Hamilton-Jacobi separable,axisymmetric, perfect-fluid solutions of Einstein's equations

In this paper we examine the Einstein equations with a perfect fluid source under the assumptions of (i) axial symmetry and time-independence, (ii) uniform rotation of the fluid about the symmetry axis, and (iii) separability of the Hamilton-Jacobi equation for the null geodesics of the space. These assumptions are made in an attempt to generalize the results of a similar investigation by Carter for the source-free case.We first extend Carter's results by showing that his additional assumption of separability of the wave equation is unnecessary, it being a consequence of the field equations.When the density of the fluid is non-zero, we are led to a particular solution discovered by Wahlquist, or to more symmetrical interior solutions with spherical equipressure surfaces. Except for the case of no rotation, these solutions cannot be matched to asymptotically flat exteriors.     

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 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.     

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.     

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.     

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka--Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of \sinh and \cosh, periodic solutions presented by trigonometric functions of \sin and \cos, and rational solutions. This method can be used to solve some other nonlinear difference--differential equations.     

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.     

Testing the quality factor with some approximate solutions to Einstein's equations

The quality factor of a metric, defined by Bel as a measure of its approximation to a solution of Einstein's equations, is tested with several well-known approximate solutions. Whereas for some metrics such as the linearized plane wave solutions the quality factor leads to some intriguing results, for others such as the Newtonian and post-Newtonian approximations for the one and two body systems, and also the linearized perturbations of the Friedmann-Robertson-Walker metrics, it seems to lead to very reasonable results.     

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.     

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.     

Explicit exact solutions to some one-dimensional conformable time fractional equations

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers), potential Korteweg-de Vries (KdV), and clannish random walker’s parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE using compatible wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution. The solutions for some specific values of the parameters covering derivative order are depicted to explain the wave propagation numerically.     

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.     

New exact static solutions of einsteins field equations

We describe a method for deriving new solutions for an ideal fluid from old and give some new solutions which may be of interest in astrophysics.     

New exact solutions of the Yang-Mills field equations

In this paper new exact solutions of the Yang-Mills SU(2) gauge field equations are obtained using the Carmeli-Charach-Kaye null-tetrad formalism. The solutions are classified and briefly discussed.     

A class of solutions of Einstein's equations which admit a 3-parameter group of isometries

The Plebanski and Stachel and Goenner and Stachel lists of metrics which are solutions of Einstein's field equations, have two double eigenvalues and admit 3-parameter groups of isometries with 2-dimensional spacelike orbits are completed by the addition of metrics which result from the use of a more general metric form.     

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 Finsler generalisation of Einstein's vacuum field equations

This paper gives a generalisation of Einstein's vacuum field equations for Finsler metrics. The given generalised field equation reproduces the Einstein equations for Riemannian metrics, and also admits non-Riemannian solutions. This is shown in detail by deriving a first order Finsler perturbation, solving the new field equation, of the Schwarzschild metric. This perturbation turns out to be time independent. The effects of the perturbation on the three Classical Tests of General Relativity are derived, and used to give limits on the size of the perturbation parameter involved.     

A study of solutions to the aesthetic field equations

We have found hundreds of solutions to the integrability equations in aesthetic field theory. The behavior of the solutions to the aesthetic field equations depends on which solution to the integrability equations we take. From computer runs down a coordinate axis we have found a type of solution where we have a maximum and a minimum, as well as the field going to zero at large distances along both directions. This kind of solution is quite prevalent. We call this type of solution a pulse solution. We have found the pulse solution in two and three dimensions as well as four dimensions. It appears regardless of whether certain symmetries are present or absent. We have taken a two- or three-dimensional and made a four-dimensional theory from it with the use of a four-dimensionale i. This process we call imbedding. We have found imbedding has not affected the overall characteristics of the solution in the cases we considered. We were able to change the character of the solutions to some degree by altering the magnitude of some of the gammas—but this did not lead to solutions with significantly more wiggles. We also found an example of an oscillatory solution. The oscillations occurred in too regular a pattern to give a realistic model for basic behavior. However, this solution indicates that aesthetic field theory has more structure then we have ever seen before. We also obtained a solution in which errors took over so fast that the computer was literally helpless in telling us what is going on. In other solutions the field appears to increase without bounds. Whether this is due to singularities or to the presence of large numbers is not clear.     

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.