### 7.3 Original Tensor-Vector-Scalar theory

The idea of the Tensor-Vector-Scalar theory of Bekenstein [33], dubbed TeVeS, is to keep the disformal
relation of Eq. 83 between the Einstein metric and the physical metric to which matter fields
couple, but to replace the above non-dynamical vector field by a dynamical vector field with an action
( being a dimensionless constant):
akin to that of the electromagnetic 4-potential vector field ( playing the role of the Faraday tensor),
but without the coupling term to the 4-current, and with a constraint term forcing the unit norm
( being a Lagrange multiplier function, to be determined as the equations are
solved). The first term in the integrand takes care of approximately aligning with the
4-velocity of matter (when simultaneously solving for (i) the Einstein-like equation of the Einstein
metric and for (ii) the vector equation obtained by varying the total action with respect to
).
Finally, the k-essence action for the scalar field is kept as in RAQUAL (Eq. 76), but with

Contrary to RAQUAL, this scalar field exhibits no superluminal propagation modes. However, [81] 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 [320]).
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 [33]). 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
Hamiltonian
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 [196]). And indeed, it was
shown in [98] 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 [81]). 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 [105], 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.