Lorentzian warped products and static space-times

Abstract

Let (M, g) be a Lorentzian manifold, (H, h) a Reimannian manifold, and let f: H [right arrow] (0, [infinity symbol] be an arbitarary smooth function. Then the product manifold M x H with Lorentzian metric g = (f[suprscript 2] g) [omega] h is called a Lorentzian warped product and denoted by M[subscript f] x H. In the case (a,b)[subscript f x H, -[infinity symbol] [less than or equal to] a < b [less than or equal to] [infinity symbol], with metric g = (-f[suprscript 2]dt[superscript 2] [omega] h, the metric is static and the warped product is called a standard static spece-time, Scharzschild space-time, universal anti-de Sitter space-time, and the Einstein static universe.