Homothetic Vector Field Research Articles

Symmetries of the nontwisting type-N solutions with cosmological constant

The conformal, homothetic, and isometric symmetries of all nontwisting type-N solutions are established. All solutions with λ allow at most the existence of two Killing vectors. All vacuum solutions, except the Robinson metric, permit maximally the existence of two (isometric and homothetic) symmetries. The Robinson solutions are the only ones which allow conformal symmetries.

Spacetimes admitting a vector field whose inner product with the Riemann tensor is zero

The equation vμRμναβ = 0 arises in various places in general relativity, in particular as the integrability conditions of the equations ℒvg = 2φg, where φ is a constant. These are the equations of a homothetic vector field v, with a zero homothetic bivector (dv = 0) in some space-time with metric g, and Rμναβ are the components of the Riemann tensor of that metric in some frame. In this paper the equation vμ Rμναβ = 0 is examined and the components of the Riemann tensor for the spacetimes which admit nonzero solutions vμ of this equation are given. The Petrov types of the Weyl tensors of these spacetimes are listed and, as a result, a correction is then made to a theorem in a paper by Collinson and Fugère about the Petrov types of spacetimes, which admit the type of separation that they require of the Hamilton–Jacobi equation for these spacetimes.

Einstein spaces and homothetic motions. II

Algebraically special, nonflat vacuum Einstein spaces with an expanding and/or twisting geodesic principal null congruence are considered. These spaces are assumed to possess locally a homothetic symmetry as well as an isometry (one Killing vector). Nine metrics are obtained, six of which are twisting Petrov type II.

Einstein spaces and homothetic motions. I

Algebraically special, nonflat vacuum Einstein spaces with an expanding and/or twisting geodesic principal null congruence are considered. These spaces are assumed to possess locally a homothetic symmetry as well as two or more Killing vectors. All metrics of such spaces are determined along with the form of the homothetic Killing vector admitted. All but one of the metrics are twist free. It is proved that two of the NUT metrics do not admit a homothetic motion.

Symmetry mappings in Einstein-Maxwell space-times

General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.

Homothetic motions with null homothetic bivectors in general relativity

It is shown that if a nonflat vacuum space-time admits a homothetic vector field with a null homothetic bivector then that space-time is algebraically special. If that homothetic vector field is a nontrivial one (not a Killing one) then the space-time is Petrov type III orN.

Homothetic motions in general relativity

Properties of homothetic or self-similar motions in general relativity are examined with particular reference to vacuum and perfect-fluid space-times. The role of the homothetic bivector with componentsH [a;b] formed from the homothetic vectorH is discussed in some detail. It is proved that a vacuum space-time only admits a nontrivial homothetic motion if the homothetic vector field is non-null and is not hypersurface orthogonal. As a subcase of a more general result it is shown that a perfect-fluid space-time cannot admit a nontrivial homothetic vector which is orthogonal to the fluid velocity 4-vector.