A class of nonfactorisable spacetimes in five dimensions

A one–parameter family of five dimensional nonfactorisable spacetimes is discussed. These spacetimes have Ricci scalar, R = 0. The warp factors, in general, are asymmetric (i.e. different warp factors for the spatial and temporal parts of the 3–brane section), though, with a specific choice of a parameter (denoted here by ν), the asymmetry can be avoided. Over a range of values of ν the energy conditions can be satisfied.The matter stress energy required to support these spacetimes decays for large values of the fifth (extra) coordinate. The projection of this stress–energy (for the symmetrically warped geometry) on the 3–brane yields an effective cosmological constant. We conclude with brief discussions on spacetimes with constant Ricci scalar and extensions to diverse dimensions. ∗Electronic address: [email protected] 1 Ever since Kaluza and Klein [1] investigated five dimensional General Relativity (GR) and derived its equivalence with four dimensional GR coupled to electromagnetism, interest in extra dimensions have drawn particle theorists and relativists towards the construction of newer models [2]. In the later part of the last century, extra dimensions arose in the context of superstrings [3] where their presence is unavoidable. Few years ago, the notion of nonfactorisable spacetimes with an extra dimension, was initiated through the work of Randall and Sundrum [4]. Nonfactorisability of a line element in this context essentially implies that the four dimensional part has a dependence on the fifth coordinate through an overall conformal factor. It was shown that there exists an exact solution of the five dimensional Einstein equation with a negative cosmological constant, where the four dimensional part is Minkowski space with a conformal factor dependent on the fifth coordinate (denoted henceforth as σ). This ‘warping’ of the four dimensional section results in a constant scaling of Minkowski space–the scale factor taking on different values with different choices of σ. The Randall–Sundrum solution is given as :