Proposed Co-ordinate System in which gaα=0 and the Line Element for Plane Gravitational Waves of Type Z=(t1-t2+t3)/(√3z)

Abstract

For the study of t/z type plane gravitational waves, we have considered the six-dimensional space-time with three time axes. We have chosen the plane wave of type Z = (t1-t2+t3)/(√3z) and made the essential investigations in the paper [6]. Adopting the same notations and results as in above paper, and using the transformations given in the proof of Theorem 1 below, we have proved that the co-ordinate system in which g_aa=0, (a=1,2 ; α = 3,4,5,6) can be obtained in the six-dimensional space-time, which is our proposed co-ordinate system. In this paper, we have proved the three necessary theorems supporting the existence of such a co-ordinate system. The whole work is on the lines of H. Takeno (1961). The consequent line element in the form ds² = g_mndx^mdxⁿ is also obtained for m,n = 1,…, 6.