7.1 Scalar-tensor k-essence

MOND is an acceleration-based modification of gravity in the ultra-weak-field limit, but since the Christoffel symbol, playing the role of acceleration in GR, is not a tensor, it is, in principle, not possible to make a general relativistic theory depend on it. Another natural way to account for the departure from Newtonian gravity in the weak-field limit and to account for the violation of the SEP inherent to the external field effect is to resort to a scalar-tensor theory, as first proposed by [38Jump To The Next Citation Point]. The added scalar field can play the role of an auxiliary potential, and its gradient then has the dimensions of acceleration and can be used to enforce the acceleration-based modification of MOND.

The relativistic theory of [38Jump To The Next Citation Point] depends on two fields, an “Einstein metric” &tidle;gμν and a scalar field ϕ. The physical metric gμν entering the matter action is then given by a conformal transformation of the Einstein metric47 through an exponential coupling function:

g ≡ e2ϕ&tidle;g . (75 ) μν μν
In order to recover the MOND dynamics, the Einstein–Hilbert action (involving the Einstein metric) remains unchanged (∫ --- d4x √ −g&tidle;&tidle;R), and the dimensionless scalar field is given a k-essence action, with no potential and a non-linear, aquadratic, kinetic term48 inspired by the AQUAL action of Eq. 16View Equation:
c4 ∫ ∘ --- Sϕ ≡ − ------- d4x − &tidle;gf(X ), (76 ) 2k2l2G
where k is a dimensionless constant, l is a length-scale, X = kl2&tidle;gμνϕ,μ ϕ,ν, and f(X ) is the “MOND function”. Since the action of the scalar field is similar to that of the potential in the Bekenstein–Milgrom version of classical MOND, this relativistic version is known as the 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:

2 [ ′ 2 2 ] c ∇. ∇ ϕf (kl |∇ ϕ| ) = kG ρ, (77 )
and the (0,0) component of the physical metric is given by g = − e2(ΦN+c2ϕ)∕c2 00, leading us precisely to the situation of Eq. 40View Equation in the weak-field, with 2 Φ = ΦN + c ϕ, with
s = (c2∕a0)|∇ ϕ| = (Xc4 ∕kl2a20)1∕2 (78 )
2 ′ &tidle;μ(s) = (4πc ∕k )f (X ), (79 )
whose finely tuned relation with the μ-function of Milgrom’s law is extensively described in Section 6.2. We note that the standard choice for X ≪ 1 is f′(X ) ∼ (X∕3 )1∕2, meaning that in order to recover &tidle;μ (s ) = s∕ ξ for small s, where ξ = GN ∕G (see Section 6.2), one must define the length-scale as
√ --- l ≡ (c2 3k)∕(4π ξa0). (80 )

It was immediately realized [38Jump To The Next Citation Point] that a k-essence theory such as RAQUAL can exhibit superluminal propagations whenever f′′(X ) > 0 [80Jump To The Next Citation Point]. Although it does not threaten causality [80Jump To The Next Citation Point], one has to check that the Cauchy problem is still well-posed for the field equations. It has been shown [80Jump To The Next Citation Point, 360] that it requires the otherwise free function f to satisfy the following properties, ∀X:

f′(X ) > 0 (81 ) f′(X ) + 2Xf ′′(X ) > 0, (82 )
which is the equivalent of the constraints of Eq. 37View Equation on Milgrom’s μ-function.

However, another problem was immediately realized at an observational level [38, 40Jump To The Next Citation Point]. Because of the conformal transformation of Eq. 75View Equation, one has that Ψ ⁄= − Φ in the RAQUAL equivalent of Eq. 73View Equation. 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, 81Jump To The Next Citation Point], 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 [441Jump To The Next Citation Point].

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