# Foliations of lightlike hypersurfaces and their physical interpretation

Open Mathematics (2012)

- Volume: 10, Issue: 5, page 1789-1800
- ISSN: 2391-5455

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topKrishan Duggal. "Foliations of lightlike hypersurfaces and their physical interpretation." Open Mathematics 10.5 (2012): 1789-1800. <http://eudml.org/doc/268944>.

@article{KrishanDuggal2012,

abstract = {This paper deals with a family of lightlike (null) hypersurfaces (H u) of a Lorentzian manifold M such that each null normal vector ℓ of H u is not entirely in H u, but, is defined in some open subset of M around H u. Although the family (H u) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (H u) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic H u. Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons.},

author = {Krishan Duggal},

journal = {Open Mathematics},

keywords = {Lightlike hypersurface; Totally umbilical hypersurface; Spacetime manifold; Isolated horizons; Black hole; light-like hypersurface; totally umbilical hypersurface; space-time manifold; isolated horizons; black hole},

language = {eng},

number = {5},

pages = {1789-1800},

title = {Foliations of lightlike hypersurfaces and their physical interpretation},

url = {http://eudml.org/doc/268944},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Krishan Duggal

TI - Foliations of lightlike hypersurfaces and their physical interpretation

JO - Open Mathematics

PY - 2012

VL - 10

IS - 5

SP - 1789

EP - 1800

AB - This paper deals with a family of lightlike (null) hypersurfaces (H u) of a Lorentzian manifold M such that each null normal vector ℓ of H u is not entirely in H u, but, is defined in some open subset of M around H u. Although the family (H u) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (H u) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic H u. Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons.

LA - eng

KW - Lightlike hypersurface; Totally umbilical hypersurface; Spacetime manifold; Isolated horizons; Black hole; light-like hypersurface; totally umbilical hypersurface; space-time manifold; isolated horizons; black hole

UR - http://eudml.org/doc/268944

ER -

## References

top- [1] Akivis M.A., Goldberg V.V., On some methods of construction of invariant normalizations of lightlike hypersurfaces, Differential Geom. Appl., 2000, 12(2), 121–143 http://dx.doi.org/10.1016/S0926-2245(00)00008-5 Zbl0965.53022
- [2] Arnowitt R., Deser S., Misner C.W., The dynamics of general relativity, In: Gravitation, Wiley, New York, 1962, 227–265 Zbl1152.83320
- [3] Ashtekar A., Beetle C., Fairhurst S., Isolated horizons: a generalization of black hole mechanics, Classical Quantum Gravity, 1999, 16(2), L1–L7 http://dx.doi.org/10.1088/0264-9381/16/2/027 Zbl0947.83027
- [4] Ashtekar A., Galloway G.J., Some uniqueness results for dynamical horizons, Adv. Theor. Math. Phys., 2005, 9(1), 1–30 Zbl1100.83016
- [5] Ashtekar A., Krishnan B., Dynamical horizons and their properties, Phys. Rev. D, 2003, 68(10), #104030 http://dx.doi.org/10.1103/PhysRevD.68.104030 Zbl1071.83036
- [6] Beem J.K., Ehrlich P.E., Global Lorentzian Geometry, Monogr. Textbooks Pure Appl. Math., 67, Marcel Dekker, New York, 1981
- [7] Bejancu A., Duggal K.L., Degenerated hypersurfaces of semi-Riemannian manifolds, Bul. Inst. Politehn. Iaşi Secţ. I, 1991, 37(41)(1–4), 13–22 Zbl0808.53015
- [8] Carter B., Extended tensorial curvature analysis for embeddings and foliations, In: Geometry and Nature, Madeira, July 30–August 5, 1995, Contemp. Math., 203, American Mathematical Society, Providence, 1997, 207–219 Zbl0877.57013
- [9] Damour T., Black-hole eddy currents, Phys. Rev. D, 1978, 18(10), 3598–3604 http://dx.doi.org/10.1103/PhysRevD.18.3598
- [10] Duggal K.L., Bejancu A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Math. Appl., 364, Kluwer Academic, Dordrecht, 1996 Zbl0848.53001
- [11] Duggal K.L., Jin D.H., Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, Hackensack, 2007 http://dx.doi.org/10.1142/6449
- [12] Galloway G.J., Maximum principles for null hypersurfaces and null splitting theorem, Ann. Henri Poincaré, 2000, 1(3), 543–567 http://dx.doi.org/10.1007/s000230050006 Zbl0965.53048
- [13] Gourgoulhon E., Jaramillo J.L., A 3 + 1-perspective on null hypersurfaces and isolated horizons, Phys. Rep., 2006, 423(4–5), 159–294 http://dx.doi.org/10.1016/j.physrep.2005.10.005
- [14] Hawking S.W., Ellis G.F.R., The Large Scale Structure of Space-Time, Cambridge Monogr. Math. Phys., 1, Cambridge University Press, London-New York, 1973
- [15] Kossowski M., The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space, Trans. Amer. Math. Soc., 1989, 316(1), 369–383 http://dx.doi.org/10.1090/S0002-9947-1989-0938920-1 Zbl0691.53046
- [16] Krishnan B., Fundamental properties and applications of quasi-local black hole horizons, Classical Quantum Gravity, 2008, 25(11), #114005 http://dx.doi.org/10.1088/0264-9381/25/11/114005
- [17] Kupeli D.N., Singular Semi-Riemannian Geometry, Math. Appl., 366, Kluwer, Dordrecht, 1996 Zbl0871.53001
- [18] Lewandowski J., Spacetimes admitting isolated horizons, Classical Quantum Gravity, 2000, 17(4), L53–L59 http://dx.doi.org/10.1088/0264-9381/17/4/101 Zbl0968.83010
- [19] Swift S.T., Null limit of the Maxwell-Sen-Witten equation, Classical Quantum Gravity, 1992, 9(7), 1829–1838 http://dx.doi.org/10.1088/0264-9381/9/7/014