Warped Product Space-times

Warped Product Space-times

Xinliang An\AffiliationUniversity of Toronto, Toronto, ON; xinliang.an@utoronto.ca \qquad Willie Wai Yeung Wong\AffiliationMichigan State University, East Lansing, MI; wongwwy@member.ams.org
GR, DG, WarpedProduct, BirkhoffTheorem, HawkingMass, KodamaVector, EinsteinMatter

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.

1 Introduction

In this paper we report on our investigation of pseudo-Riemannian warped product manifolds with 2 dimensional base. We prove, in this context, a microcosm of results that either generalize those previously were shown in the setting of a spherically symmetric ansatz in general relativity, or that specialize (with much simpler proofs) results known in greater generality.

Our two main observations are:

  1. Many Lorentzian geometric results relating to spherically-symmetric space-times do not, in fact, make full use of the spherical symmetry ansatz. In particular, they can be reproduced in the warped product context with essentially arbitrary fibers. These include the rigidity portion of Birkhoff’s theorem, the realization of the Hawking energy as a dual potential of a geometric vector field, as well as much of causal theory under the dominant energy condition.

  2. When coupled to matter fields in the general relativistic setting, the warped product ansatz often impose strong restrictions on the allowable solutions, reducing the theory to be essentially similar to the spherically symmetric case. This include our novel generalized Birkhoff theorem classifying warped product electrovacuum solutions (previous results only treat vacuum solutions), as well as our observation that in the case of scalar field and fluid matter, the warped product ansatz forces the fiber manifold to be Einstein (but not necessarily homogeneous).

One of our original goals in the present paper was to develop a framework suitable for future investigations of the dynamical properties of solutions to Einstein’s field equations, under the simplifying warped-product ansatz. The spherical symmetry ansatz has led to remarkable discoveries [Christ1995, Christ1999, Daferm2003, Daferm2005, Daferm2012, BurLef2014], and similarly planar or toroidal symmetries [LefTch2011, LefSmu2010]. There has been furthermore recent results concerning general surface symmetric spacetimes [DafRen2016]. One of our conclusions, unfortunately for our original motivation, is that for some common matter models, the warped-product ansatz is no more general than the surface symmetric ansatz.

Below we first introduce the warped product ansatz and then summarize the main results of this paper.

1.1 Pseudo-Riemannian warped products

Let be a two-dimensional pseudo-Riemannian manifold and be a pseudo-Riemannian manifold with dimension . We consider the warped product where is a positive real-valued function on . The warped product metric on is given by


where by an abuse of notation we identify and with their pull-backs onto via the canonical projections. We refer to as either the warping function or, following the relativity literature, the area radius.

Throughout we will use

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