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.

Living Rev. Relativity 15, (2012), 10
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