### 7.7 Generalized Einstein-Aether theories

All theories reviewed so far are best expressed in the “Einstein frame”, and involve a form for the
physical metric to which matter couples (an form expressed as a function of the Einstein metric and of
the other additional fields). However, the work of [513] has shown that, for instance, TeVeS
(Sections 7.3 and 7.4) is expressible as a pure Tensor-Vector theory in the matter frame, and that the
physical metric then both satisfies the Einstein–Hilbert action and couples minimally to the
matter fields, just like in GR. In fact, the modification of gravity in TeVeS thus only comes
from the coupling of the physical metric to the vector field. The idea of Zlosnik et al. [514]
was then that a similar, but simpler, modification of gravity could be obtained by devising
a simple tensor-vector theory in the matter frame, with no a priori on the geometry of the
physical metric. Starting from the extensively studied Einstein-Aether theories [206, 207], with a
vector action of the type of Eq. 86, the idea is to make the k-essence free function (the
“MOND function” of Eq. 76) act directly on the vector field rather than on an additional scalar
field. This leads to vector k-essence, or Generalized Einstein-Aether (GEA) theories (also called
non-canonical Einstein-Aether theories), in which the Einstein–Hilbert and matter actions remain
as in GR, but with an additional unit-norm vector field with the following action [431, 514]:
where (see Eq. 87 and replacing by )
The unit-norm constraint fixes the vector field in terms of the metric, and from there we have that, in the
weak-field limit, , with defined as in Eq. 73. The Einstein equation in the weak-field
limit then yields a BM type of Poisson equation (Eq. 17) for the full gravitational potential , with
and [431]. In the deep-MOND limit, the usual choice for
is of the type , and the length-scale must be fixed as:
Let us note that this weak-field limit of GEA theories is different from that of RAQUAL or TeVeS, where
only the scalar field obeys a BM-like equation governed by an interpolating function , and where
the total potential is given by Eq. 40.
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 [431] 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 [65] or for black hole
solutions [451].

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 [183]. 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 [206] 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].