The remarkable feature of GEA theories allowing for the desired enhancing of gravitational lensing without any on the form of the physical metric is that, writing the metric as in Eq. 73, it can be shown  that in the limit the action of Eq. 93 is only a function of and is thus invariant under disformal transformations , of the type of Eq. 83. These GEA theories are currently extensively studied, mostly in a cosmological context (see Section 9), but also for their parametrized post-Newtonian coefficients in the solar system  or for black hole solutions .
Interestingly, it has been shown that all these vector field theories (TeVeS, BSTV, GEA) are all part of a broad class of theories studied in . Yet other phenomenologically-interesting theories exist among this class, such as, for instance, the models considered by Zhao & Li [502, 506, 510] with a dynamical norm vector field, whose norm obeys a potential (giving it a mass) and has a non-quadratic kinetic term à-la-RAQUAL, in order to try reproducing both the MOND phenomenology and the accelerated expansion of the universe, while interpreting the vector field as a fluid of neutrinos with varying mass [504, 505]. This has the advantage of giving a microphysics meaning to the vector field. Such vector fields have also been argued to arise naturally from dimensional reduction of higher-dimensional gravity theories [34, 261], or, more generally, to be necessary from the fact that quantum gravity could need a preferred rest frame  in order to protect the theory against instabilities when allowing for higher derivatives to make the theory renormalizable (e.g., in Hořava gravity [64, 195]). Inspired by this possible need of a preferred rest frame in quantum gravity, relativistic MOND theories boiling down to particular cases of GEA theories in which the vector field is hypersurface-orthogonal have, for instance, been proposed in [61, 396].
Living Rev. Relativity 15, (2012), 10
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