Pure Radiation Metrics Research Articles

A Ricci Soliton Characterization of Pure Radiation Metrics Conformal to a Vacuum Space-Time

A Ricci Soliton Characterization of Pure Radiation Metrics Conformal to a Vacuum Space-Time

Curvature properties of a special type of pure radiation metrics

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 the 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 a 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.

Symmetry analysis of radiative spacetimes with a null isotropy using GHP formalism

A complete and simple invariant classification of the conformally flat pure radiation metrics with a negative cosmological constant that were obtained by integration using the generalised invariant formalism is presented. We show equivalence between these metrics and the corresponding type O subclass of the more general spacetime studied by Siklos. The classification procedure indicates that the metrics possess a one degree of null isotropy freedom which has very interesting repercussions in the symmetry analysis. The Killing and homothetic vector analysis in GHP formalism is then generalised to this case were there is only one null direction defined geometrically. We determine the existing Killing vectors for the different subclasses that arise in the classification and compare these results to those obtained in the symmetry analysis performed by Siklos for a larger class of metrics with Ricci tensor representing a pure radiation field and a negative cosmological constant. It is also shown that there are no homothetic Killing vectors present.

Conformal pure radiation with parallel rays

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays. Then, we derive conditions in terms of the tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class. We also give analogous results for n-dimensional pseudo-Riemannian pp-waves.

Invariant classification of metrics using invariant formalism

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates. By performing this classification we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we are able to obtain complete agreement with the results obtained in intrinsic coordinates. In particular, we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.

On conformally flat and type N pure radiation metrics

We study pure radiation spacetimes of algebraic types O and N with a possible cosmological constant. In particular, we present explicit transformations which put these metrics, that were recently re-derived by Edgar, Vickers and Machado Ramos, into a general Ozsvath-Robinson-Rozga form. By putting all such metrics into the unified coordinate system we confirm that their derivation based on the GIF formalism is correct. We identify only few trivial differences.

Type O pure radiation metrics with a cosmological constant

In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant Λ, and \(\tau \ne 0\) , by considering the case \(\Lambda +\tau\bar\tau \ne 0\) . This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case \(\Lambda +\tau\bar\tau = 0\)) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors.

Using the generalised invariant formalism: a class of conformally flat pure radiation metrics with a negative cosmological constant

We demonstrate an integration procedure for the generalised invariant formalism by obtaining a subclass of conformally flat pure radiation spacetimes with a negative cosmological constant. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N respectively. This subclass of spacetimes turns out to have one degree of isotropy freedom, so in this paper we have extended the integration procedure for the generalised invariant formalism to spacetimes with isotropy freedom,

Obtaining a class of type O pure radiation metrics with a negative cosmological constant, using invariant operators

Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.

Obtaining a class of type N pure radiation metrics using invariant operators

We develop further the integration procedure in the generalized invariant formalism, and demonstrate its efficiency by obtaining a class of Petrov type N pure radiation metrics without any explicit integration, and with comparatively little detailed calculations. The method is similar to the one exploited by Edgar and Vickers when deriving the general conformally flat pure radiation metric. A major addition to the technique is the introduction of non-intrinsic elements in the generalized invariant formalism, which can be exploited to keep calculations manageable.

Classifying geometries in general relativity:III. Classification in practice

This is the third in a series of papers concerning a project to set up acomputer database of exact solutions in general relativity which can beaccessed and updated by the user community. In this paper, we describehow the Cartan-Karlhede method for classifying a geometry isaccomplished in practice. We give as an example the classification ofthe Edgar-Lugwig conformally flat pure radiation metrics.

A generalized Lie derivative and homothetic or Killing vectors in the Geroch-Held-Penrose formalism

A generalized Lie derivative operator suitable for use within the GHP formalism and the notion of preferred GHP tetrads relative to a vector are introduced. The usual homothetic or Killing equations are then replaced by an equivalent but much more manageable set of equations involving the commutators of this new operator with the four GHP derivative operators. This allows for an efficient treatment of the homothetic or Killing condition when constructing new solutions of Einstein's field equations or when obtaining the homothetic and/or Killing vectors for a given metric. Two applications are given. The first sheds new light on the vacuum twisting type N problem with one or two homothetic/Killing vectors. In the second we find the subclass of all type N, and of all conformally flat, pure radiation metrics (with 0) which possess one or more homothetic or Killing vectors.

Interpreting a conformally flat pure radiation spacetime

A physical interpretation is presented of the general class of conformally flat pure radiation metrics that has recently been identified by Edgar and Ludwig. It is shown that, at least in the weak-field limit, successive wave surfaces can be represented as null (half) hyperplanes rolled around a two-dimensional null cone. In the impulsive limit, the solution reduces to a pp-wave whose direction of propagation depends on retarded time. In the general case, there is a coordinate singularity which corresponds to an envelope of the wave surfaces. The global structure is discussed and a possible vacuum extension through the envelope is proposed.

Conformally Ricci flat pure radiation metrics

Wils has recently investigated pure radiation metrics which are conformal to a vacuum spacetime. It is shown here that, contrary to his claim, he did not find all such solutions. The metrics he found were of type N (or O) and the conformal factor was independent of the radial coordinate. The form of the metric depended on whether the spin coefficient vanished or not. We find that in the case there are additional type O solutions, some of which have only a radially dependent conformal factor and that in the case, in addition to the type N and type O solutions found by Wils, there are type III pure radiation metrics which are conformally Ricci flat. It is proved that this exhausts the class of conformally Ricci flat pure radiation solutions of Einstein's equations. For all type O solutions when , and for some type O solutions in the case, two conformal factors are found, one dependent on the radial coordinate, the other independent of it, indicating that these spaces possess a conformal symmetry.

Integration in the GHP Formalism III: Finding Conformally Flat Radiation Metrics as an Example of an “Optimal Situation”

Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the `optimal situation', when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a metric which he claims is the only conformally flat, pure radiation metric which is not a plane wave; this metric has been shown by Koutras to admit no Killing vectors, in general. Therefore, as a simple illustration of the GHP integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics. Our result shows that the conformally flat, pure radiation metrics, which are not plane waves, are a larger class than Wils has obtained.

Homogeneous and conformally Ricci flat pure radiation fields

Some unsolved problems concerning pure radiation fields in general relativity are studied. The general solutions for homogeneous and for conformally Ricci flat metrics are given. They are all of Petrov type N or O, and contain a hitherto unidentified conformally flat metric. Non-aligned Petrov type III pure radiation metrics are shown not to exist.