No-Go theorem in spacetimes with two commuting spacelike killing vectors

Four-dimensional Riemannian spacetimes with two commuting spacelike Killing vectors are studied in Einstein's theory of gravity, and found that no outer apparent horizons exist, provided that the dominant energy condition holds.     

Homogeneous solutions of the Einstein equations with an ideal charged fluid

The space-times V₄ with an ideal charged fluid as source allowing the motion group G_r, r 4, are investigated. It is assumed that the fluid velocity vector is directed along the timelike vector of the Killing group. In the case of groups G₄, acting on V₄, as well as groups of higher mobility, a complete investigation is performed of the space-times by using the system of Einstein-Maxwell equations. Exact solutions are found with fourth- and fifth-order groups.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 74–79, December, 1984.     

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 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.     

Twistfree and Papapetrou vacuum spacetimes with a spacelike killing vector

We show that, for the case of vacuum solutions of the Einstein equations with a spacelike hypersurface orthogonal Killing vector /³ and associated metricds ² =e ^2U (dx ³)² +e ^–2U _ab dx ^a dx ^b whereU is not a constant, there exists at every point of the quotient 3-space a plane of vectorsK ^a such that £_KR_ab=0 andK ^a R_ab=0 whereR{inab} is the Ricci tensor formed from _ab . Then in the case whereU{in,a} is a timelike or spacelike vector in the quotient 3-space, Petrov type I solutions of the vacuum field equations are obtained. In the simpler case whereU{in,a} is a null vector in the quotient 3-space, the complete solution of the vacuum field equations is obtained. It is shown that this solution is Petrov type III of Kundt's class. For the case of Papapetrou solutions where there is a twist potential which is a function ofU, solutions corresponding to the twistfree solutions are given.     

Nonexistence of quantum fields associated with two-dimensional spacelike planes

It is shown that in a relativistic quantum field theory satisfying Wightman's axioms, there are no nontrivial field-like operators, or even bilinear forms, associated to a two (or less)-dimensional spacelike plane in Minkowski space. This generalizes Wightman's result that fields can not be defined as operators at a point and stands in contrast to Borchers' result that field operators can be associated with one-dimensional timelike planes.     

Einstein equations for a fluid sphere and the mass function

A mass function is used to study the Einstein equations in the case of a spherically symmetric ideal fluid. A new system of Einstein equations is proposed and it is shown that this system is simpler than the standard system for the given solutions. The mass function is used to obtain certain exact solutions, including the general solution for a uniform fluid sphere.Dnepropetrovsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 90–95, April, 1995.     

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 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.     

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].     

The structure of the space of solutions of Einstein’s equations coupled with scalar fields

Following the work of Arms, Fischer, Marsden and Moncrief, it is proved that the space of solutions of Einstein’s equations coupled with self-gravitating mass-less scalar fields has conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a nontrivial set of simultaneous Killing fields, either all spacelike or including one (independent) timelike.     

On certain new exact solutions of the Einstein equations for axisymmetric rotating fields

We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations （PDEs） and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.     

Einstein equations and Fierz-Pauli equations with self-interaction in quantum gravity

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, together with the covariant Hilbert-gauge condition, where W denotes the covariant wave operator and G _ik the Einstein tensor of the metric g _ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions _ik(z)with g ^ik Z,_i Z,_k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,p(g*)²(x)³h hG/ c ³ concerning the total nonlinear gravitational field g ^*ik =g ^k + ^k .     

Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid

Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vector such that _;v + _v; =2g _v are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.This work was supported in part by the United States Atomic Energy Commission under contract number AT(04-3)-34 Project Agreement No. 125. It was completed when one of the authors (A.H.T.) was on sabbatical leave from the University of California, Berkeley and in residence at the Department of Applied Mathematics and Theoretical Physics, Cambridge University.     

Stationary axisymmetric solutions of the Einstein equations with rigidly rotating perfect fluid and nonlinear charged sources

A class of stationary, rigidly rotating perfect fluids coupled with nonlinear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found for the equation of state 3p+=const. We use a structure function for the Born-Infeld nonlinear electrodynamics which is invariant under duality rotations and a metric possessing a four-parameter group of motions. The solution is of Petrov type D and the eigenvectors of the electromagnetic field are aligned to the Debever-Penrose vectors.     

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.     

Solutions of the Einstein equations with zero coupling

A study is made of the algebraic properties and geometric structure of solutions of the Einstein equations the metric tensors of which differ by the product of two identical isotropic vectors. Proof is offered for a theorem which states that when the congruence of isotropic lines with a tangent vector field used for the coupling is geodesic, both spaces are algebraically special in the Petrov-Penrose sense. A noncoordinate transformation of this type can be used to find new exact solutions from known solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 32–37, October, 1971.The author thanks Professor V. I. Rodichev for interest in the study and for valuable discussions.