Projective collineations in Einstein spaces

Abstract

Spacetimes admitting a group of (local) projective collineations are considered. In an n-dimensional proper Einstein space it is shown that any vector field xi i generating a proper projective collineation (that is one which is not an affine collineation) is the gradient of a scalar field phi (up to the addition of a Killing vector field). Then a four-dimensional Einstein spacetime admitting a proper projective collineation is shown to have constant curvature. For an n-dimensional space of non-zero constant curvature, the scale field phi satisfies a system of third-order linear differential equations. The complete solution of this system is found in closed form and depends on (n+1)(n+2)/2 arbitrary constants. All gradient vector fields xi i generating projective collineations are found explicitly and together with the n(n+1)/2 Killing vector fields generate a Lie algebra of dimension n(n+2).