From M. Milgrom,
Mon. Not. R. Astron. Soc. 437, 2531 (2014). MOND laws of galactic dynamics preprint (
https://arxiv.org/pdf/1212.2568.pdf) pages 16-19:
"4.6.The external-field effect
Unlike ND, MOND is nonlinear even in the NR regime. It generally does not satisfy the strong equivalent principle; so effects of an overall acceleration on the internal dynamics of a system are generically expected. To be able to say what constraints the basic tenets impose on such effects I have to confine myself here to theories whereby only the instantaneous value of the external acceleration matters. This excludes from the discussion a large class of MI theories that are time nonlocal (Milgrom 1994). In these, the full(external)trajectory of the system enters, which complicates the discussion. Some of the possible consequences of such nonlocality are discussed briefly in Milgrom(2011),but what follows here does not apply to such theories.
Consider then a system of mass m(‘system m’), and extent r, that is falling in the field of a mother system with acceleration whose instantaneous value is g0. Assume that the theory and conditions are such that, to a good enough approximation, all the information about the mother system enters the dynamics within m only through g0. One can then write
a=a(m,r,a0,G,g0 ,n0 ,α), (15)
where a stands for the internal acceleration runs of elements of m, namely the full acceleration in the field of the mother system minus g0 (suppressing the dependence on position, time, and particle index). It is written as a function of all the available dimensioned independent parameters, as well as of n0 ,the unit vector in the direction of g0, and of α, which stands for the many dimensionless parameters that characterize the configuration, such as all the mass ratios, and all the geometrical parameters(angles, ratios of all distances tor, etc.). Here I am only interested in scaling laws of the dimensioned parameters–for example, in how |a| depends on the dimensioned system attributes–so I shall suppress the dependence on n0 and α. Since a/g0 is dimensionless, it can depend only on dimensionless quantities; so we can write, most generally
a=g0 F∗(η,θ),η≡mG r2g0 ∼gN g0 ,θ≡g0 a0 . (16)
When g0≪|a|, its effects can be neglected. So here I shall be interested in the opposite case, of external-acceleration dominance,g0≫|a|.35 The above choice of dimensionless variables is useful for this case. Clearly, F∗(0,θ)=0. So we are interested in the behaviour of F∗ to lowest order in η. We shall see below that external-field dominance requires η≪1 when θ≳1, and the SI condition η≪θ when θ≲1(in which case the whole problem is in the DML; η and θ both scale likeλ−1 underscaling); so we can write this condition generally as η≪min(1,θ).
We do not know that a MOND theory is necessarily expandable in powers of η near η=0. But assuming that it does, I write
a≈g0ηqf(θ),η≪min(1,θ). (17)
(I assume that q does not depend on θ; see below.)
To constrain q and f(θ) I now employ the basic tenets of MOND. The limit a0→0, namely whena0≪|a|≪g0, is strongly Newtonian for all accelerations, and is within the validity domain of eq.(17). For this region, g0 anda0 have to disappear from expression(17). This implies that q=1, and that|f(θ≫1)|∼1, such that(mG/r2)f(∞) is the internal Newtonian acceleration.
It is difficult to make general statements about the intermediate case where the two accelerations are of the same order field within m. This means that the internal dynamics is Newtonian for any value of gN when θ≫1; i.e.,also wheng N≪a0. In other words: whenever the external field is highly Newtonian and dominates over the internal field, the latter is necessarily Newtonian. This result holds also when q depends on θ, because then we still must have q(θ→∞)→1.
More generally, in as much as q=1 for all θ, we can write eq.(17)in its full validity domain (external-field dominance) as
a=mG r2 f(θ). (18)
This means that when the external field is dominant, the internal dynamics is always quasiNewtonian, in the sense that the accelerations scale as mG/r2, only with an enhanced effective constant Geff∼G|f(θ)|, and with not-quite-Newtonian geometrical aspects that stem from the fact that f has different geometric properties than f(∞): for example, f depends on the direction relative to n0 ,and on the theory at hand, while f(∞) does not.
When θ≪1 the whole system is in the DML, where the basic tenets dictate that eq.(18) becomes SI. Underscaling, θ scales like g0, namely θ→λ−1θ (since g0 is a DML acceleration of the mother system it scales as g0→λ−1g0). This means that f must become proportional to θ−1: f(θ≪1)≈θ−1¯f. We see then that f(θ) has the same asymptotic behaviours as 1/µ(θ), where µ is the interpolating function appearing in present MOND theories. If q does depend on θ, the EFE does not conform to the standard results.
For example, in the DML we could have 0<q(0)=1, in which case SI dictates f(θ≪1)≈θ−q(0)ˆf. Then a∼g0(gN a0/g2 0 )q(0)=g0(η/θ)q(0). We see that, as stated above, the condition for external-field dominance, a≪g0, whenθ<1, is indeed always η≪θ. For example, if q(0)=1/2, this gives the standard scaling of the MOND acceleration in isolated systems a∼(gN a0)1/2; i.e.,there is no EFE, except for effects in¯f(0). So the basic tenets lead to the standard EFEresults(indeed to an EFE) only if some additional analytic properties are assumed. The toy DML theory described by eq.(6), which satisfies scale invariance (but which does not combine with an appropriate Newtonian limit), does not lead to an EFE. The above analytic assumptions do hold in all the MOND formulations considered to date: e.g., in the original, pristine formulation in (Milgrom 1983),in the formulation of Bekenstein & Milgrom (1984), and in QUMOND(Milgrom 2010a). For example, in QUMOND, we can write schematically (ignoring the vectorial nature of the quantities involved)
a/g0 ∼ν[θµ(θ)+θη][µ(θ)+η]−1, (19)
where ν(y) is the QUMOND interpolating function, and µ(x) is such that ν[xµ(x)]µ(x)=1. We have µ(θ≪1)≈θ,µ(θ≫1)≈1; so we see explicitly why the condition η≪min(1,θ) is tantamount to a dominant external field. And, clearly the next to zeroth-order term is a/g0∼ η(1+ˆν)/µ(θ), where−1/2<ˆν<0 is the logarithmic derivative of ν.
In summary, the fact that an external field |g0|≫a0 renders the internal dynamics Newtonian, follows from only the basic tenets of MOND, provided that only the instantaneous external field enters the internal dynamics (not necessarily true in MI, time-nonlocal theories). This is relevant, for example, to experimental results in the laboratory and Solar system, and to the dynamics of star clusters near the sun. On the other hand, the specific form of the EFE when|g0|≪a0, even its very existence, is not strictly dictated by the basic tenets alone. Its basic features do follow under another plausible assumption, shared by all full-fledged theories considered to date: that the expansion power in eq.(17) does not depend on θ.
There is no EFE in the DM paradigm."
