On homogeneous solutions of Einstein's field equations

In the first part of this paper we describe a formalism capable of finding all homogeneous solutions of Einstein's field equations with any arbitrary energy-impulse tensor. In the second part we find all homogeneous vacuum solutions.     

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.Supported in part by NFR, the Swedish Academy of Sciences and the Gustavsson Foundation     

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.     

Two simple solutions to Einstein's field equations

Metrics of the formds ²=dx ²+dy ²–dt ²+N ² dz ² are considered and found to contain rotating dust solutions as well as pure radiation fields.     

On solutions of Einstein's equations with scalar zero-mass source

It is shown that most, if not all, previously known solutions of the gravitational field equations with a zero-mass scalar sourceø can be found without solving the field equations by applying known theorems. Penney's recent solution, characterized in part by having a conformally flat metricds ₂=e ² _ij dx ⁱ dx ^j is shown to be insufficiently general when his vectora _i is null. The problem is reformulated and new solutions of the conformally flat type are found. These are, in general, such that ø and are no longer linearly related.     

On the strength of Einstein's unified field equations

Einstein's concept of the strength of a system of field equations, which has been related in a simple way to the amount of initial data required for the system, is examined for his last unified field theory. The apparently surprising weakness of this system is traced to the high order of the associated electromagnetic field equations. These equations allow the existence of purely electric longitudinal waves, in spite of the absence of any photon mass.     

On the existence of solutions to Einstein's equation with non-zero Bondi news

It is shown that theC-metric (with parameters chosen to lie in suitable intervals) admits a conformal completion such that the space of generators of null infinity, , is a 2-sphere. This structure of is both necessary and sufficient for the analysis of gravitational radiation in exact general relativity. Bondi news (as well as the electromagnetic radiation field, in the charget case) is examined and found to be non-zero. Thus the issue of existence of exact solutions to the Einstein (and Einstein-Maxwell) equations admitting radiation (in the sense of Bondi, Sachs, and Penrose) is resolved. In addition, the analysis clarifies the sense in which the vacuumC-metric represents the gravitational field of two accelerating black-holes.     

Generating solutions of Einstein's field equations by typing mistakes

A solution to Einstein's field equations is presented that represents a Petrov type II electromagnetic null field with one Killing vector. This solution generalizes a vacuum solution previously discovered by Hoenselaers. The solution was found by the peculiar method of generalizing a member of this class inadvertently discovered by making a typing error when checking the vacuum solution with the computer algebra system SHEEP.     

Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant

The initial value problem for Einstein's field equations with positive cosmological constant is analysed where data are prescribed at past conformal infinity. It is found that the data on past conformal infinity are given, up to arbitrary conformal rescalings, by a freely specifyble Riemannian metric and a trace-free, symmetric tensorfield of valence two, which satisfies a divergence equation. For each initial data set exists a unique (semi-global) past asymptotically simple solution of Einstein's equations. The case is discussed where in such a space-time exists a Killing vector field with a time-like trajectory which approaches a point p on conformal infinity. It is shown that in a neighbourhood of the trajectory near p the space-time is conformally flat.     

On the existence of a general rotating solution of Einstein's equations

In an earlier paper we considered a power-series expansion of the metric for a rotating field in terms of a parameter and constructed a solution of Einstein's equations to the first few orders in terms of two harmonic functions. We encountered a pair of Poisson-type equations which were apparently insoluble explicitly. The form of the metric considered was the Weyl-Lewis-Papapetrou form. In this paper we consider a power-series expansion of the most general form of a rotating metric and show that one encounters the same two Poisson equations as before. If these equations are insoluble explicitly, as seems likely, then a general solution depending on two harmonic functions cannot exist in closed form.     

