On a general static axisymmetric solution of the Einstein vacuum equations

It is shown that a generalization of the procedure given in [1] for construction of the gravitational multipoles leads to the same Newtonian limit as the generalized Erez-Rosen solution [2].     

An extension of the plane-symmetric electrovac general solution to Einstein equations

We present the general solution to Einstein-Maxwell equations representing plane-symmetric metrics associated with electromagnetic fields that are not fully plane-symmetric. There are two classes in the general solution, the first approaches Taub's static metric or Kasner's spatially homogeneous one as the electromagnetic field goes to zero, while the second approaches the fiat metric.     

On a static axisymmetric solution of the Einstein equations

A family of static, axisymmetric, asymptotically flat solutions of the Einstein equations is discussed. A source with an exterior described by a member of this family initially could have an area smaller than that of a n appropriately defined Schwarzchild surface. Intuition does not suggest the fate of the collapsing source.     

On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations

Let Σ be a compact three-manifold with non-positive Yamabe invariant. We prove that in any long time constant mean curvature Einstein flow over Σ, having bounded C ^α space–time curvature at the cosmological scale, the reduced volume [g(k) is the evolving spatial three-metric and k the mean curvature] decays monotonically towards the volume value of the geometrization in which the cosmologically normalized flow decays. In more basic terms, we show that there is volume collapse in the regions where the injectivity radius collapses (i.e. tends to zero) in the long time limit. We conjecture that under the curvature assumption above the Thurston geometrization is the unique global attractor. We prove this in some special cases.     

On an exact classical wave solution of the Einstein equations

The J. Scherk and J. H. Schwarz generalization of a Peres solution is shown to be the Peres solution in a different system of coordinates.     

Self-consistent solutions of the semiclassical Einstein equations with cosmological constant

The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parameters/=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable when and have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 < < 1/9. In both cases these solutions have a final de Sitter stage.     

Mixed problem for the Einstein equations

A question in the theory of gravitation which shares importance with attempts to experimentally observe gravitational waves and to solve various astrophysical problems by means of the general theory of relativity is the question of finding all possible formulations of the various problems, e. g., the Cauchy problem and the mixed problem with various modifications of the boundary conditions.     

On the type N twisting vacuum solution of the Einstein equations

One of the problems in the catalogue of solutions to the vacuum solutions of the Einstein equations is the dearth of solutions to what is known as the type N twisting metric. Were it to be found in its general form it would then, according to the Peeling theorem of Sachs (Proc R Soc Lond A270:103, 1962), describe the asymptotic gravitational field of an isolated source. The only known mathematical solution was found by Hauser (Phys Rev Lett 33:1112, 1974) and (Phys Rev Lett 33:1525, 1974). In this article the general equations are reduced to one third order complex equation for one complex variable plus one simple condition.     

A mixed problem for the Einstein equations and an approximate method of solution

The present paper poses a mixed problem for the Einstein equations. A combined method for solving the problem is introduced. The method consists of a combination of the finite-difference method for the time coordinate, and Galerkin's method for solving the system of equations so obtained. Existence and uniqueness conditions are found for the mixed problem in an appropriately introduced functional space. The convergence conditions for the method are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 113–118, May, 1976.     

A general exact solution of the Einstein-Dirac equations with the cosmological constant in homogeneous space

We have obtained a general exact solution of the system of Einstein-Dirac equations with the cosmological constant in homogeneous Riemannian space of the first type according to the Bianchi classification.     

Basic properties of the Einstein equations with a Ricci-Scalar-Dependent cosmological term

Basic properties of the Einstein equations modified by a cosmological Λ-term dependent on the Ricci scalar R are considered. We show that in addition to a nonzero divergence of the energy-momentum tensor of the matter and the consequent cold matter mass nonconservation as the Universe expands, this model suggests a significant modification of the equations for the gravitational potential and particle acceleration in the Newtonian approximation. These circumstances allow the necessary criteria for possible functional dependences Λ(R) to be formulated. Nevertheless, by introducing a variable Λ-term, we can look at the problems of dark matter and dark energy anew. In particular, we show that the model in which the cosmological term depends linearly on the Ricci scalar (this corresponds to the approximation of a more complex dependence in the case of low matter densities) makes it possible to satisfactorily describe the rotation curves of galaxies without invoking the dark matter hypothesis and to construct a cosmological model with a variable vacuum energy density, in qualitative agreement with the present views of the early Universe.     

On the universality of Einstein equations

It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalent to Einstein's general relativity interacting with additional matter fields.     

Investigation of the problem of singularities in general relativity

Recently Addy and Datta have obtained a linearized solution for isentropic motions of a perfect fluid by assigning Cauchy data on the hypersurfacex ⁴=0 and by imposing a restriction on the equation of state. In the present paper we pursue this study and discuss the problem of singularities from the standpoint of a local observer for which a singularity is defined as a state with an infinite proper rest mass density. It is shown that for a closed universe with any distribution of matter whatsoever there occurred a singularity in the past in the nonrotating parts of the universe and it must recur in the future. Furthermore, the collapse of a rotating fluid to a singularity seems inevitable when the relativistic equation of state is considered.     

Cauchy problem for the Einstein equations. III

The equations of small perturbations of the metric in synchronous coordinates are equivalent to a first-order system for which the Cauchy problem is correct only upon the satisfaction of several conditions imposed on the lowest symbols of the operator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 105–109, November, 1982.     

Brane world solution to the cosmological constant problem

We consider a model with two parallel (positive tension) 3-branes separated by a distance L in five-dimensional spacetime. If the interbrane space is anti--de Sitter, or is not precisely anti--de Sitter but contains no event horizons, the effective four-dimensional cosmological constant seen by observers on one of the branes (chosen to be the visible brane) becomes exponentially small as L grows large.     

Generalised cosmological Friedmann equations and the de Sitter solution

It is shown that the generalised cosmological Friedmann equations in the gauge theory of gravitation obtained earlier have solutions with the de Sitter metric and vanishing as well as nonvanishing torsion.