On space–time carrying a total hyperbolic skew symmetric Killing vector field
✍ Scribed by I. Mihai; R. Rosca; L. Verstraelen
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 94 KB
- Volume
- 40
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
The complex vectorial formalism on a general space-time (M, g) was constructed by Cahen, Debever and Defrise. This formalism is based on the local isomorphism I : L(4) → SO 3 (C), where L(4) is the four-dimensional Lorentz group acting on the tangent spaces T p M and SO 3 (C) is the three-dimensional complex rotation group. In this framework, the congruence of Debever plays a distinguished role. Its properties determine the general space-time M, in terms of Petrov's classification.
In the present paper, we assume that any hyperbolic vector field X on M is a skew symmetric Killing vector field having a spatial vector field Y as generative. The existence of such a vector field X is determined by an exterior differential system in involution. It is shown that M is the local Riemannian product M = M h × M s , where M h (resp. M s ) is a totally geodesic and totally pseudo-isotropic hyperbolic (resp. spatial) surface (the Gauss map is ametric). Any such M is a space-time of type D in Petrov's classification.
It is proved that the congruence of Debever is of electric type; in particular, it is geodesic and shear 1-free. Other geometric properties on such a general space-time are obtained.