Einstein–Maxwell equations and the groups of homothetic motion

Abstract

The Einstein–Maxwell field equations for a source-free, non-null electromagnetic field are studied under the assumption of admitting a nontrivial homothetic conformal motion, generating a homothetic bivector which is also non-null. It is shown that a space-time, whether vacuum or not, cannot admit a non-null homothetic vector field as a geodesic tangent. It is also shown that if a common principle null direction of the electromagnetic and the homothetic bivectors is geodesic and shear free, then the space-time must be algebraically special. Furthermore, it is found that if the electromagnetic and the homothetic bivectors have common principal null directions, then the vector field generating the homothetic bivector cannot be hypersurface orthogonal, unless it is a Killing vector field. Moreover, if these common principle null directions are also geodesics, then there exists no solution to the combined Einstein–Maxwell equations, unless, the non-null homothetic vector field is a Killing vector field. Finally, an example of a space-time admitting a non-null, nontrivial homothetic vector field generating a homothetic bivector which is also non-null is given.