Cauchy problem for perfectly conducting viscous magnetofluid in general relativity

In recent years evidence has been accumulating that shows that in the early universe matter was in the dense plasma state, possibly associated with a strong primordial magnetic field. It has also been shown that the effect of particle creation at this epoch is equivalent in macroscopic theory to the effect of viscosity, of which shear viscosity was much more important and which may have removed some anisotropy present initially. The aim of this paper is to study the Cauchy problem for a perfectly conducting viscous magnetofluid assuming the pressure and coefficients of viscosity to be functions of the density alone. We also study the consistency conditions satisfied by four unknown quantities on the initial Cauchy hypersurface and show that hydrodynamic and Alfvén waves cannot be differentiated, unlike what occurs in a perfect magnetofluid.     

Convective deformations of the matter tensor for the relativistic perfect magnetofluid

Maugin's Scheme for a relativistic perfect magnetofluid is used to study the convection-free stress and convective deformations of the matter tensor for the magnetofluid. It is proved that the convection-free stress of the magnetofluid implies the conservation of the pressure, the density, and the magnitude of the magnetic field along the flow vector. The relation between convective deformations of the matter tensor for the magnetofluid and deformation tensor is obtained.     

On space-time permeated by the perfect magnetofluid

A space-time permeated by the self-gravitating perfect fluid with infinite electrical conductivity and constant magnetic permeability (perfect magnetofluid) is investigated. For aC space defined as the space in which the divergence of conformal curvature vanishes, it is proved that the rotation explicitly depends on the magnetic field. In aJ space characterized by the vanishing of the divergence of Petrov space-matter tensor, the invariance of the energy density, the isotropic pressure, and the magnitude of the magnetic field along the divergence-free magnetic lines is established. It is found that if the stress-energy tensor of the perfect magnetofluid is a Killing tensor, the energy density, the isotropic pressure, and the magnitude of the magnetic field are constant. Moreover it is shown that the stream lines are expansion-free and the magnetic lines are divergence-free. It is proved that the complexion of the field of the perfect magnetofluid remains invariant along the magnetic lines if and only if these lines are normal to the lines of vorticity.     

On the Cauchy problem of the relativistic Boltzmann equation

It is shown that the relativistic Boltzmann equation has a local solution through an initial distribution function, if the scattering cross section is bounded for high energies and if the initial distribution falls off exponentially with the energy.Supported by the National Aeronautics and Space Administration under Grant No. NGR 44-004-042.On leave from the University of Heidelberg.     

On the geometry of relativistic magnetofluid flows

This paper deals with the construction of “magnetic vorticity” vector using Greenberg's theory of spacelike congruences for the trajectories of magnetic fields. A set of propagation equations is derived for the geometrical invariants associated with the congruences of magnetic field lines and fluid flow lines. Some applications of these propagation equations are made. A generalization of Ferraro's law of isorotation is obtained employing the propagation equation forω ² along the magnetic field lines.     

The Hamiltonian structure of general relativistic perfect fluids

We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual canonical symplectic structure. The evolution is governed by a Hamiltonian which is equivalent to that obtained from a canonical analysis. The relationship of our Hamiltonian structure with other approaches in the literature, such as Clebsch potentials, Lagrangian to Eulerian transformations, and its use in clarifying linearization stability, are discussed.Research supported in part by NSF grant MCS 81-08814(A02)Research supported in part by NSF grant MCS 81-07086     

On general relativistic vortex-dynamics

The general theory of relativity gives an absolutely covariant formulation of Helmholtz's laws of vorticity which is valid in arbitrary reference systems. For small relative velocities u_i, (with u_i,u²) these generally covariant laws deliver Helmholtz's first law for a vorticity _i in a rigidly rotating references system with the angular velocity of the rotation.     

Global aspects of the Cauchy problem in general relativity

It is shown that, given any set of initial data for Einstein's equations which satisfy the constraint conditions, there exists a development of that data which is maximal in the sense that it is an extension of every other development. These maximal developments form a well-defined class of solutions of Einstein's equations. Any solution of Einstein's equations which has a Cauchy surface may be embedded in exactly one such maximal development.     

