7 Relativistic MOND Theories

In Section 6, we have considered the classical theories of MOND and their predictions in a vast number of astrophysical systems. However, as already stated at the beginning of Section 6, these classical theories are only toy-models until they become the weak-field limit of a relativistic theory (with invariant physical laws under differentiable coordinate transformations), i.e., an extension of general relativity (GR) rather than an extension of Newtonian dynamics. Here, we list the various existing relativistic theories boiling down to MOND in the quasi-static weak-field limit. It is useful to restate here that the motivation for developing such theories is not to get rid of dark matter but to explain the Kepler-like laws of galactic dynamics predicted by Milgrom’s law (see Section 5). As we shall see, many of these theories include new fields, so that dark matter is often effectively replaced by “dark fields” (although, contrary to dark matter, their energy density can be subdominant to the baryonic one; note that, even more importantly, in a static configuration these dark fields are fully determined by the baryons, contrary to the traditional dark matter particles, which may, in principle, be present independent of baryons).

These theories are great advances because they enable us to calculate the effects of gravitational lensing and the cosmological evolution of the universe in MOND, which are beyond the capabilities of classical theories. However, as we shall see, many of these relativistic theories still have their limitations, ranging from true theoretical or observational problems to more aesthetic problems, such as the arbitrary introduction of an interpolating function (Section 6.2) or the absence of an understanding of the 2 Λ ∼ a0 coincidence. What is more, the new fields introduced in these theories have no counterpart yet in microphysics, meaning that these theories are, at best, only effective. So, despite the existing effective relativistic theories presented here, the quest for a more profound relativistic formulation of MOND continues. Excellent reviews of existing theories can also be found in, e.g., [34Jump To The Next Citation Point, 35, 81Jump To The Next Citation Point, 100, 136Jump To The Next Citation Point, 183Jump To The Next Citation Point, 318Jump To The Next Citation Point, 429Jump To The Next Citation Point, 431Jump To The Next Citation Point].

The heart of GR is the equivalence principle(s), in its weak (WEP), Einstein (EEP) and strong (SEP) form. The WEP states the universality of free fall, while the EEP states that one recovers special relativity in the freely falling frame of the WEP. These equivalence principles are obtained by assuming that all known matter fields are universally and minimally coupled to one single metric tensor, the physical metric. It is perfectly fine to keep these principles in MOND, although certain versions can involve another type of (dark) matter not following the same geodesics as the known matter, and thus effectively violating the WEP. Additionally, note that the local Lorentz invariance of special relativity could be spontaneously violated in MOND theories. The SEP, on the other hand, states that all laws of physics, including gravitation itself, are fully independent of velocity and location in spacetime. This is obtained in GR by making the physical metric itself obey the Einstein–Hilbert action. This principle has to be broken in MOND (see also Section 6.3). We now recall how GR connects with Newtonian dynamics in the weak-field limit, which is actually the regime in which the modification must be set in order to account for the MOND phenomenology of the ultra–weak-field limit. The action of GR written as the sum of the matter action and the Einstein–Hilbert (gravitational) action43:

4 ∫ √ --- SGR ≡ Smatter[matter, gμν] +--c--- d4x − gR, (69 ) 16 πG
where g denotes the determinant of the metric tensor gμν with (− ,+,+, + ) signature44, and μν R = R μνg is its scalar curvature, R μν being the Ricci tensor (involving second derivatives of the metric). The matter action is a functional of the matter fields, depending on them and their first derivatives. For instance, the matter action of a free point particle Spp writes:
∫ ∫ ∘ ------------- Spp ≡ − mcds = − mc − gμν(x)vμvνdt, (70 )
depending on the positions x and on their time-derivatives v μ. Varying the matter action with respect to (w.r.t.) matter fields degrees of freedom yields the equations of motion, i.e., the geodesic equation in the case of a point particle:
d2x μ μ dx αdx β ---2-= − Γαβ--------, (71 ) dτ dτ dτ
where the proper time τ = s is approximately equal to ct for slowly moving non-relativistic particles, and Γ μαβ is the Christoffel symbol involving first derivatives of the metric. On the other hand, varying the total action w.r.t. the metric yields Einstein’s field equations:
Rμν − 1-Rgμν = 8πG-Tμν, (72 ) 2 c4
where Tμν is the stress-energy tensor defined as the variation of the Lagrangian density of the matter fields over the metric.

In the static weak-field limit, the metric is written as (up to third-order corrections in 1∕c3)45:

( ) 2Φ 2Ψ g0i = gi0 = 0, g00 ◟=◝◜◞ − 1 − c2-, gij ◟=◝◜◞ 1 + -c2 δij, (73 ) Taylor Taylor
where, in GR,
Φ = ΦN and Ψ = − ΦN , (74 )
and ΦN is the Newtonian gravitational potential. From the (0,0) components of the weak-field metric, one gets back Newton’s second law for massive particles 2 i 2 i i d x ∕dt = − Γ 00 = − ∂ ΦN ∕dx from the geodesic equation (Eq. 71View Equation). On the other hand, Einstein’s equations (Eq. 72View Equation) give back the Newtonian Poisson equation ∇2 ΦN = 4πG ρ. Thus, the metric plays the role of the gravitational potential, and the Christoffel symbol plays the role of acceleration. Note, however, that if time-like geodesics are determined by the (0, 0) component of the metric, this is not the case for null geodesics. While the gravitational redshift for light-rays is solely governed by the g00 component of the metric too, the deflection of light is, on the other hand, also governed by the gij components (more specifically by Φ − Ψ in the weak-field limit). This means that, in order for the anomalous effects of any modified gravity theory on lensing and dynamics to correspond to a similar46 amount of “missing mass” in GR, it is crucial that Ψ ≃ − Φ in Eq. 73View Equation.

 7.1 Scalar-tensor k-essence
 7.2 Stratified theory
 7.3 Original Tensor-Vector-Scalar theory
 7.4 Generalized Tensor-Vector-Scalar theory
 7.5 Bi-Scalar-Tensor-Vector theory
 7.6 Non-minimal scalar-tensor formalism
 7.7 Generalized Einstein-Aether theories
 7.8 Bimetric theories
 7.9 Dipolar dark matter
 7.10 Non-local theories and other ideas

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