Covariant derivatives on null submanifolds. (English) Zbl 1235.83016
Summary: The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch’s work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.
MSC:
83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
53Z05 | Applications of differential geometry to physics |
Keywords:
Null hypersurfaces; Null submanifolds; Covariant derivative; Conformal transformation; Asymptotically flat spacetime; Killing normal vectorReferences:
[1] | Geroch R.: Asymptotic structure of space-time. In: Esposito, F.P., Witten, L. (eds) Asymptotic Structure of Space-Time, pp. 1–105. Plenum Press, New York and London (1976) |
[2] | Spivak M.: A Comprehensive Introduction to Differential Geometry. Vol. 3, 2nd edn. Publish or Perish, Houston (1979) · Zbl 0439.53003 |
[3] | Duggal K.L., Bejancu A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer, Dordrecht (1996) · Zbl 0848.53001 |
[4] | Hickethier, D.: Covariant derivatives on null submanifolds. PhD Thesis, Oregon State University (2010) (Available online at: http://hdl.handle.net/1957/19547 .) · Zbl 1235.83016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.