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We still want a relativistic extension of MOND. Such a theory is needed for conceptual completion of the MOND idea. But, it is doubly needed because we already have observed relativistic phenomena that show mass discrepancies, and we must ascertain that there too the culprit is not dark matter but modified dynamics.

Because of the near values of a0 and the Hubble acceleration, there are no local black holes that are in the MOND regime. The only system that is strongly general relativistic and in the MOND regime is the Universe at large. This, however means that we would need a relativistic extension of MOND to describe cosmology. In fact, as I have indicated, MOND itself may derive from cosmology, so it is possible that the two problems will have to be tackled together as parts and parcels of a unified concept. And, because the cosmological expansion is strongly coupled with the process of structure formation this too will have to await a modified relativistic dynamics for its treatment.

Several relativistic theories incorporating the MOND principle have been discussed in the literature, but none is wholly satisfactory (see, e.g. [Bekenstein & Milgrom (1984)], [Bekenstein 1988], [Sanders 1997], and references therein).

There have also been attempts to supplement MOND with extra assumptions that will enable the study of structure formation, so as to get some glimpse of structure formation in MOND. For these see [Milgrom (1989)], [Sanders (2001)], and [Nusser (2001)].

Gravitational light deflection, and lensing, is another phenomenon that requires modified relativistic dynamics. It is tempting to take as a first approximation the deflection law of post-Newtonian General Relativity with a potential that is the non-relativistic MOND potential (see e.g. analyzes by [Qin & al 1995], and [Mortlock & Turner 2001] based on this assumption). This, however, is in no way guaranteed. In GR this is only a post-Newtonian approximation, and perhaps it would turn out to be a post-Newtonian approximation of MOND (i.e. an approximation of MOND in the almost Newtonian, a > > a0 regime). But, there is no reason to assume that it is correct in the deep-MOND regime. Even in the framework of this assumption one needs to exercise care. For example, the thin-lens hypothesis, by which it is a good approximation to assume that all deflecting masses are projected on the same plane perpendicular to the line of sight, breaks down in MOND. For example, n masses, M, arranged along the line of sight (at inter-mass distances larger that the impact parameter) bend light by a factor n1/2 more than a single mass nM.

Also note that we may expect surprises in mondified inertia where we cannot even speak of the modified, MOND potential, as alluded to above.

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