Some solutions of Einstein's equations with viscous fluids

If we admit the bulk and shear viscosity coefficients to be homogeneous in all the stages of the cosmic evolution and the fact that these can be expressed in terms of the metric coefficients, we obtain some cosmological models that are exact solutions of Einstein's equations. The metric utilized is the one of Szekeres's class II and the curvature source is a viscous fluid without heat flux.     

Anisotropic and inhomogeneous solutions of the Einstein-Boltzmann equations for Friedmann-Robertson-Walker spacetimes

It is shown how the techniques that yield anisotropic and inhomogeneous solutions of the Einstein-Liouville equations for Friedmann-Robertson-Walker (FRW) spacetimes can be used to demonstrate the existence of anisotropic and inhomogeneous FRW solutions of the Einstein-Boltzmann equations, thereby providing a possible relativistic kinetic theory basis (and consequently a more self-consistent basis) for tilting imperfect fluid FRW cosmological models (based on a set of phenomenological laws of relativistic thermodynamics).     

The universal solution of Einstein's equations of general relativity

Einstein's equations of general relativity are solved in terms of gravitational potential derivatives, withT equal to mass and/or field energy such thatT 0 outside a body. The line element equation then describes the variance of test particle internal geometrical structure and time-rate due to work done in a field, not the space-time curvature. Specific properties of gravitational fields and bodies come from this new solution: (a) The gravitational field consists of electromagnetic spin 2 gravitons which produce the gravitational force through the magnetic vector. (b) The gravitational mass is the Newtonian mass, not the relativistic mass, of a moving body. (c) An action principle exists in gravitation theory. (d) Attractive gravity exists between matter and antimatter. (e) Unification with quantum physics appears possible.     

Anisotropic stellar structure equations for magnetized strange stars

The fact that a fermion system in an external magnetic field breaks the spherical symmetry suggests that its intrinsic geometry is axisymmetric rather than spherical. In this work we analyze the impact of anisotropic pressures, due to the presence of a magnetic field, in the structure equations of a magnetized quark star.We assume a cylindrical metric and an anisotropic energy momentum tensor for the source. We found that there is a maximum magnetic field that a strange star can sustain, closely related to the violation of the virial relations.     

Einstein's greatest mistake?

Recent astronomical observations have re-opened the old paradox that the universe appears to be younger than some of the objects within it. I suggest that a natural resolution of this paradox lies with the re-introduction of Einstein's cosmological constant, and that sentiment against this constant fails to take into account its connection with quantum field theory.     

Conformally flat solutions to the Einstein-Maxwell equations

The integration of the Einstein-Maxwell equations for an anisotropic charged fluid sphere acting as a source of the Reissner-Nordström metric is considered, under the assumption of a conformally flat interior metric. The solutions asymptotically tend to static configurations. In the isotropic pressure limiting case, the non-static solutions are found to be incompatible with charged models.     

Asymptotic solutions of the viscous solar wind equations

There is one and only one solution of the viscous solar wind equations which is asymptotically represented for large distances from the sun by a formal solution of Y. C. Whang, C. K. Liu, and C. C. Chang as a power series in the inverse one third power of the distance.     

Convergent solutions of the inviscid solar wind equations

Formal solutions of the inviscid solar wind equations as power series in the inverse one seventh power and in the inverse one fifth power of the distance from the sun are shown to be convergent.     

Viscous solutions of the two-fluid solar wind equations

The two-fluid equations describing the ideal, steady, viscous solar wind are examined, and supersonic solutions are sought in which the electron and ion temperatures vary as inverse powers of the radial distance from the Sun. Just two solutions are found, and these are analogous to those found by Whang et al. (1966) and Dahlberg (1970) in one-fluid theory.     

The exact solution of Einstein's disc

In this Letter, Einstein's equations of gravitational field have been used to calculate the metric of Einstein's disc. The known result has been obtained, but its physical significance is made very clear.     

On stationary solutions of some canonical systems of differential equations

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used.     

Exact two-soliton solutions to the Einstein gravitational field equations

Following our previous work on the existence of 1-soliton solution to the Einstein gravitational field equations in the presence of a spherically-symmetric static background field, we have found six sets of analytical 2-soliton solutions to the Einstein field equations under a certain ansatz in the absence of the stated background field. Numerical analysis shows that if the two solitons of the transverse nature are injected at space variable z±, the longitudinal field componentg ₃₃ will acquire non-zero values for a bounded spatial region at later time. The nature of the solitons becomes rather complex when they interact. The amplitudeg of each soliton may change its magnitude resulting from the interaction. We have found that we might interpret the evolution of one field component as the gravitational instanton in our solutions. We must remark also that the total energy of the interacting solitons remains constant, as expected, at all time. These solutions correspond to the situation where the Riemann tensor is in general non-zero and are truly non-trivial solutions.     

A Shearing Non-Adiabatic Solution of Einstein's Equations

An analytical solution of the Einstein's equations is found for a collapsing radiating body, consisting of an isotropic fluid with shear undergoing radial heat flow with outgoing radiation. The behavior of the density, pressure, heat flux, mass and luminosity are analyzed for a body of 6 M⊙. This revised version was published online in July 2006 with corrections to the Cover Date.     

The solutions of certain generalized anomalous diffusion equations of fractional order

In this paper we have proposed certain generalizations of anomalous diffusion equations for fractional order. These diffusion equations are solved by the method of Laplace transform with respect to the time variable and Fourier transform with respect to the space variable. The solutions of some known diffusion equations are also shown to be derived here.     

On static spherical symmetric solutions of the Bach-Einstein gravitational field equations

For field equations of 4th order, follwing from a Lagrangian “Ricci scalar plus Weyl scalar”, it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = o nad r = ∞ resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme.     

Singularity-free static centrally symmetric solutions of some fourth order gravitational field equations

The fourth order field equations proposed by TREDER with a linear combination of BACH 's tensor and EINSTEIN 's tensor on the left-hand side admit static centrally symmetric solutions which are analytical and non-flat in some neighborhood of the centre of symmetry.