Teleparallelism as a universal connection on null hypersurfaces in general relativity

Details Teleparallelism as a universal connection on null hypersurfaces in general relativity Teleparallelism as a universal connection on null hypersurfaces in general relativity

Aug 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986gregr..18..781m&link_type=abstract

General Relativity and Gravitation (ISSN 0001-7701), vol. 18, Aug. 1986, p. 781-804. NASA-supported research.

Computer Science

2

Null Hypothesis, Relativity, Surfaces, Degrees Of Freedom, Gauss Equation, Gravitational Fields, Space-Time Functions

Scientific paper

It is shown that a close relationship between the inner geometry of a null hypersurface N3 and the Newman-Penrose (NP) (1962, 1963) spin coefficient formalism exists. Projecting the null complex NP tetrad onto N3, two triads of basis vectors in N3 are obtained. The inner geometry of N3 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 of N3, 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 of N3. The resulting generalized Gauss-Codazzi equations are identical to nine of the NP equations, i.e., to the half of the Ricci identities. This result gives a geometrical meaning to the entire formalism. Finally a general proof of Penrose's theorem that the shear of the null generators of N3 is the only initial null datum for a gravitational field on N3 is presented.

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