On Metric Perturbations in Brane-World Scenarios

In this note we reconsider linearised metric perturbations in the one-brane Randall-Sundrum Model. We present a simple formalism to describe metric perturbations caused by matter perturbations on the brane and remedy some misconceptions concerning the constraints imposed on the metric and matter perturbations by the presence of the brane. An interesting alternative to standard Kaluza-Klein compactification is to view our 4-dimensional Universe as a 3-brane embedded in a bigger space with large, or infinite extra dimensions, but such that matter fields are localised on the brane [1]-[12]. Considerable effort has been devoted to investigate possible observable consequences of such scenarios. For this one has to determine how the gravitational dynamics, including matter is affected by the extra dimensions in which gravity can propagate. The linearised metric perturbation in Brane-World Scenarios have been considered by numerous authors [8, 9], [13]-[24]. In [13, 16] the authors considered linear perturbations of a Minkowski brane in coordinates in which the four-dimensional metric is transverse-traceless. In this gauge the brane appears “bent” in the presence of matter, that is, the coordinates are discontinuous on the brane. In [17] and [22] the same analysis was carried out in alternative gauges for which the coordinates are continuous across the brane. On the other hand linear perturbations were analysed for an expanding Universe on the brane [14]-[24] and some cosmological consequences were analysed using a generalisation of the 3 + 1 decomposition of the metric perturbations [25, 26, 27]. In this note we present a simple generalisation of the formalism [22] which allows us to relate and correct some of the results obtained