Cylindrically symmetric Brans-Dicke fields. I

A class of rigorous solutions for the Brans-Dicke scalar-tensor theory for Einstein-Rosen nonstatic cylindrically symmetric metric is obtained when only scalar field is present (vacuum solutions of Brans-Dicke theory). As the solutions of Brans-Dicke vacuum fields are conformal to either zero-mass scalar field or vacuum solutions of Einstein's gravitational theory, a set of solutions conformal to the above which correspond to zero-mass scalar field has also been obtained.     

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity (II)

It is shown that all torsion-free vacuum solutions of the model of de Sitter (dS) gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian (F ² + T ²), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations.     

Use of conformal,mapping to construct new solutions of the Einstein equations

A method is obtained for constructing new solutions of the Einstein equations from known solutions, beginning with the structure of the Einstein tensor and using conformai mapping. The sollowing cases are considered: transformation of vacuum solutions into hydrodynamic solutions and the transformation of hydrodynamic solutions into solutions of the same type. A necessary and sufficient condition, imposed on the vacuum metric, is found which guarantees that solutions with a hydrodynamic energy-momentum tensor will be produced from vacuum solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 110–114, November, 1973.The author is deeply grateful to Ya. I. Pugachev and to the members of the seminar directed by him for discussion of the results of this paper.     

Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent \({\gamma}\) lies in the stability regime \({(4/3, 2)}\), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is \({C^{{1}/{2}}}\)-Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent \({\gamma}\) lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.     

Electromagnetic Duality in General Relativity

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to those in electromagnetic theory. It turns out that the duality transformation implies the Einstein vacuum equation without the cosmological term. The vacuum equation is invariant under interchange of active and passive electric parts, giving rise to the same vacuum solutions but with the opposite sign for the gravitational constant. Further, by modifying the equation it is possible to construct interesting dual solutions to vacuum as well as to flat spacetimes.     

Quark Gluon Condensate,Virtuality and Susceptibility of QCD Vacuum

We study vacuum of QCD in this work. The structure of non-local quark vacuum condensate, values of various local quark and gluon vacuum condensates, quark-gluon mixed vacuum condensate, quark and gluon virtuality in QCD vacuum state, quark dynamical mass and susceptibility of QCD vacuum state to external field are predicted by use of the solutions of Dyson-Schwinger equations in “rainbow” approximation with a modeling gluon propagator and three different sets of quark-quark interaction parameters. Our theoretical predictions are in good agreement with the correspondent empirical values used widely in literature, and many other theoretical calculations. The quark propagator and self-energy functions are also obtained from the numerical solutions of Dyson-Schwinger equations. This work is centrally important for studying non-perturbative QCD, and has many important applications both in particle and nuclear physics.     

Cosmological Friedman models with the charged vacuum

Exact solutions corresponding to the stable charged vacuum are obtained for cosmological Friedman models with a charged scalar field. It is demonstrated that the presence of a massive vector field for the vacuum changes the law of expansion at early stages of evolution of the universe. A method for construction of solutions with the charged vacuum for an O(N) invariant scalar multiplet is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 19–24, December, 1985.     

A family of Jordan-Brans-Dicke Kerr solutions

A family of solutions of the vacuum Jordan-Brans-Dicke or scalar-tensor gravitational field equations is given. This family reduces to the Kerr rotating solution of the vacuum Einstein equations when the scalar field is constant. The family does not have spherical symmetry when the rotation is zero and the scalar field is not constant. The method used to generate the new solutions can also be used to obtain vacuum Jordan-Brans-Dicke solutions from any given vacuum stationary, axisymmetric solution.     

Axisymmetric vacuum fields in general projective relativity

A method has been derived which enables one to obtain solutions to the stationary, axially symmetric vacuum fields in general projective relativity developed by Arcidiacono from known solutions of the vacuum field in Einstein's theory. The analogue of the Kerr solution in general projective relativity has been obtained as an example. Finally, a relation between the stationary and static axially symmetric vacuum fields in general projective relativity has been derived.     

Linearization procedure for cylindrically and axially symmetric space-times

An investigation of the vacuum Einstein gravitational field equations for cylindrically and axially symmetric space-times is presented which leads to an equivalent differential system involving a simple nonlinearity only. The case when this equivalent system is linear is analyzed in detail and two methods for generating solutions of the Einstein vacuum equations are set up. As a result, in the axially symmetric case the linearity of the equivalent system characterizes completely the Kramer-Neugebauer transforms of Papapetrou line elements. Accordingly, Weyl solutions are shown to generate exhaustively both Lewis and van Stockum solutions. Analogous results are obtained also in the cylindrically symmetric case.     

Fermionic zero modes for dyons and chiral symmetry breaking in QCD

Dyonic classical solutions of Yang-Mills theory are considered and the complete set of fermionic zero modes of these solutions are studied. Representing the QCD vacuum as a gas of dyons, one obtains chiral symmetry breaking due to zero modes similarly to the case of instantonic vacuum.     

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.     

Rotating Dilaton Solutions in 2+1 Dimensions

We report a three parameter family of solutions for dilaton gravity in 2+1 dimensions with finite mass and finite angular momentum. These solutions are obtained by a compactification of vacuum solutions in 3+1 dimensions with cylindrical symmetry. One class of solutions corresponds to conical singularities and the other leads to curvature singularities.     

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.     

Kundt solutions of Einstein's equations with one non-null Killing vector

The Kundt class of algebraically special solutions of Einstein's field equations, representing vacuum and electromagnetic null fields with one twisting, non-null Killing vector, is discussed. This generalizes the case with a hypersurface-orthogonal Killing vector field which is discussed by Kramer and Neugebauer [3]. The solutions are shown to be equivalent to the Hoenselaers (vacuum) and Hoenselaers-Skea (electromagnetic null) solutions, once some small corrections and the relevant coordinate transformations are made to the latter solutions.     

New axially symmetric solutions of the Einstein-Maxwell equations

New exact solutions of the algebraic form for the static Einstein-Maxwell equations representing the exterior gravitational field of a massive magnetic dipole are derived. They are then used for construction of the stationary electrovacuum solutions reducing to the Schwarzschild metric in a pure vacuum limit.     

Almost Birkhoff theorem in general relativity

We extend Birkhoff’s theorem for almost LRS-II vacuum spacetimes to show that the rigidity of spherical vacuum solutions of Einstein’s field equations continues even in the perturbed scenario.     

Static solutions of the Brans-Dicke equations

It is shown that, given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection. Spherically and axially symmetric fields are considered explicitly. It is demonstrated how some solutions of the Brans-Dicke equations may be obtained without having to solve any field equations explicitly at all.     

On Chelnokov-Zeitlin solutions to the vacuum Einstein equations

The equivalence of the Chelnokov-Zeitlin solutions to the vacuum Einstein equations with a special class of Lewis solutions is established in a direct way. Also, an oversight on the signature of the solutions is pointed out and corrected.     

General solutions for a static isotropic metric in the Brans-Dicke gravitational theory

The complete set of vacuum solutions for the metric tensor of a static spherically symmetric field is given, some of these solutions showing the remarkable feature of not agreeing-even in first order-with the classically well-known weak-field solutions of the Brans-Dicke (B.D.) equations. The existence of a particular two-parameter family of solutions raises severe doubts about the so-called Machian aspect of B.D. theory.