Conformal invariant gravity coupled to a gauged scalar field and warped spacetimes

Abstract

Abstract We investigate the conformal invariant Lagrangian of the self-gravitating U(1) scalar-gauge field on the time-dependent Bondi–Marder axially symmetric spacetime. By considering the conformal symmetry as exact at the level of the Lagrangian and broken in the vacuum, a consistent model is found with an exact solution of the vacuum Bondi–Marder spacetime, written as g μ ν = ω 2 g μ ν , where ω is the conformal factor and g μ ν the ‘un-physical‘ spacetime. Curvature could then be generated from Ricci-flat g μ ν by suitable dilaton fields and additional gauge freedom. If we try to match this vacuum solution on the interior vortex solution of the coupled Einstein-scalar-gauge field, we need, besides the matching conditions, constraint equations in order to obtain a topological regular description of the small-scale behaviour of the model. Probably, one needs the five-dimensional warped counterpart model, where the warp factor determines the large-scale behaviour of the model. The warped five-dimensional model can be reformulated by considering the warp factor as a dilaton field conformally coupled to gravity and embedded in a smooth M 4 ⊗ R manifold. Dark energy and the cosmological constant could then be emergent in this model. The dilaton field has a dual meaning. At very early times, when ω → 0 , it describes the small-distance limit, while at later times it manifests itself as a warp (or scale) factor that determines the dynamical evolution of the universe. However, as expected, the conformal invariance is broken (trace-anomaly) by the appearance of a mass term and a quadratic term in the energy–momentum tensor of the scalar-gauge field, arising from the extrinsic curvature terms of the projected Einstein tensor. By considering the dilaton field and Higgs field on equal footing on small scales, there will be no singular behaviour, when ω → 0 and one can deduce constraints to maintain regularity of the action.