### 7.1 Scalar-tensor k-essence

MOND is an acceleration-based modification of gravity in the ultra-weak-field limit, but since the
Christoffel symbol, playing the role of acceleration in GR, is not a tensor, it is, in principle, not possible to
make a general relativistic theory depend on it. Another natural way to account for the departure from
Newtonian gravity in the weak-field limit and to account for the violation of the SEP inherent to
the external field effect is to resort to a scalar-tensor theory, as first proposed by [38]. The
added scalar field can play the role of an auxiliary potential, and its gradient then has the
dimensions of acceleration and can be used to enforce the acceleration-based modification of
MOND.
The relativistic theory of [38] depends on two fields, an “Einstein metric” and a scalar field . The
physical metric entering the matter action is then given by a conformal transformation of the Einstein
metric
through an exponential coupling function:

In order to recover the MOND dynamics, the Einstein–Hilbert action (involving the
Einstein metric) remains unchanged (), and the dimensionless scalar field
is given a k-essence action, with no potential and a non-linear, aquadratic, kinetic
term
inspired by the AQUAL action of Eq. 16:
where is a dimensionless constant, is a length-scale, , and is the
“MOND function”. Since the action of the scalar field is similar to that of the potential in the
Bekenstein–Milgrom version of classical MOND, this relativistic version is known as the Relativistic
Aquadratic Lagrangian theory, RAQUAL.
Varying the action w.r.t. , the scalar field yields, in a static configuration, the following modified
Poisson’s equation for the scalar field:

and the component of the physical metric is given by , leading us precisely to
the situation of Eq. 40 in the weak-field, with , with
and
whose finely tuned relation with the -function of Milgrom’s law is extensively described in Section 6.2.
We note that the standard choice for is , meaning that in order to recover
for small , where (see Section 6.2), one must define the length-scale as
It was immediately realized [38] that a k-essence theory such as RAQUAL can exhibit superluminal
propagations whenever [80]. Although it does not threaten causality [80], one
has to check that the Cauchy problem is still well-posed for the field equations. It has been
shown [80, 360] that it requires the otherwise free function to satisfy the following properties, :

which is the equivalent of the constraints of Eq. 37 on Milgrom’s -function.
However, another problem was immediately realized at an observational level [38, 40]. Because of the
conformal transformation of Eq. 75, one has that in the RAQUAL equivalent of Eq. 73. In other
words, as it is well-known that gravitational lensing is insensitive to conformal rescalings of the metric,
apart from the contribution of the stress-energy of the scalar field to the source of the Einstein
metric [40, 81], the “non-Newtonian” effects of the theory respectively on lensing and dynamics do not at
all correspond to similar amounts of “missing mass”. This is also considered a generic problem with any
local pure metric formulation of MOND [441].