Finally, the k-essence action for the scalar field is kept as in RAQUAL (Eq. 76), but with noted that such superluminal propagation might have to be re-introduced in order to avoid excessive Cherenkov radiation and suppression of high-energy cosmic rays (see also ).
The static weak-field limit equation for the scalar field is precisely the same as Eq. 77, and the scalar field enters the static weak field metric Eq. 73 as meaning that lensing and dynamics are compatible, with being a factor depending on and on the cosmological value of the scalar field (see Eq. 58 of ). This can be normalized to yield at redshift zero. Again, all the relations between the free function and Milgrom’s -function can be found in Section 6.2 (see also [145, 431]).
This theory has played a true historical role as a proof of concept that it was possible to construct a fully relativistic theory both enhancing dynamics and lensing in a coherent way and reproducing the MOND phenomenology for static configurations with the dynamical 4-vector pointing in the time direction. However, the question remained whether these static configurations would be stable. What is more, although a classical Hamiltonian49 unbounded from below in flat spacetime would not necessarily be a concern at the classical level (and even less if the model is only “phenomenological”), it would inevitably become a worry for the existence of a stable quantum vacuum (see however ). And indeed, it was shown in  that models with such “Maxwellian” vector fields having a TeVeS-like Lagrange multiplier constraint in their action have a corresponding Hamiltonian density that can be made arbitrarily large and negative (see also Section IV.A of ). What is more, even at the classical level, it has been shown that spherically-symmetric solutions of TeVeS are heavily unstable [412, 413], and that this type of vector field causes caustic singularities , in the sense that the integral curves of the vector are timelike geodesics meeting each other when falling into gravity potential wells. Thus, another form was needed for the action of the TeVeS vector field.
Living Rev. Relativity 15, (2012), 10
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