### 7.2 Stratified theory

A solution to the above gravitational lensing problem due to the conformal rescaling of the metric
in RAQUAL has been presented in [385]. Inspired by “stratified” theories of gravity [334],
Sanders [385] suggested, in addition to the scalar field of RAQUAL, the use of a non-dynamical
timelike vector field with unit-norm (in terms of the Einstein
metric), in order to enforce a disformal relation between the Einstein and physical metrics:
The second term only affects the component, and it then appears immediately that in the
weak-field limit (rhs terms of Eq. 73), and the problem of lensing is cured. However, the prescription that a
4-vector points in the time direction is not a covariant one, and the theory should involve strong preferred
frame effects, although these can now be fully suppressed, as well as any deviation from GR at small
distances, with an appropriate additional “Galileon” term in addition to the asymptotic deep-MOND
k-essence term in the action of the scalar field [22] (the other advantage being that the interpolating
function then does not have to be inserted by hand). In any case, endowing the vector field with
covariant dynamics of its own has been the next logical step in developing relativistic MOND
theories.