Uniqueness of the metric line element in dimensionally reduced theories

Abstract

The conformal ambiguity in the definition of the four-dimensional reduced metric arises in any theory whose higher-dimensional metric contains scalar fields. If d=5 the reduced metric can be uniquely determined by its relation to the total five-dimensional gravitational energy. For d)5 the notion of gravitational energy is as yet undefined and for one scalar field several authors have used various arguments in order to single out one (and the same in all cases) reduced metric. It is shown that a unique choice of the reduced metric (i.e. a unique conformal factor) follows from the positive-energy theorem in four dimensions. For a generic case when the spacetime evolution of the internal homogeneous space is described by n scalar fields, the author proves the existence of an analogous conformal factor, ensuring that the total kinetic energy in four dimensions satisfies the dominant energy condition.

Gravitational wave astronomy: The definitive test for the “Einstein frame versus Jordan frame” controversy

“…The "Einstein frame versus Jordan frame" controversy started because some authors claimed that scalar-tensor gravity is unreliable in the Jordan frame, leading to the problem of negative kinetic energies [23,24,25]. On the other hand, the Einstein frame version of scalar-tensor gravity, which is obtained by the conformal rescaling of the metric [26,27,28,29] g ab = ϕg ab (1) and a nonlinear scalar field redefinition [26,28] …”

“…Thus, the fundamental question is: which is the physical frame of observations? Using of conformal transformations to perform analyses in the Einstein frame abounds in the literature, with divergence of opinions between different authors [5,6,17,22,23,24,25,28,29]. The motion in the Einstein frame is not geodesic [26], a key point which strongly endorses deviations from equivalence principle and non-metric gravity theories in the Einstein frame [6,26,31,32].…”

“…The first of Eqs. (24) shows that perturbations have the speed of light, the second the transverse effect of the field.…”

Backreaction mechanism in multifluid and extended cosmologies

“…In addition, the relation between averaged quantities in the two frames remains unclear since the domain volumes in different frames don't have a clear correspondence either. On the other hand, besides the mathematical tool that the conformal transformation may provide, the physical meaning of both frames has to be analysed carefully, where the positivity of the energy and the behaviour of the ground state may play a crucial role to discriminate between frames, as pointed out in Refs [51,52].…”

“…Moreover, as pointed out in Refs. [51,52], also the physical meaning of each frame can not be easily established. An analysis of the ground state and the positivity of the energy in both frames is an important tool to determine whether the frames, and in particular the Jordan one, are well defined.…”