M. Ziad

The classification of plane symmetric spacetimes by isometries

A complete classification of plane symmetric Lorentzian manifolds according to their additional isometries and metrics (or classes of metrics) is obtained by solving the Killing equations. We obtain all metrics (or classes of metrics), that admit the group of motions Gr (where r=3,4,5,6,7 and 10) containing SO(2)⨷R2, the minimal symmetry inherited by the plane symmetric manifolds.

The classification of static plane symmetric space–times according to their Ricci collineations

Static plane symmetric space–times are classified according to their Ricci collineations (RCs). Their relation with isometries of the space–times is established. Unlike the spherically symmetric space–times, some metrics are found for a nondegenerate Ricci tensor which have more RCs than isometries.

Static spherically symmetric space‐times with six Killing vectors

It had been proved earlier that spherically symmetric, static space‐times have ten, seven, six, or four independent Killing vectors (KV’s), but there are no cases in between. The case of six KV’s is investigated here. It is shown that the space‐time corresponds to a hyperboloid cross a sphere, reminiscent of Kaluza–Klein theory, with a compactification from four down to two dimensions. In effect, there is a unique metric for this space‐time corresponding to a uniform mass distribution over all space.

Homotheties of cylindrically symmetric static manifolds and their global extension

Cylindrically symmetric static manifolds are classified according to their homotheties and metrics. In each case the homothety vector fields and the corresponding metrics are obtained explicitly by solving the homothety equations. It turns out that these metrics admit homothety groups Hm , where m = 4,5,7,11. This classification is then used to identify the cylindrically symmetric static spaces admitting the local homotheties, which are globally prohibited due to their topological construction. Einsteins field equations are then used to identify the physical nature of the spaces thus obtained.

Spherically symmetric manifolds which admit five isometries

Spherically symmetric space–times which admit a five‐dimensional group of motions are studied and it is shown that it can never be locally a maximal group of motions but globally it could be a maximal group.

UNIQUENESS OF THE McVITTIE SOLUTION AS PLANE SYMMETRIC SOURCELESS ELECTROMAGNETIC FIELD SPACETIMES

It is proved that plane symmetric Lorentzian manifolds representing a sourceless electromagnetic field admit only a four-dimensional maximal symmetry group and consist of only the McVittie5 spacetime and its non-static analogue.4