Garry Ludwig

Classification of the Ricci tensor

This paper contains a classification of the Ricci tensorRαβ. The method of derivation is analogous to the spinor version of the Petrov classification of the Weyl tensor. It is shown how the various classes are related to the number and type of eigenvectors and eigenvalues ofRαβ. The classification is useful in the geometrization of various fields. The case of a real scalar field is treated in detail.

All conformally flat pure radiation metrics

The complete class of conformally flat, pure radiation metrics is given, generalizing the metric given recently by Wils.

On asymptotically flat space-times

The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penroses conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.

Geometrodynamics of electromagnetic fields in the Newman-Penrose formalism

The “already unified” field theory of Rainich, Misner, and Wheeler is rederived in the spin-coefficient formalism of Newman and Penrose. Conditions equivalent to the Rainich algebraic conditions are obtained by classifying the tracefree Ricci tensor according to its principal null directions. The case of a null electromagnetic field is also treated fully. Necessary and sufficient conditions are given for a Riemannian geometry to have an electromagnetic field, null or non-null, as its source.

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 conformally flat, pure radiation metric, which has been shown by Koutras to admit no Killing vectors, in general. In order to present a simple illustration of the ghp integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics, which are not plane waves. Our result shows that the conformally flat, pure radiation metrics are a larger class than Wils has obtained.

Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field ξ is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ξ; this result is translated into the GHP formalism using a new generalised Lie derivative operator Łξ with respect to a vector field ξ. We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator Łξ in the presence of a Killing vector field ξ. This new operator Łξ also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if ξ is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.

Integration in the GHP Formalism II: An Operator Approach for Spacetimes with Killing Vectors, with Applications to Twisting Type N Spaces

Held has proposed a coordinate- and gauge-free integration procedure within the ghp formalism built around four functionally independent zero-weighted scalars constructed from the spin coefficients and the Riemann tensor components. Unfortunately, a spacetime with Killing vectors (and hence cyclic coordinates in the metric, and in all quantities constructed from the metric) may be unable to supply the full quota of four scalars of this type. However, for such a spacetime additional scalars may be supplied by the components of the Killing vectors. As an illustration we investigate the vacuum type N spaces admitting a Killing vector and a homothetic Killing vector. In a direct manner, we reduce the problem to a pair of ordinary differential operator “master equations”, making use of a new zero-weighted ghp operator. In two different ways, we show how these master equations can be reduced to one real third-order operator differential equation for a complex function of a real variable—but still with the freedom to choose explicitly our fourth coordinate. It is then easy to see there is a whole class of coordinate choices where the problem reduces essentially to one real third-order differential equation for a real function of a real variable. It is also outlined how the various other differential equations, which have been derived previously in work on this problem, can be deduced from our master equations.

The GHP conformai Killing equations and their integrability conditions, with application to twisting type N vacuum spacetimes

The conformai Killing equations in resolved form and their first and second integrability conditions are obtained in the compact spin coefficient formalism for arbitrary spacetimes. To facilitate calculations an operatorL is introduced which agrees with the Lie derivative only when operating on quantities with GHP weights (0,0). The resulting equations are used to find the conditions for the existence of a two dimensional non-Abelian group of homothetic motions in a twisting typeN vacuum spacetime. The equivalence of two such sets of metrics is established, metrics that were recently the subject of independent investigations by Herlt on the one hand and by Ludwig and Yu on the other.

On algebraically special space-times with twisting rays

The reduced gravitational field equations are derived for algebraically special space-times with twisting geodesic and shear-free rays for a large class of Ricci tensors. These equations coincide with those derived by Trim and Wainwright under more restrictive assumptions on the Ricci tensor. Penroses conformal technique is used to facilitate computation and interpretation of the results. The remaining coordinate freedom and freedom in the choice of tetrad is discussed.

Integration in the GHP formalism I: A coordinate approach with applications to twisting type N spaces

The compacted spin coefficient (ghp) formalism is clearly more concise and efficient than the older Newman-Penrose formalism. Yet few people use it when integration of the field equations is involved, Held being the notable exception. However, to most workers in the field, Helds approach seems far removed from the usual Newman-Unti (nu) type integration procedure. This paper and a subsequent one are concerned with integration within theghp formalism. In this first paper we develop aghpcoordinate-style integration procedure modelled closely on thenu procedure whereas in the second paper we present aghpoperator-style integration procedure along the lines suggested by Held. For simplicity of illustration we restrict the discussion to algebraically special vacuum spacetimes. We show clearly the similarities and differences between the two approaches, and compare their respective efficiencies. To deal with a concrete example, we illustrate the two methods by once more considering the problem of twisting typeN vacuum solutions to Einsteins field equations. TheGhp approach enables us to have a comprehensive overview of this much discussed problem and gain new insight into the relationship between various results derived in a number of different formalisms.