Republication of: Exact solutions of the field equations of the general theory of relativity

This is an English translation of a paper by Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, first published in 1960. The original paper was part 1 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. (The other parts of the series will be printed as Golden Oldies in the future.) The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. It is accompanied by an editorial note written by G. F. R. Ellis, and by the biographies of the authors: P. Jordan (written by A. Krasiński) and W. Kundt (written by himself). The biography of J. Ehlers is contained elsewhere in the same issue of GRG, which is devoted to his memory. An editorial note to this paper and a biography can be found in this issue preceding this Golden Oldie and online via doi:. Original paper: Pascual Jordan, Jürgen Ehlers, Wolfgang Kundt, Strenge L?sungen der Feldgleichungen der Allgemeinen Relativit?tstheorie. Akademie der Wissenschaften und der Literatur, Abhandlungen der Mathematisch-naturwissenschaftliche Klasse Nr 2S (1960), pp. 21–105. Reprinted with the kind permission of the Academy of Sciences and Literature, Mainz, and of the authors: Jürgen Ehlers and Wolfgang Kundt. Translated by Anita Ehlers, Anita.Ehlers@t-online.de, and by Manfred Trümper, manfred@truemper.fr, with ample help from Wolfgang Kundt. P. Jordan (Deceased July 31, 1980) J. Ehlers (Deceased May 20, 2008)

---

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.     

Exact solutions to Einstein field equations

In a recent paper Reboucas and d'Olival obtain an ordinary differential equation for a Bianchi type II metric with a rotating timelike congruence of geodesics, and obtain a particular solution of the differential equation. This paper completely integrates the differential equation.     

Exact solutions for the massless plane symmetric scalar field in general relativity,with cosmological constant

The Klein–Gordon equations are solved for the case of a plane-symmetric static massless scalar field in general relativity with cosmological constant, generalizing the solutions found by Taub, Novotny and Horsky, and Singh. A separate class of solutions is obtained in which the metrics reduce to flat space in the limit that .The static solutions can be used to generate time-dependent cosmological solutions, one of which exhibits rapid inflation followed by continued exponential expansion at all later times.     

Republication of: Contributions to the theory of pure gravitational radiation. Exact solutions of the field equations of the general theory of relativity II

This is an English translation of a paper by Pascual Jordan, Juergen Ehlers and Rainer Sachs, first published in 1961 in the proceedings of the Academy of Sciences and Literature in Mainz (Germany). The original paper was part 2 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. (Parts 1 and 4 of the series have already been reprinted, parts 3 and 5 will be printed as Golden Oldies in near future.) This second paper discusses the geometry of geodesic null congruences, the algebraic classification of the Weyl tensor by spinor methods, and applies these to a study of the propagation of gravitational and electromagnetic radiation. It has been selected by the Editors of General Relativity and Gravitation for republication in the Golden Oldies series of the journal. The republication is accompanied by an editorial note written by Malcolm A. H. MacCallum and Wolfgang Kundt.     

New exact cylindrical solutions of the Einstein-Maxwell equations of general relativity

We find new exact cylindrical solutions of the Einstein-Maxwell equations, which are cylindrical waves. In some of these solutions Painlevé transcendental functions of type III and V appear. There are maxima in the electromagnetic energy density.     

Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Coordinate-dependent 3 + 1 formulation of the general relativity field equations

This paper presents a coordinate-dependent 3+ 1 decomposition of the general relativity field equations in terms of a scalar potentialc ²[(–g ₄₄)^1/2–1], a vector potentialA _icg _4i/(–g₄₄)^1/2, and the three-space metric _ijg _ij–g_4i g _4j/g ₄₄. The equations are exact and the form of the decomposed equations is valid in any coordinate system.     

Comments on models of scalar-tensorial field equations in general relativity

In this paper we consider some of the proposed models for introducing the long-range scalar interaction in Riemannian space-times. The relationship among these models is discussed. Particular emphasis is placed on the introduction of the scalar interaction via a conformal mapping on the original Riemannian geometry. Following this method we introduce a spinorial model for the coupled system. We also discuss the meaning of the identities satisfied by the left-hand side of the coupled field equations, which are present for any model derivable from an action principle.     

Killing vector fields and the Einstein-Maxwell field equations in general relativity

A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector F_ij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor F_ij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient α_ij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,_i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case F_ij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

Slow-motion approximations in general relativity I. Decomposition of the field equations

A new (3+1)-dimensional decomposition of the Einstein gravitational field equations is obtained for a general spacetime. The metric is taken in the form $$ds^2 = e^{ - 2u} k_{ab} (dx^a + \xi ^a dt)(dx^b + \xi ^b dt) - c^2 e^{2u} dt^2 $$ and the resulting equations treatk _ab as the metric in the space-like hypersurfacest=constant. It is shown that this decompostion forms a more convenient starting point for slow motion approximations than does their usual 4-dimensional formulation. This is illustrated by a derivation of the first post-Newtonian approximation to the field equations, the simplicity there resulting fromk _ab being still flat to this order.     

On axially symmetric soliton solutions of the coupled scalar-vector-tensor equations in general relativity

The inverse scattering theory used by Belinsky and Zakharov to obtain soliton solutions of the of the Einstein equations is here applied to the case of a five-dimensional space and interpreted in the framework of the Jordan-Kaluza-Klein theory. For two solitons exact, stationary, axially symmetric and asymptotically flat solutions are obtained.     

Cosmological solutions foe a spinor field in the general theory of relativity

We consider models of the universe containing linear and nonlinear spinor matter. It is assumed that the linear spinor matter is described by the generally covariant Dirac equation, and the nonlinear by the generally covariant Ivanenko-Heisenberg equation.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 68–71, December, 1973.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

Exact solutions of the Altarelli-Parisi equations

Evolution equations for single species parton distributions are solved analytically. For infrared-convergent theories the solutions can be expressed in a closed form as convolutions over Bessel functions. For infrared-divergent theories, such as QCD, the exact solution involves an infinite number of convolutions. The accuracy of various approximations to this solution is estimated.