Canonical decomposition of the general relativistic initial value problem

A canonical transformation is employed to implement a conformal transformation of the configuration variables of general relativity. The transformation is so chosen that the spatial constraints become algebraic in the trace of the momentum density. The temporal constraint is then found to have the form of York and O'Murchadha. The role played by the York coordinate condition in decoupling the constraint equations is examined, and a procedure to solve the constraint equations without employing such a coordinate condition is sketched.     

On the Cauchy problem foe Einstein's equations

The mechanism of nonuniqueness of the solution to the problem with initial conditions for the equations of the free gravitational field is elucidated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 32–34, September, 1980.I thank V. L. Bonch-Bruevich, N. S. Kardashov, and I. D. Novikov for interest and assistance.     

On freeze-out problem in relativistic hydrodynamics

A finite unbound system which is equilibrium in one reference frame is in general nonequilibrium in another frame. This is a consequence of the relative character of the time synchronization in the relativistic physics. This puzzle was a prime motivation of the Cooper-Frye approach to the freeze-out in relativistic hydrodynamics. Solution of the puzzle reveals that the Cooper-Frye recipe is far not a unique phenomenological method that meets requirements of energy-momentum conservation. Alternative freeze-out recipes are considered and discussed. The text was submitted by the authors in English.     

A global analysis approach to the general relativistic fluid ball problem

That a self-gravitating perfect fluid in empty space has a spherical equilibrium configuration if it is static-i.e., nonrotating-is considered physically evident, but has not yet been rigorously derived from Einstein's field equations together with suitable asymptotic conditions. In this paper the global analysis techniques developed recently mainly by Fischer, Marsden, and Cantor are used to derive the result that if a family of static perfect fluid solutions with fixed total gravitational massm and fixed equation of state(p) satisfying 0 p and 0 d/dp < depends differentiably on a parameter and contains the spherically symmetric solution then it must consist of solutions diffeomorphic to the spherically symmetric one.Partially supported by the National Sciences and Engineering Research Council, grant No. A8059.     

Thermodynamics of relativistic rotating perfect fluids

A new formulation of thermodynamics for special and general relativistic rotating perfect fluids is developed. Both isolated systems and portions of isolated systems electrically uncharged or charged are treated. Exploiting the symmetry of motion of stationary axisymmetric fluids, the global thermodynamic functions, including total energy and spin, are defined as free scalars, represented by hypersurface integrals of conserved vectors. Local equilibrium parameters such as local temperature and chemical potential are scalar functions. There also exist global equilibrium parameters, global temperature and global chemical potential, which are free scalars. The connection between local and global conditions of thermodynamic equilibrium is made clear and explicit. Thermodynamic potentials are introduced in the context of treating open systems in a relativistically invariant way.     

Application of the method of kinemetric invariants to the Cauchy problem in general relativity

A class of metrics in which the spatial part describes flat space is considered. The algebraic condition of reduction of the three-dimensional part of an arbitrary metric to Cartesian form is found. The Cauchy problem for these metrics is considered in terms of kinemetrically invariant quantities. The results are used to solve the Cauchy problem for a spherically symmetric gravitational field. A solution is also obtained for the tachyon twin of the Schwarzschild field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 48–52, April, 1976.     

On free general relativistic initial data on the light cone

We provide a simple explicit parameterization of free general relativistic data on the light cone.     

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.     

On the problem of a cyclotron for particles with relativistic increase in mass

The paper deals with the problem of circular accelerators with time constant magnetic field and constant frequency of the accelerating voltage. An analysis is made of the possibility of compensating the change in time of revolution (caused by the increase in mass during energy growth) by simultaneous axial and radial displacement of the equilibrium orbit. It is found that the problem can be solved only with certain approximations. The approximate numerical parameters of the accelerator are given and the approximations used are discussed. The paper is based on [1].

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In conclusion the author thanks the director of the institute Ing. J. Váa for permitting publication and Dr. M. Seidl, Ing. J. Teichman, Ing. R. Klíma and Ing. Z. Sedláek who, apart from solving other problems of the accelerator, cooperated with the author on its very conception.     

On the spherical symmetry of static perfect fluids in general relativity

We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide class of general relativistic, static perfect-fluid models provided there exists a spherically symmetric model with the same equation of state and surface potential. The method of proof, which is inspired by recent work of Masood-ul-Alam, is illustrated by demonstrating uniqueness of a class of solutions due to Buchdahl which correspond to an extreme case of the inequality on the equation of state required by our theorem.