Curvature properties of a special type of pure radiation metrics

Abstract

A spacetime denotes a pure radiation field if its energy momentum tensor represents a situation in which all the energy is transported in one direction with the speed of light. In 1989, Wils and later in 1997 Ludwig and Edgar studied the physical properties of pure radiation metrics, which are conformally related to a vacuum spacetime. In the present paper we investigate the curvature properties of special type of pure radiation metrics presented by Ludwig and Edgar. It is shown that such a pure radiation spacetime is semisymmetric, Ricci simple, R-space by Venzi and its Ricci tensor is Riemann compatible. It is also proved that its conformal curvature 2-forms and Ricci 1-forms are recurrent. We also present a pure radiation type metric and evaluate its curvature properties along with the form of its energy momentum tensor. It is interesting to note that such pure radiation type metric is Ein(3) and 3-quasi-Einstein. We also find out the sufficient conditions for which this metric represents a generalized pp-wave, pure radiation and perfect fluid. Finally we made a comparison between the curvature properties of Ludwig and Edgar's pure radiation metric and pp-wave metrics.

The Jordan algebras of Riemann, Weyl and curvature compatible tensors

“…Examples of K-compatible tensors were obtained by Shaikh et al (see for example [22,21]) starting from specific metrics. Bourguignon [1] proved that if b ij is a Codazzi tensor thenR jklm = R jkrs b r l b s m is a GCT.…”

“…Since ∇ i R i j = 1 2 ∇ j R, the scalar curvature is constant. A manifold is a constant curvature manifold if the Riemann tensor has the form (22). Such manifolds are Einstein manifolds.…”

Curvature properties of Vaidya metric

“…In 1997 Ludwig and Edgar [36] presented a pure radiation metric which is conformally related to vacuum space or Ricci flat space. The pure radiation metric of Ludwig and Edgar ([36], [66]) in (u; r; x; y)-coordinates (r > 0 and x > 0) is given by…”

“…where w is a smooth function of u, x, y, and p is a non-zero constant. Recently, Shaikh et al [66] have studied the curvature properties of the Ludwig-Edgar pure radiation metric (5.1). Since both the Vaidya metric (1.2) and Ludwig-Edgar metric (5.1) represent pure radiation fields, in this section we are mainly interested to make a comparisons between the geometric properties of Vaidya metric (1.2) and Ludwig-Edgar pure radiation metric (5.1).…”

Three-dimensional Lorentzian $Ein(2)$ Lie groups

“…They got a complete classification of algebraic Ricci solitons of three-dimensional Lorentzian Lie groups and they proved that, contrary to the Riemannian case, Lorentzian Ricci solitons needed not be algebraic Ricci solitons. In [2], [7], [8], the definition of Ein(2) manifolds was introduced. In [6], some examples of Ein(2) manifolds were given.…”