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Zbl 0786.53014
Mars, Marc; Senovilla, José M.M.
Geometry of general hypersurfaces in spacetime: junction conditions.
(English)
[J] Classical Quantum Gravity 10, No.9, 1865-1897 (1993). ISSN 0264-9381; ISSN 1361-6382/e

The authors study the geometry of a general hypersurface $\Sigma$ 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 $TM = \text{span}\{l\}\oplus T\Sigma$. They find a condition for the volume form on $\Sigma$ 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 $\Sigma$ which is inverse to the restriction of the contravariant metric $g\sp{ab}$ of spacetime to $\Sigma$. They establish how -- for a given rigging -- these connections relate. Finally, they derive junction conditions for joining pieces of spacetime across general hypersurfaces.
[M.Kriele (Berlin)]
MSC 2000:
*53B30 Lorentz metrics, indefinite metrics
53B25 Local submanifolds

Keywords: rigging; general hypersurface; connections; volume form; Gauss-- Codazzi equations; metric connection; spacetime; junction conditions

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