An exact stationary solution of Einstein's equations

We give an exact stationary solution ofEinstein's empty space field equations. It represents two massless sources possessing angular momentum, and held in position by stresses. The solution conforms withMach's principle in the following sense: the spinning sources cause a rotation of the local inertial frame relative to test particles at infinity.     

A static three-center solution to Einstein's gravitational field equations

In this paper the physical situation of three collinear, axisymmetric masses is studied within the framework of Einstein's general theory of relativity. A solution is found, and the feasibility of the existence of coupled negative-positive masses is demonstrated in direct correspondence to such a feasibility in Newtonian theory.     

An exact solution of Einstein's equation with a dilaton field

An exact solution of Einstein gravity coupled to a dilaton field is found. The solution is conformally flat and is invariant under Lorentz transformations. The singularities and conformal structure of the metric are examined.     

Compatible Klein-Gordon equations for a system of two mutually interacting particles in an external field

We start from a system of two compatible coupled Klein-Gordon equations describing a system of two relativistic and mutually interacting spinless particles, and we introduce into each equation an additional term describing a supplementary interaction of the corresponding particle with an external field. The compatibility of the equations, destroyed by the introduction of the new coupling terms, is restored by adding two counter-terms, computed from the mutual and external interaction terms chosen as input, and vanishing when either the mutual interaction or the external field is switched off. It seems that the same method could be used (with the necessary adaptations) in the building of other systems of coupled compatible equations. When used in the problem of introducing a mutual coupling into a pair of independent Klein-Gordon equations, it leads to the usual transversal variables depending coupling terms.     

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 axially symmetric solutions of Einstein's field equations in matter and an example of static gravitational shielding

A model system, consisting of a thin spherical shell with radiusR and massM and a point massm at a distances>R from the center of the sphere, held fixed by an appropriate strut, is solved to ordermM. The stresses in the shell are not of the canonical Weyl type, and it is argued that the same is true for more realistic situations, e.g., rotating matter. Owing to the nonlinearity of Einstein's field equations, the field of the point mass is shielded from the interior of the shell by a factor lying between 1–3M/R and 1–2M/R, and the field outside the shell explicitly depends onR.     

Relativistic kinetic equations for inelastically interacting particles in a gravitational field

The relativistic canonical formalism is used to construct the kinetic equations for a gas in a gravitational field, whose particles interact with one another via numerous inelastic collisions. Boltzmann's H-theorem is proved for T-invariant interactions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 19–23, August, 1983.     

Matter tensor for a freely falling oscillating bar in a gravitational wave

Matter tensor and motion of a test body is obtained. The test body is an oscillating bar; the background metric is a Bondi plane wave. Amplitude and frequency of bar and wave are arbitrary and not necessarily constant.     

An exact solution for uniformly accelerated particles in general relativity

Recently an exact solution of Einstein's empty-space equations referring to four uniformly accelerated particles was given. The relation of this to static axially symmetric metrics of the Weyl and Einstein-Rosen classes is investigated in the present paper. A physical interpretation of the singularity along half of the axis of symmetry of the uniformly accelerated metric in Weyl's form is given. An exact solution corresponding to an expanding (contracting) singular null surface is obtained by a limiting process from that for uniformly accelerated particles.     

An exact solution for uniformly accelerated particles in general relativity

We present an exact solution of the Einstein empty-space equations referring to four particles in relative motion. The particles move with different uniform accelerations relative to a co-ordinate system which is Minkowskian at infinity, except in certain directions. If positive and negative masses are allowed, the particles can move freely under their own gravitation; if all four masses are positive, stresses extending to infinity are needed to cause the motion, but two of the particles can move freely. There are three results of interest. First, the field can be described in terms of a classical potential which is the average of retarded and advanced potentials corresponding to the particles. Secondly, the field at spatial infinity is entirely different from that of a static mass, and theg _ik fall off like the inversesquare of the distance. Thirdly, the world-lines of free particles are geodesics of the space-time.     

Exact one-body solution of the gauge-covariant gravitational field equations

Dirac has recently introduced a new theory of gravity which includes his large numbers hypothesis. The exact one-body vacuum solution consistent with multiplicative matter creation is obtained and discussed.Supported by a grant from the National Research Council of Canada.     

An exact anisotropic solution of the Einstein-Liouville equations

A family of exact solutions of the Einstein-Liouville equations are presented, in which the space-time geometry is that of ak=0 ork=+1 Robertson-Walker space-time but the particle distribution function is anisotropic (and can be inhomogeneous). In some of these solutions, the fluid average (barycentric) velocity is not the timelike eigenvector of the fluid stress tensor. Then a “fundamental observer” moving with the average (barycentric) velocity will not observe these universes to be isotropic.     

A nonstatic plane symmetric solution of Einstein's field equations for zero-rest mass scalar fields

An exact solution of Einstein's field equations of general relativity is given for a plane symmetric zero-rest-mass scalar field in the nonstatic case. The solution generalizes Singh's solution, which itself extends Taub's empty space-time.     

New exact solutions of Einstein's field equations: Gravitational force can also be repulsive!

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are given.     

Elliptic solutions of equations of motion of the interacting particles in an external field

We consider the problem of finding analytic solutions to the equations of motion of completely integrable systems of two interacting particles on a straight line in an external field of an anharmonic oscillator. It is shown that for all values of the parameters of the interaction hamiltonian there exist solutions dependent on time through the jacobian elliptic functions.     

An exact solution of the renormalization-group equations for the mean field theory of stable and metastable states

An exact solution of the renormalization-group equations corresponding to the mean field theory of stable and metastable states is given which yields the correct free energies for these states. An unusual feature of this solution is that the renormalized Hamiltonian in the two-phase region becomes a multivalued function of the order parameter for all values of the length rescaling parameter beyond a certain critical value. This is closely related to the multivaluedness of the free energy as a function of magnetic field which characterizes the classical theory of metastable and unstable states. As a consequence of this multivaluedness, the trajectory flow in the space of coupling constants exhibits unusual bifurcation. This leads to difficulties in evaluating the metastable and unstable free energies by a trajectory integral of the spin-independent term, which can be resolved by an extension of the standard formalism.This work was supported by NSF grant #550-346-01 (JDG) and a U.S.-Japan Cooperative Science grant (KK and JDG).     

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.