Warped 5D spacetimes, cosmic strings and conformal invariance

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 BondiMarder spacetime, written as g(mu nu) = omega 2 (g) over bar (mu nu), where. is the conformal factor and (g) over bar (mu nu) the 'un-physical' spacetime. Curvature could then be generated from Ricci-flat (g) over bar (mu nu) by suitable dilaton fields and additional gauge freedom. If we try to match this vacuum solution onto 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 behavior of the model. This warp factor is determined by the Einstein field equations for the five-dimensional warped space, where only gravity can propagate into the bulk. 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 circle times R manifold. It is conjectured that the four-dimensional conformal factor is related to the dilaton field of the five-dimensional counterpart model. The dilaton field (alias warp factor), has a dual meaning. At very early times, when omega -> 0, it describes the small-distance limit, while at later times it is 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. These terms can be interpreted as a constraint in order to maintain conformal invariance and the tracelessness of the energy-momentum tensor could then be maintained by a contribution from the bulk. By considering the dilaton field and Higgs field on equal footing on small scales, there will be no singular behavior, when omega -> 0 and one can deduce constraints to maintain regularity of the action. We also present a numerical solution of the model and calculate the (time-dependent) trace-anomaly. The solution depends on the mass ratio of the scalar and gauge fields, the parameters of the model and the vortex charge n.