarf satellites of the Milky Way and Andromeda are in the regime where the external field of the giant host dominates over the internal field of the dwarf. A test particles in orbit around such dwarfs at many optical radii (e.g., tidal debris) would show a Keplerian fall-off around a central mass inferred to have lots of dark matter.
      This is a cautionary example where it would be very easy to wrongly interpret such an observation as falsifying MOND because V(r) does not remain flat and the edge of dark matter subhalo appears to be encompassed.
    • The Milky Way
      Andromeda is the next large system in the Local Group, and is a bit more massive than the Milky Way. It should take over around 300 kpc, so we would infer an "edge" to the dark matter halo of that order (the exact radius depending on how we chose to define the "edge"). In general, all galaxies would be inferred to have an edge to their phantom dark matter halos (as they are via gravitational lensing) simply because there comes a point when the rest of the universe takes over.
    • Lyα clouds
      These insubstantial wisps of gas are usually so diffuse that their internal accelerations are less than the external acceleration imposed by the neighboring large scale structure.
See Milgrom's explanation for more on why this happens in MOND.

Not the Internal Field Effect

The External Field Effect is often confused with what I'll call the Internal Field Effect (IFE) to distinguish it. The EFE is the effect of the field of a very large system on a much smaller sub-system. The putative IFE is the effect of the accelerations within a sub-system on how that sub-system behaves within the larger system.

An obvious example is a solar system orbiting within a galaxy. The solar system has Newtonian accelerations, so isn't it in the Newtonian regime, and therefore immune to MONDian effects? The answer is no - the internal accelerations of the sub-system are irrelevant to how that sub-system responds to an external field. Only the field of the parent system at the position of the center of mass of the sub-sytem is relevant to that. Much as we do not need to know the quantum mechanical motion of all the particles that compose a baseball to understand the macrosopic motion of its center of mass, so too is the internal structure (and accelerations) of a solar system irrelevant to how the center of mass of the system orbits in a galaxy. A Newtonian solar system is a billiard ball to a MONDian galaxy. There is a boundary condition that defines the edge of the billiard ball, roughly where gin ~ gex. This occurs at about 7,000 AU for our solar system. Everything inside of that boundary is part of the internal structure of the system. While the EFE can reach inside that boundary, the IFE cannot reach outside it. Only the center of mass of such systems matters to determining their orbits in MOND, not their internal structure, nor the magnitude of their internal accelerations. That is to say, MOND respects the weak equivalence principle: the motion of a particle - be it a billiard ball or a solar system - is independent of its internal structure or composition.

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