6.3 The external field effect

The above return to a rescaled Newtonian behavior at very large radii and in the central parts of isolated systems, in order to avoid theoretical problems with the interpolating function, would happen anyway, even with the interpolating function going to zero, for any non-isolated system in the universe (and this return to Newtonian behavior could actually happen at much lower radii) because of a very peculiar aspect of MOND: the external field effect, which appeared in its full significance already in the pristine formulation of MOND [293Jump To The Next Citation Point].

In practice, no objects are truly isolated in the Universe and this has wider and more subtle implications in MOND than in Newton–Einstein gravity. In the linear Newtonian dynamics, the internal dynamics of a subsystem (a cluster in a galaxy, or a galaxy in a galaxy cluster for instance) in the field of its mother system decouples. Namely, the internal dynamics is always the same independent of any external field (constant across the subsystem) in which the system is embedded (of course, if the external field varies across the subsystem, it manifests itself as tides). This has subsequently been built in as a fundamental principle of GR: the Strong Equivalence Principle (see Section 7). But MOND has to break this fundamental principle of GR. This is because, as it is an acceleration-based theory, what counts is the total gravitational acceleration with respect to a pre-defined frame (e.g., the CMB frame32). Thus, the MOND effects are only observed in systems where the absolute value of the gravity both internal, g, and external, ge (from a host galaxy, or astrophysical system, or large scale structure), is less than a0 . If ge < g < a0 then we have standard MOND effects. However, if the hierarchy goes as g < a0 < ge, then the system is purely Newtonian33, and if g < ge < a0 then the system is Newtonian with a renormalized gravitational constant. Ultimately, whenever g falls below g e (which always happens at some point) the gravitational attraction falls again as 2 1∕r. This is most easily illustrated in a thought experiment where one considers MOND effects in one dimension. In Eq. 17View Equation, one has ∇ Φ = g + ge and 4πG ρ = ∇. (gN + gNe ), which in one dimension leads to the following revised Milgrom’s law (Eq. 7View Equation) including the external field:

( ) [ ( ) ( ) ] gμ g-+-ge + ge μ g-+-ge- − μ ge- = gN, (59 ) a0 a0 a0
such that, when g → 0, we have Newtonian gravity with a renormalized gravitational constant Gnorm ≈ G ∕[μe (1 + Le )] where μe = μ(ge∕a0) and Le = (d lnμ ∕dlnx )x=ge∕a0, assuming, as before, that the external field only varies on a much larger scale than the internal system. Similarly, for QUMOND (Eq. 30View Equation) in one dimension, one gets the equivalent of Eq. 10View Equation:
( ) [ ( ) ( ) ] gN-+-gNe- gN-+-gNe- gNe- g = gNν a + gNe ν a − ν a . (60 ) 0 0 0
When dealing in the future with very extended rotation curves whose last observed point is in the extreme weak-field limit, it could be interesting, as a first-order approximation, to use the latter formulae34, adding the external field as an additional parameter of the MOND fit to the external parts of the rotation curve. Of course, this would only be a first-order approximation because it would neglect the three-dimensional nature of the problem and the direction of the external field.

Now, in three dimensions, the problem can be analytically solved only in the extreme case of the completely–external-field–dominated part of the system (where g ≪ ge) by considering the perturbation generated by a body of low mass m inside a uniform external field, assumed along the z-direction, g = g 1 e e z. Eq. 17View Equation can then be linearized and solved with the boundary condition that the total field equals the external one at infinity [38Jump To The Next Citation Point] to yield:

Gm-- Φ(x,y,z ) = − μ &tidle;r , (61 ) e
with
2 2 2 1∕2 &tidle;r = r(1 + Le(x + y )∕r ) , (62 )
squashing the isopotentials along the external field direction. Thus, this is the asymptotic behavior of the gravitational field in any system embedded in a constant external field. Similarly, in QUMOND (Eq. 30View Equation), one gets
Gm νe Φ(x,y, z) = − -----, (63 ) &tidle;r
with
&tidle;r = r∕[1 + (L ∕2 )(x2 + y2)∕r2], (64 ) Ne
where LNe = (d lnν∕d lny )y=gNe∕a0.

For the exact behavior of the MOND gravitational field in the regime where g and ge are of the same order of magnitude, one again resorts to a numerical solver, both for the BM equation case and for the QUMOND case (see Eq. 25View Equation and Eq. 35View Equation). For the BM case, one adds the three components of the external field (no longer assumed to be in the z-direction only) in the argument of μ M1 which becomes 2 2 2 1∕2 {[(Φ (B ) − Φ(A ))∕h − gex] + [(Φ(I) + Φ(H ) − Φ (K ) − Φ (J))∕(4h) − gey] + [(Φ(C ) + Φ (D ) − Φ (E &#, and similarly for the other Mi and Li points on the grid (Figure 17View Image). One also adds the respective component of the external field to the term estimating the force at the Mi and Li points in Eq. 25View Equation. With M1, for instance, one changes (Φi+1,j,k − Φi,j,k) → (Φi+1,j,k − Φi,j,k − hgex) in the first term of Eq. 25View Equation. One then solves this discretized equation with the large radius boundary condition for the Dirichlet problem given by Eq. 61View Equation instead of Eq. 20View Equation. Exactly the same is applicable to calculating the phantom dark matter component of QUMOND with Eq. 35View Equation, except that now the Newtonian external field is added to the terms of the equation in exactly the same way.

This external field effect (EFE) is a remarkable property of MONDian theories, and because this breaks the strong equivalence principle, it allows us to derive properties of the gravitational field in which a system is embedded from its internal dynamics (and not only from tides). For instance, the return to a Newtonian (Eq. 61View Equation or Eq. 63View Equation) instead of a logarithmic (Eq. 20View Equation) potential at large radii is what defines the escape speed in MOND. By observationally estimating the escape speed from a system (e.g., the Milky Way escape speed from our local neighborhood; see discussion in Section 6.5.2), one can estimate the amplitude of the external field in which the system is embedded, and by measuring the shape of its isopotential contours at large radii, one can determine the direction of that external field, without resorting to tidal effects. It is also noticeable that the phantom dark matter has a tendency to become negative in “conoidal” regions perpendicular to the external field direction (see Figure 3 of [490Jump To The Next Citation Point]): with accurate-enough weak-lensing data, detecting these pockets of negative phantom densities could, in principle, be a smoking gun for MOND [490Jump To The Next Citation Point], but such an effect would be extremely sensitive to the detailed distribution of the baryonic matter. A final important remark about the EFE is that it prevents most possible MOND effects in Galactic disk open clusters or in wide binaries, apart from a possible rescaling of the gravitational constant. Indeed, for wide binaries located in the solar neighborhood, the galactic EFE (coming from the distribution of mass in our galaxy) is about 1.5 × a0. The corresponding rescaling of the gravitational constant then depends on the choice of the μ-function, but could typically account for up to a 50% increase of the effective gravitational constant. Although this is not, properly speaking, a MOND effect, it could still perhaps imply a systematic offset of mass for very-long–period binaries. However, any effect of the type claimed to be observed by [188] would not be expected in MOND due to the external field effect.


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