Warped Product Space-times

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.

References in corpus (6)

• A generalization of Hawking's black hole topology theorem to higher dimensions
• Black holes without spacelike singularities
• Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes
• Spherically symmetric spacetimes with a trapped surface
• Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter
• The formation of trapped surfaces in spherically-symmetric Einstein-Euler spacetimes with bounded variation

Cited by in corpus (6)

• Properties of the Null Distance and Spacetime Convergence
• Extension of Kodama vector and quasilocal quantities in three-dimensional axisymmetric spacetimes
• IDEAL characterization of higher dimensional spherically symmetric black holes
• Geometrical origin of the Kodama vector
• Non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics: Birkhoff-type results in linearized gravity
• Warped Product Einstein Manifolds in Four Dimensions