On Invariants ofm-Vector in Lorentzian Geometry

Abstract

LetGbe the groupM (n, 1)generated by all pseudo-orthogonal transformations and translationsof Lorentzian spaceEn1orG = SM (n, 1)is the subgroup ofM (n, 1)generated by rotations andtranslations ofEn1. We describe the correlations between Gram determinantdetG(x1, . . . , xm)of thesystem{x1, . . . , xm}and the number of linearly independent null vectors in the system{x1, . . . , xm}.Using methods of invariant theory and these results, the system of generators of the polynomialring of allG-invariant polynomial functions of vectorsx, x2, . . . , xminEn1is obtained for groupsG = M (n, 1)andG = SM (n, 1).