Conformal Invariance and Warped 5-Dimensional Spacetimes

Abstract

We show that the Einstein field equations for a five-dimensional warped spacetime, where only gravity can propagate into the bulk, determine the dynamical evolution of the warp factor of the four-dimensional brane spacetime. This can be explained as a holographic manifestation. The warped 5D model can be reformulated by considering the warp factor as a dilaton field ($ω$) conformally coupled to gravity and embedded in a smooth $M_4 \otimes R$ manifold. On the brane, where the U(1) scalar gauge fields live, the dilaton field manifests itself classically as a warp factor and enters the evolution equations for the metric components and matter fields. We write the Lagrangian for the Einstein-scalar gauge fields in a conformal invariant setting. 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 in the projected Einstein tensor. These terms can be interpreted as a constraint in order to maintain conformal invariance. By considering the dilaton field and Higgs field on equal footing on small scales, there will be no singular behavior, when $ω\rightarrow 0$ and one can deduce constraints to maintain regularity of the action. Our conjecture is that $ω$, alias warp factor, has a dual meaning. At very early times, when $ω\rightarrow 0$, 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. 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 $n$.