Finally, the k-essence action for the scalar field is kept as in RAQUAL (Eq. 76), but with
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
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 [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.
http://www.livingreviews.org/lrr-2012-10 |
Living Rev. Relativity 15, (2012), 10
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