Paper detail

Pseudo-Riemannian Ricci-flat and Flat Warped Geometries and New Coordinates for the Minkowski metric

It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann curvature, only if its base is flat and its fibre is maximally symmetric, i.e. of constant curvature. Applying this result systematically all possible warped forms of the Euclidean, Minkowski, and flat metrics of arbitrary signature can be obtained in closed form up to disjoint diffemorphisms of the base and fiber metrics. In particular, we obtained four new time-dependent forms of the Minkowski metric in four dimensions in addition to reproducing all of its known warped forms.