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Direct Research Journals
Vol.3 (1), pp. 1-6, February 2015
ISSN 2354-4155

LIGHTLIKE PARALLEL VECTOR FIELDS AND EINSTEIN’S EQUATIONS OF GRAVITY: A PARTIAL SOLUTION
Article Number: DRJA17086173
DOI:

Original Research Article


Muremyi Roger


I.C.K (Catholic Institute of Kabgayi), in Rwanda.


date Received: November 12, 2015     date Accepted: December 23, 2014     date Published: February 28, 2015

Abstract

The purpose of this paper was to find out the lightlike parallel vector field and Einstein’s equation of gravity, specifically a partial solution. The pseudo –Riemannian manifold and gravitational field through Einstein’s equation have been used , the curvature tensor in 4 dimensional space and the  Christoffel symbols in 3- dimensional are equal to Christoffel symbols in 4- dimensional space  by using contraction properties of  the tensors and as the results  the existence of lightlike parallel vector field implies that the space time is flat, this leads us to non existence of  the lightlike parallel vector field in a non trivial gravitational field.


Key words:  Tensors, Parallel vector field, differential manifold, smooth manifold, Riemannian connection, Christoffel symbols.