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 [385Jump To The Next Citation Point]. Inspired by “stratified” theories of gravity [334], Sanders [385] suggested, in addition to the scalar field ϕ of RAQUAL, the use of a non-dynamical timelike vector field U μ = (− 1,0,0,0) with unit-norm 2 U = − 1 (in terms of the Einstein metric), in order to enforce a disformal relation between the Einstein and physical metrics:
gμν ≡ e− 2ϕ &tidle;gμν − 2 sinh (2ϕ)U μUν. (83 )
The second term only affects the g00 component, and it then appears immediately that Ψ = − Φ in the weak-field limit (rhs terms of Eq. 73View Equation), 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 [22Jump To The Next Citation Point] (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.
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