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Geometry of general hypersurfaces in spacetime: junction conditions. (English) Zbl 0786.53014
The authors study the geometry of a general hypersurface Σ which is embedded in a spacetime. They choose a transverse vector field l which they call a ‘rigging’ and construct then two (in general different) connections on this hypersurface. The first connection is induced by the splitting TM=span{l}TΣ. They find a condition for the volume form on Σ induced by l to be parallel with respect to this condition. They also derive Gauß and Codazzi equations for their setup. The second connection is a metric connection. Here they use the rigging in order to single out a special metric on Σ which is inverse to the restriction of the contravariant metric g ab of spacetime to Σ. They establish how – for a given rigging – these connections relate. Finally, they derive junction conditions for joining pieces of spacetime across general hypersurfaces.
MSC:
53B30Lorentz metrics, indefinite metrics
53B25Local submanifolds