7.4 Generalized Tensor-Vector-Scalar theory

The generalization of TeVeS was proposed by Skordis [428]. Inspired by the fact that Einstein-Aether theories [206Jump To The Next Citation Point, 207Jump To The Next Citation Point] also present instabilities when the unit-norm vector field is “Maxwellian” as above, it was simply proposed to use a more general Lagrangian density for the vector field, akin to that of Einstein-Aether theories:
4 ∫ ∘ --- SU ≡ − -c---- d4x − &tidle;g[K αβμνU β,αU ν,μ − λ(&tidle;gμνU μUν + 1)], (86 ) 16πG
αβμν αμ βν αβ μν αν βμ α μ βν K = c1&tidle;g &tidle;g + c2&tidle;g &tidle;g + c3&tidle;g &tidle;g + c4U U &tidle;g (87 )
for a set of constants c1,c2,c3,c4. Interestingly, spherically-symmetric solutions depend only on the combination c − c 1 4, not on c 2 and c 3 that can, in principle be chosen to avoid the instabilities of the original TeVeS theory. Of course, the original unstable theory is also included in this generalization through a specific combination of the four ci (see, e.g., [431Jump To The Next Citation Point]).

Thus, this generalized version is the current “working version” of what is now called TeVeS: a tensor-vector-scalar theory with an Einstein-like metric, an Einstein-Aether-like unit-norm vector field, and a k-essence-like scalar field, all related to the physical metric through Eq. 83View Equation. It has been extensively studied, both in its original and generalized form. It has for instance been shown that, contrary to many gravity theories with a scalar sector, the theory evidences no cosmological evolution of the Newtonian gravitational constant G and only minor evolution of Milgrom’s constant a0 [145Jump To The Next Citation Point, 39]. However, the fact that the latter is still put in by hand through the length-scale of the theory l ∼ c2∕a0, and has no dynamical connection with the Hubble or cosmological constant is perhaps a serious conceptual shortcoming, together with the free function put by hand in the action of the scalar field (but see [22Jump To The Next Citation Point] for a possible solution to the latter shortcoming). The relations between this free function and Milgrom’s μ can be found in [145Jump To The Next Citation Point, 431Jump To The Next Citation Point] (see also Section 6.2), the detailed structure of null and timelike geodesics of the theory in [431Jump To The Next Citation Point], the analysis of the parametrized post-Newtonian coefficients (including the preferred-frame parameters quantifying the local breaking of Lorentz invariance) in [173, 372, 391Jump To The Next Citation Point, 450], solutions for black holes and neutron stars in [244, 245, 247, 246, 374, 438, 439], and gravitational waves in [216, 214, 215, 373]. It is important to remember that TeVeS is not equivalent to GR in the strong regime, which is why it can be tested there, e.g., with binary pulsars or with the atomic spectral lines from the surface of stars [122], or other very strong field effects50. However, these effects can always generically be suppressed (at the price of introducing a Galileon type term in the action [22Jump To The Next Citation Point]), and such tests would never test MOND as a paradigm. It is by testing gravity in the weak field regime that MOND can really be put to the test.

Finally, let us note that TeVeS (and its generalization) has been shown to be expressible (in the “matter frame”) only in terms of the physical metric gμν, and the vector field Uμ [513Jump To The Next Citation Point], the scalar field being eliminated from the equations through the “unit-norm” constraint in terms of the Einstein metric &tidle;gμνU U = − 1 μ ν, leading to gμνU U = − e− 2ϕ μ ν. In this form, TeVeS is sometimes thought of as GR with an additional “dark fluid” described by a vector field [503].

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