7.10 Non-local theories and other ideas
All the models so far somehow invoke the existence of new “dark fields”, notably because for local pure
metric theories, the Hamiltonian is generically unbounded from below if the action depends on a finite
number of derivatives [81, 136, 441]. A somewhat provocative solution would thus be to consider
non-local theories. A non-local action could, e.g., arise as an effective action due to quantum
corrections from super-horizon gravitons [440]. Deffayet, Esposito-Farèse & Woodard [123
]
have notably exhibited the form that a pure metric theory of MOND could take in order to
yield MONDian dynamics and MONDian lensing for a static, spherically-symmetric baryonic
source.
In such a static spherically-symmetric geometry, the Einstein–Hilbert action of Eq. 69 can be rewritten
in the weak-field expansion as [123
]:
where
and
are the weak-field
and
components of the static weak-field
metric, respectively. The MOND modification to this action implies to obtain
as
a solution in the deep-MOND limit, where the first equality ensures that lensing and dynamics are
consistent, leading to the following tentative action in the ultra-weak-field limit [123
]:
where
and
is an arbitrary constant. While it is impossible to express this form of the action
as a local functional of a general metric, Deffayet et al. [123
] showed that it was entirely possible to do so in
a non-local model, making use of the non-local inverse d’Alembertian and of a TeVeS-like vector field,
introduced not as an additional “dark field”, but as a non-local functional of the metric itself (by, e.g.,
normalizing the gradient of the volume of the past light-cone). A whole class of such models is
constructible, and a few examples are given in [123], for which stability analyses are still needed,
though.
As already mentioned in Section 6.1.1, this non-locality was also inherent to classical toy
models of “modified inertia”. In GR, this would mean making the matter action of a point
particle (Eq. 70) depend on all derivatives of its position, but such models are very difficult to
construct [300] and no fully-fledged theory exists along these lines. However, a few interesting
heuristic ideas have been proposed in this context. For instance, Milgrom [304
] proposed that
the inertial force in Newton’s second law could be defined to be proportional to the difference
between the Unruh temperature and the Gibbons–Hawking one. It is indeed well known that, in
Minkowski spacetime, an accelerated observer sees the vacuum as a thermal bath with a temperature
proportional to the observer’s acceleration
[110, 470], where
is the Planck
constant and
the Boltzmann constant. On the other hand, a constant-accelerated observer
in de Sitter spacetime (curved with a positive cosmological constant
) sees a non-linear
combination of that vacuum radiation and of the Gibbons–Hawking radiation (with temperature
[174
]) due to the cosmological horizon in the presence of a positive
. Namely,
the Unruh temperature of the radiation seen by such an accelerated observer in de Sitter spacetime is [174]
. The idea of Milgrom [304] is to then define the right-hand side of the
norm of Newton’s second law as being proportional to the difference between the two temperatures:
which trivially leads to
with
(which is, however, observationally too
large by a factor
) and the interpolating function
having the exact form of Eq. 54. In short,
observers experiencing a very small acceleration would see an Unruh radiation with a small temperature
close to the Gibbons–Hawking one, meaning that the inertial resistance defined by the difference
between the two radiation temperatures would be smaller than in Newtonian dynamics, and
thus the corresponding acceleration would be larger. However, no relativistic version (if at all
possible) of this approach has been developed yet: a few difficulties arise due to the direction of
the acceleration, or by the fact that stars in galaxies are free-falling objects along geodesics,
and not accelerated by a non-gravitational force, as in the case of basic Unruh radiation. It
was interestingly noted [308
] that the de Sitter spacetime could be seen as a 4-dimensional
pseudo-sphere embedded in a 5-dimensional flat Minkowski space, and that the acceleration
of a constant-accelerated observer in this flat space would be exactly
.
Then, MOND could arise from symmetry arguments in this 5-dimensional space similar to those
leading to special relativity in Minkowski space [308]. Interestingly, arguments very similar to
this whole vacuum radiation approach have also recently been made in the context of entropic
gravity [191, 192, 224, 476]. Finally, another interesting idea to get MOND dynamics has
been the tentative modification of special relativity, making the Planck length and the length
two new invariants, in addition to the speed of light, an attempt known as Triply
Special Relativity [233]. In any case, despite all these attempts, there is still no fully-fledged theory of
MOND at hand, which would derive from first principles, and the quest for such a formulation of MOND
continues.