The relativistic theory of  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 metric47 through an exponential coupling function:k-essence action, with no potential and a non-linear, aquadratic, kinetic term48 inspired by the AQUAL action of Eq. 16: 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:
It was immediately realized  that a k-essence theory such as RAQUAL can exhibit superluminal propagations whenever . Although it does not threaten causality , 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, :
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 .
Living Rev. Relativity 15, (2012), 10
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