On solving Einstein's field equations in the Newman-Penrose formalism

Using the Newman-Penrose formalism and Penrose's conformai rescaling a method is presented for finding systematically solutions of (or, at least, reduced equations for) the general field equations. These solutions are necessarily (locally) asymptotically flat and are represented in a coordinate system based on a geodesic, twist-free, expanding null congruence. All redundant equations are disposed of and the freely specifiable data are clearly exhibited. Although the few equations that remain to be solved are, in general, intractable, well-known theorems guarantee the existence and uniqueness of solutions. The method applies to spaces and spaces as well as to real space-times.     

Exact solutions of Einstein's field equations for a massive point-particle with scalar point-charge

The exact static and spherically symmetric solution of Einstein's field equations for a massive point-particle with a scalar point-charge as source of a massless scalar field is derived in Schwarzschild coordinates. There exists no longer a Schwarzschild horizon. Only at the point-particle metric and scalar field are singular (naked singularity).     

A class of stationary charged dust solutions of Einstein's field equations

We consider a special class of stationary rotating charged dust solutions of Einstein's field equations without cosmological constant. In these space-times, the motion of freely falling particles and of light rays can be visualized by the motion of charged particles in an appropriate model magnetic field. Any curl-free magnetostatic field, given on an open subset of Euclidean 3-space, can serve as a model magnetic field for a charged dust solution in this sense. The simplest example, corresponding to a homogeneous model magnetic field, is given by Som-Raychaudhuri space-time. Some other examples are worked out.     

Discontinuities in the solutions of Einstein's equations

The quantities that determine the jump conditions for discontinuities in the second derivatives of the components of the metric tensor are obtained.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 10, pp. 40–43, October,1981.I take this opportunity to express my deep gratitude to L. R. Volevich for his constant attention to my work.     

On the hyperbolicity of Einstein's and other gauge field equations

It is shown that Einstein's vacuum field equations (respectively the conformal vacuum field equations) in a frame formalism imply a symmetric hyperbolic system of reduced propagation equations for any choice of coordinate system and frame field (and conformal factor). Certain freely specifiable gauge source functions occurring in the reduced equations reflect the choice of gauge. Together with the initial data they determine the gauge uniquely. Their choice does not affect the isometry class (conformal class) of a solution of an initial value problem. By the same method symmetric hyperbolic propagation equations are obtained from other gauge field equations, irrespective of the gauge. Using the concept of source functions one finds that Einstein's field equation, considered as second order equations for the metric coefficients, are of wave equation type in any coordinate system.Work supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft     

Globally regular solutions of Einstein's equations

This is a review, covering known globally regular solutions describing either vacuum or fields with physically reasonable sources. The largest class is that with static spherical symmetry, but many others are known, even with A = 0. If A 0 there is a variety of regular cosmological solutions.     

A study of some exact solutions to Einstein's field equations which yield separable Hamilton-Jacobi equations

Exact solutions to Einstein's field equations, which give rise to a Stäckel-separable Hamilton-Jacobi equation of the form $$,y,z)\left[ {X(x)\left( {\frac{{\partial S}}{{\partial x}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial x}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - 2\left( {\frac{{\partial S}}{{\partial y}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) + Z(z)\left( {\frac{{\partial S}}{{\partial z}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial z}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - F(x,y,z)\left( {\frac{{\partial S}}{{\partial t}}} \right)^2 } \right] = \lambda $$ are considered. It is shown that there are no solutions for whichD is a function ofx orz, orx andz. The exact solutions are of Petrov typeN and are plane polarized waves without rotation. Some of the solutions are given explicitly, up to two arbitary functions. For these solutions the Hamilton-Jacobi equation is reduced to an uncoupled set of first-order ordinary differential equations.     

On the existence of algebraic differential equations with noded periodic solutions

It is proved that a system of three autonomous differential equations with polynomial (of the third degree and higher) right sides may have a noded closed curve as its solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 389–392, March, 1972.In conclusion the author deeply thanks Professor Yu. I. Neimark for his statement of the problem and the great interest that he manifested in its solution.     

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.This research was supported through a fellowship from the Consejo Nacional de Investigaciones Cientificas y Technicas de la Republica Argentina.