General Relativity and Gravitation

, Volume 18, Issue 8, pp 781–804 | Cite as

Teleparallelism as a universal connection on null hypersurfaces in general relativity

  • P. O. Mazur
  • L. M. Sokolowski
Research Articles


We show that there exists a close relationship between inner geometry of a null hypersurfaceN3 and the Newman-Penrose (NP) spin coefficient formalism. Projecting the null complexNP tetrad ontoN3 we get two triads of basis vectors inN3. Inner geometry ofN3 is based on the assumption that these vectors are parallelly transported along the surface; this gives rise to the teleparallel connection as a metric nonsymmetric affine connection. The gauge freedom for the choice of the basis triads is given by the isotropy subgroup of the local Lorentz group leaving invariant the direction of the null generators ofN3, and teleparallelism is determined by the equivalence class of the basis triads with respect to the global gauge group. Nine of the twelve NP coefficients are identified as the triad components of the torsion and the second fundamental form ofN3. The resulting generalized Gauss-Codazzi equations are identical to 9 of the NP equations, i.e., to the half of the Ricci identities. This result gives a geometrical meaning to the entire formalism. Finally we present a general proof of Penrose's theorem that the shear of the null generators ofN3 is the only initial null datum for a gravitational field onN3.


Isotropy Subgroup Affine Connection Null Hypersurface Global Gauge Coefficient Formalism 


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Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • P. O. Mazur
    • 1
  • L. M. Sokolowski
    • 2
  1. 1.Institute for Theoretical PhysicsUniversity of CaliforniaSanta Barbara
  2. 2.Astronomical ObservatoryJagellonian UniversityKrakowPoland

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