7.3 Original Tensor-Vector-Scalar theory

The idea of the Tensor-Vector-Scalar theory of Bekenstein [33Jump To The Next Citation Point], dubbed TeVeS, is to keep the disformal relation of Eq. 83View Equation between the Einstein metric g&tidle;μν and the physical metric gμν to which matter fields couple, but to replace the above non-dynamical vector field by a dynamical vector field Uμ with an action (K being a dimensionless constant):
c4 ∫ ∘ --- [K ] SU ≡ − ------ d4x − &tidle;g --&tidle;g αβ&tidle;gμνU[α,μ]U[β,ν] − λ (&tidle;gμνUμU ν + 1) , (84 ) 16πG 2
akin to that of the electromagnetic 4-potential vector field (U[μ,ν] 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 μ μν U U ν = &tidle;g UμU ν = − 1 (λ being a Lagrange multiplier function, to be determined as the equations are solved). The first term in the integrand takes care of approximately aligning U μ with the 4-velocity of matter (when simultaneously solving for (i) the Einstein-like equation of the Einstein metric &tidle;gμν and for (ii) the vector equation obtained by varying the total action with respect to U μ).

Finally, the k-essence action for the scalar field is kept as in RAQUAL (Eq. 76View Equation), but with

XTeVeS = kl2(&tidle;gμν − UμU ν)ϕ,μ ϕ,ν . (85 )
Contrary to RAQUAL, this scalar field exhibits no superluminal propagation modes. However, [81Jump To The Next Citation Point] 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. 77View Equation, and the scalar field enters the static weak field metric Eq. 73View Equation as Φ = − Ψ = Ξ ΦN + c2ϕ meaning that lensing and dynamics are compatible, with Ξ being a factor depending on K and on the cosmological value of the scalar field (see Eq. 58 of [33Jump To The Next Citation Point]). This can be normalized to yield Ξ = 1 at redshift zero. Again, all the relations between the free function f and Milgrom’s μ-function can be found in Section 6.2 (see also [145Jump To The Next Citation Point, 431Jump To The Next Citation Point]).

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 [81Jump To The Next Citation Point]). 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.

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