Spherically symmetric perfect fluid solutions of Einstein's equations in noncomoving coordinates

Almost all known spherically symmetric perfect fluid solutions of Einstein's equations have been obtained in comoving coordinates and nearly all are shear-free. In this paper we study two solutions in noncomoving coordinates and show that they contain shear.     

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.     

A class of solutions of Einstein's equations for the interior of a rigidly rotating perfect fluid

A general class of solutions of Einstein's equations for the interior of a rigidly rotating axisymmetric perfect fluid is presented, which depends on an arbitrary function. To get solutions explicitly one has to calculate two integrals involving the arbitrary function. The equipressure surfaces of all solutions of the class are spheres or planes. A family of solutions, which depend on four arbitrary real constants, is calculated explicitly. The solution of the family, which is obtained if we assign a specific value to one of its parameters, and which was found before, is futher generalized with the addition of one more parameter.     

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.     

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.     

On the existence of a general rotating solution of Einstein's equations

In an earlier paper we considered a power-series expansion of the metric for a rotating field in terms of a parameter and constructed a solution of Einstein's equations to the first few orders in terms of two harmonic functions. We encountered a pair of Poisson-type equations which were apparently insoluble explicitly. The form of the metric considered was the Weyl-Lewis-Papapetrou form. In this paper we consider a power-series expansion of the most general form of a rotating metric and show that one encounters the same two Poisson equations as before. If these equations are insoluble explicitly, as seems likely, then a general solution depending on two harmonic functions cannot exist in closed form.     

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.     

A plane symmetric solution of Einstein's field equations of general relativity containing zero-rest-mass scalar fields

An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.     

Einstein's equations and locality

The problem of formulating a local quantum theory of Einstein equations is examined. It is proved that Einstein equations cannot hold as operator equations if written in terms of a potentialh _µ(D) which is a weakly local field. This result is independent of the kind of metric chosen in the Hilbert space and it doesn't require covariance ofh _µ.As a consequence, the peculiar features of the radiation gauge method, i.e. non locality and non covariance, appear as necessary features of any solution not involving unphysical particles.On leave of absence from: Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy.     

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.     

Two mass solutions of Einstein's vacuum equations: The double Kerr solution

The newly developed transformations which allow one to generate solutions of Einstein's vacuum equations from given ones have become in the meantime very powerful tools for constructing new solutions. We prove the equivalence of Kramer-Neugebauer and HKX transformations and construct the double Kerr solution by means of four HKX transformations. The resulting solution is analysed in detail. We employ an analytic continuation of the parameters to pass from underextreme to hyperextreme constituents and pose conditions for asymptotic flatness, for existence of an axis between the massive objects, and for balance caused by the repulsive interaction of the angular momenta. We define mass and angular momentum of the single constituents and compute these quantities explicitly in addition to the total mass and the total angular momentum of the solution. The distance and force between the massive objects are defined and given in a suitable approximation. We prove that two black holes—constituents possessing a horizon—cannot be in balance. The above-mentioned conditions are solved for equal hyperextreme constituents which balance due to spin-spin repulsion. We give the distance of balance. Finally, we compare rotating and nonrotating two-mass systems of equal masses and equal distance between them and estimate the change of force between the masses caused by the spin-spin repulsion.     

A solution for a family of perfect fluid cylinders in general relativity

A static solution of Einstein's equations representing perfect fluid cylinders of finite radius is presented. The density and pressurep are finite and monotonic decreasing from the axis to the boundary. The ratiop/ (< 1) also decreases steadily. The regularity condition at the axis is satisfied and the solution is free of singularity. The equation of state is found and special cases considered. It is shown that the solution can be joined to the vacuum cylindrical metric at the boundary. There is then at least one free parameter left in the family of solutions.     

Some properties of Einstein's equations

For the matrix Q of the characteristic determinant of the field equations, left null vectors of general form are found. Some special solutions are constructed and some of their properties investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 7–9, October, 1980.I am grateful to V. L. Bonch-Bruevich, N. S. Kardashov, and I. D. Novikov for assistance.     

Characteristic manifolds of Einstein's equations

Several classes of coordinate conditions admitting the existence of different types of waves are studied. The velocity of propagation of the latter is not necessarily equal to the fundamental velocity.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 9, pp. 99–102, September, 1981.     

The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

Einstein's equations as constitutive relations

The aim of this article is to present one more possible derivation of the Einstein's equations of general relativity. The new route is based on old concepts of continuum mechanics, extended to four spacetime dimensions. Our main idea is to consider Einstein's equations as the constitutive relations of the spacetime manifold. Moreover, for us, the spacetime manifold is a nonlinear elastic medium with extrinsic defects (extra-matter). This idea dates back to Cauchy's treatment of elasticity. In Cauchy's words: the crucial ingredient in solid mechanics is the equation which expresses how the force intensity at any point in a body is related to the change of shape near that point. We will try to generalize them.     

Einstein's equations near spatial infinity

A new class of space-times is introduced which, in a neighbourhood of spatial infinity, allows an expansion in negative powers of a radial coordinate. Einstein's vacuum equations give rise to a hierarchy of linear equations for the coefficients in this expansion. It is demonstrated that this hierarchy can be completely solved provided the initial data satisfy certain constraints.Work supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project no. 4069