Warped Product Space-times

TL;DR

This paper explores the geometric and physical properties of warped product space-times in relativity, providing new theorems, analyzing energy conditions, and constructing cosmological solutions with diverse spatial geometries.

Contribution

It offers a systematic geometric framework for warped product space-times, generalizes Birkhoff-type theorems, and constructs novel cosmological solutions with non-isotropic spatial geometries.

Findings

Generalized Birkhoff-type theorems for warped products

Non-existence results related to Yamabe class of fibers

Construction of cosmological solutions with non-isotropic spatial geometry

Abstract

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking-Galloway-Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein-Euler equations whose spatial geometry is generally not isotropic.