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.     

Non-existence of stationary,axisymmetric dust solutions of Einstein's equations on spatially compact manifolds

By using a theorem on the classifacation of SO(2)-actions on three-dimensional manifolds, it is shown that axisymmetric, stationary solutions of Einstein's equations do not exist on spacetimes with compact, space-like hypersurfaces, provided that the energy-momentum tensor is trace-free in the two-dimensional space of trajectories.     

Matching in a class of stationary axisymmetric perfect fluid solutions of Einstein's equations

The class of previously found stationary axisymmetric perfect fluid solutions of Einstein's equations is written inh-orthogonal coordinates,h being a space-like coordinate. Matching of a big number of solutions of the class with each other seems to be possible for a proper choice of some parameters. The exterior solutions of the class are matched explicitly with interior solutions. Also, interior solutions are matched explicitly with each other.     

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.     

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.     

Non-radiative solutions of Einstein's equations for dust

The quasi-spherical collapsing space-time of Szekeres is investigated. The arbitrary functions can be chosen so that it has positive density, and no Killing vectors; yet a ballr<r ₀ of it can joined to the Schwarzschild metric, and is therefore non-radiative.     

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.     

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.     

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.     

Classification of the type D perfect-fluid solutions of the Einstein equations

A classification scheme is given for the known exact solutions of Petrov type D of the Einstein field equations with perfect-fluid source. The scheme is based on the kinematic properties of the fluid and on the geometric properties of the principal null congruences of the Weyl tensor.     

Some real and complex solutions of Einstein's equations

The construction of some real solutions of Einstein's vacuum field equations from certain half flat holomorphic metrics is described.     

Exact anisotropic viscous fluid solutions of Einstein's equations

A method for obtaining anisotropic, rotationless viscous fluid matter solutions of Bianchi type I and Segré type [1, 111] with the barotropic equation of state is presented. Solutions for which the anisotropy decreases exponentially or with a power law as well as solutions with average Hubble parameterH t ^–1 are discussed. Also, a class of solutions with constant anisotropy and Bianchi type VI_h is found. The dominant energy condition holds and the transport coefficients show the right sign.     

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.     

Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

The quality factor of approximate solutions of Einstein's equations

I define the quality factor of arbitrary metrics to be used as a measure of their approximation to a solution of Einstein's vacuum equations.     

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.