Newtonian gravity from the Higgs field: the sublimation of aether

We illustrate why a space-time structure as in General Relativity is not in contradiction with a dynamical origin of gravity from a scalar field. Further, we argue that the recently discovered gap-less mode of the singlet Higgs field represents the most natural dynamical agent of Newtonian gravity. 1. Following the original induced-gravity models [1], one may attempt to describe gravity as an effective force induced by the vacuum structure, in analogy with the attractive interaction among electrons that can only exist in the presence of an ion lattice. Looking for such a ‘dynamical’ mechanism one should also be able to account for those peculiar ‘geometrical’ properties at the base of the classical space-time description of General Relativity. The idea that gravitation requires somehow an underlying scalar field arises naturally when considering the non-trivial ambiguity [2] associated with the operative definition of the infinitesimal transformation to the reference frame of a freely falling observer. Indeed, within General Relativity, the Equivalence Principle is used to relate a frame in which a constant force is acting to a frame in a Riemannian space uniformly accelerating with respect to an inertial frame. An alternative and equivalent description [2] is obtained by considering infinitesimal acceleration transformations of the conformal group. In this case, the equivalence is not with a frame in a Riemannian space but with a frame in a more general Weyl space, i.e. a space where the affine connections are not given by the Christoffel symbols [2] since at each point x, besides a symmetric tensor gμν(x), there is a vector κμ(x). One usually considers the Weyl space in relation to the original Weyl’s theory including electromagnetism. That theory predicts that the length of a rod depends on his history, which may seem physically unteinable. However, the idea of a Weyl space cannot be rejected a priori so that one is still faced with the interpretation of the infinitesimal acceleration transformation and as such of the basic ingredient underlying any space-time description in gravitational fields. In this sense, the formal foundation of General Relativity contains an arbitrary assumption, at least beyond the weak-field limit where it has been experimentally tested. In an alternative approach, one could have argued as follows. A priori, each of the two possibilities represents an arbitrary choice. Therefore, it may be natural to solve the ambiguity by adopting the more general conformal-transformation point of view but restricting to that special class of Weyl spaces that are equivalent to Riemannian spaces. This equivalence can be expressed as |g| ≡ 1 and κμ 4π = −∂μσ̃ (1) where σ̃ is a scalar field. In this case, in fact, the Weyl connections constructed with gμν and κμ are the Christoffel symbols of the metric tensor [2] ĝμν = e gμν (2)