Relativity: the special theory. By J. L. Synge. Pp. 450. 76s. 1956. (North Holland Publishing Co. Amsterdam)

We find all higher dimensional solutions of Einstein-Maxwell theory that are the topological product of two manifolds of constant curvature. These solutions include the higher dimensional Nariai, Bertotti-Robinson and anti-Nariai solutions and the anti--de Sitter Bertotti-Robinson solutions with toroidal and hyperbolic topology (Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski-Hacyan solutions). We give explicit results for any dimension $Dg~4.$ These solutions are generated from the appropriate extremal limits of the higher dimensional near-extreme black holes in de Sitter and anti--de Sitter backgrounds. Thus, we also find the mass and charge parameters of higher dimensional extreme black holes as a function of the radius of the degenerate horizon.

Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions

A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson and type D (anti-)Nariai or Plebanski–Hacyan spaces, and their generalizations. All possibilities are discussed and interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type D pure radiation or explicit type II electrovacuum waves in the (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein–Maxwell fields with a non-vanishing cosmological constant.

Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes

Cosmological models for Bianchi type III and Kantowski-Sachs space-times within the framework of Lyra geometry are obtained. The physical behavior of the models is also discussed.

Bianchi type III and Kantowski-Sachs cosmological models in Lyra geometry

An exact Bianchi-type II cosmological model with matter and an electromagnetic field

We define “locally isotropic” spaces, as spaces in which there exists, in the tangent space at each pointP, a subgroupA (P) (of dimension at least 1) of the Lorentz groupL

 +
 ↑
 , leaving the Riemann tensor and its 2 first covariant derivatives invariant; the subgroupsA(P) are assumed to be conjugate inL

 +
 ↑
 . These spaces admit a group of local isometriesG. IfI

 P
 denotes the subgroup ofG leavingP fixed, thendA (P)=I

 P
 . All spaces of petrov type D, admitting local isotropy are determined.

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Michel Cahen

Papers

Lorentzian $4$ dimensional manifolds with ``local isotropy''