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, , and external,
(from a host galaxy, or astrophysical
system, or large scale structure), is less than
. If
then we have standard
MOND effects. However, if the hierarchy goes as
, then the system is purely
Newtonian33,
and if
then the system is Newtonian with a renormalized gravitational constant. Ultimately,
whenever
falls below
(which always happens at some point) the gravitational attraction falls again
as
. This is most easily illustrated in a thought experiment where one considers MOND effects in
one dimension. In Eq. 17
, one has
and
, which in one
dimension leads to the following revised Milgrom’s law (Eq. 7
) including 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 ) by considering the perturbation
generated by a body of low mass
inside a uniform external field, assumed along the
-direction,
. Eq. 17
can then be linearized and solved with the boundary condition that the total field
equals the external one at infinity [38
] to yield:
For the exact behavior of the MOND gravitational field in the regime where and
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. 25
and Eq. 35
). For the BM case, one adds the three components of the external
field (no longer assumed to be in the
-direction only) in the argument of
which becomes
,
and similarly for the other
and
points on the grid (Figure 17
). One also adds the respective
component of the external field to the term estimating the force at the
and
points in Eq. 25
.
With
, for instance, one changes
in the first term of
Eq. 25
. One then solves this discretized equation with the large radius boundary condition
for the Dirichlet problem given by Eq. 61
instead of Eq. 20
. Exactly the same is applicable
to calculating the phantom dark matter component of QUMOND with Eq. 35
, 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. 61 or Eq. 63
) instead of a logarithmic (Eq. 20
) 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 [490
]): with accurate-enough
weak-lensing data, detecting these pockets of negative phantom densities could, in principle,
be a smoking gun for MOND [490
], 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
. The corresponding rescaling of the gravitational constant then depends on the
choice of the
-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.
http://www.livingreviews.org/lrr-2012-10 |
Living Rev. Relativity 15, (2012), 10
![]() This work is licensed under a Creative Commons License. E-mail us: |