Linear Perturbations for the Vacuum Axisymmetric Einstein Equations

In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.     

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.     

Instantons and gravitation theory

Some problems related to using nonperturbative quantization methods in theories of gauge fields and gravitation are studied. The unification of interactions is considered in the context of the geometric theory of gauge fields. The notion of vacuum in the unified interaction theory and the role of instantons in the vacuum structure are considered. The relation between the definitions of instantons and the energymomentum tensor of a gauge field and also the role played by the vacuum solutions to the Einstein equations in the definition of vacuum for gauge fields are demonstrated. The Schwarzschild solution, as well as the entire class of vacuum solutions to the Einstein equations, is a gravitational instanton even though the signature of the space-time metric is hyperbolic. Gravitation, oncluding the Einstein version, is considered a special case of an interaction described by a non-Abelian gauge field. Translated from Teoreticheskaya i Matematicheskaya. Fizika. Vol. 115, No. 2, pp. 312–320, May. 1998.     

On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.     

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the “Double Null Foliation Gauge”

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region much larger than the one provided by the Cauchy–Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the a priori estimates for the Weyl equations, associated with the “Bel-Robinson norms”. In particular, if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a “geometric” way the Einstein equations, then we show how the Cauchy–Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally, we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.     

On wave solutions of gauge-invariant generalization of field theories with asymmetric fundamental tensor in a generalized peres space-time

Summary The gauge invariant generalization of field theories with asymmetric fundamental tensor developed by Buchdahl has been considered and its plane wave-like solutions in the sense of Takeno are investigated in generalized Peres space-time, recently considered by the author. It has been shown that under certain conditions these solutions become identical with those of strong field equations of Einstein in the same space-time. It has been also shown that this space-time satisfying the field equations of Buchdahl admits a parallel null vector field and is gravitationally null which further, transforms to other well known forms of space-time under a new time coordinate Z=z-t. Entrata in Redazione il 2 afosto 1976. Work is supported by State Council of Science and Technology (U.P.), India.     

Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

This paper is devoted to the study of a transonic shock in three-dimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptic-hyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

     

Remarks on the local version of the inverse scattering method

It is very likely that all local holomorphic solutions of integrable (1+1)-dimensional parabolic-type evolution equations can be obtained from the zero solution by formal gauge transformations that belong (as formal power series) to appropriate Gevrey classes. We describe in detail the construction of solutions by means of convergent gauge transformations and prove an assertion converse to the above conjecture; namely, we suggest a simple necessary condition for the existence of a local holomorphic solution to the Cauchy problem for the evolution equations under consideration in terms of scattering data of initial conditions.     

Wedge Dislocation in the Geometric Theory of Defects

We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.     

Oscillatory Singularities in Bianchi Models with Magnetic Fields

An idea which has been around in general relativity for more than 40  years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time, progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to cases where the Einstein equations are coupled to matter fields, focussing on the example of a dynamical system arising from the Einstein–Maxwell equations with symmetry of Bianchi type VI₀. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration than in the vacuum case. The difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.     

Exact stationary wave patterns in three coupled nonlinear Schrödinger/Gross–Pitaevskii equations

The evolution of a Bose–Einstein condensate (BEC) with an internal degree of freedom, i.e., spinor BEC, is governed by a system of three coupled mean-field equations. The system admits the application of the inverse scattering transform and Hirota bilinear method under appropriate conditions, which makes it possible to generate exact analytical solutions relevant to physical applications. Here, we produce six families of exact periodic solutions, directly constructed in terms of Jacobi elliptic functions. Solitary-wave limit forms, obtained from these solutions in the long-wave limit, are presented too.     

The relationship between the Einstein and Yang-Mills equations

In this paper we prove that special requirements to Yang-Mills equations on a 4-dimensional conformally connected manifold allow one to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. We propose an algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang-Mills conformally connected manifold. As an application of the algorithm, we prove that the metric of any Einstein space and the Robertson-Walker metric are embeddable into the specified manifold.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

Solutions globales des équations d'Einstein–Maxwell avec jauge harmonique et jauge de Lorenz

Adapting a method of Lindblad and Rodnianski, we prove existence of global solutions for the Einstein–Maxwell equations with small enough smooth and asymptotically flat initial data. We use harmonic gauge and Lorenz gauge. To cite this article: J. Loizelet, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Einstein Equations on a 5-Manifold with a Causal Structure in the Absence of Matter Fields

A 5-manifold with a restricted smooth structure and an appropriate group of coordinate transformations including general relativity and gauge transformations is considered. An explicit expression for the Riemannian curvature of a 4-dimensional vector distribution is obtained, which implies the classical Einstein and Maxwell equations. Bibliography: 14 titles.     

D-dimensionalp-brane cosmological models associated with a Lie algebra of the typeA m

We study a D-dimensional cosmological model on the manifold M=ℝ×M₀×˯×M_n describing an evolution of n+1 Einstein factor spaces M_i in a theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so-called pseudo-Euclidean Toda-like system. Assuming that the characteristic vectors related to the configuration of p-branes and their couplings to the dilatonic scalar fields can be interpreted as the root vectors of a Lie algebra of the type A_m≡sl(m+1,ℂ), we reduce the model to an open Toda chain, which is integrable by the customary methods. The resulting metric has the form of the Kasner solution. We single out the particular model describing the Friedmanlike evolution of the three-dimensional external factor space M₀ (in the Einsteinian conformal gauge) and the contraction of the internal factor spaces M₁,…,M_n. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 374–394, June, 2000.     

Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields

The isovector fields (infinitesimal generators of Lie groups) of Einstein vacuum equations for stationary axially symmetric rotating fields, in conventional form, that is a coupled system of nonlinear partial differential equations (PDEs) of second order are derived using the geometric prolongation technique. Some symmetry transformations and similarity (exact) solutions of Einstein vacuum equations are obtained.     

Dynamical Symmetry Breaking in Geometrodynamics

Using a first-order perturbative formulation, we analyze the local loss of symmetry when a source of electromagnetic and gravitational fields interacts with an agent that perturbs the original geometry associated with the source. We had proved that the local gauge groups are isomorphic to local groups of transformations of special tetrads. These tetrads define two orthogonal planes at every point in space–time such that every vector in these local planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. Because the local gauge symmetry in Abelian or even non-Abelian field structures in four-dimensional Lorentzian space–times is manifested by the existence of local planes of symmetry, the loss of symmetry is manifested by a tilt of these planes under the influence of an external agent. In this strict sense, the original local symmetry is lost. We thus prove that the new planes at the same point after the tilting generated by the perturbation correspond to a new symmetry. Our goal here is to show that the geometric manifestation of local gauge symmetries is dynamical. Although the original local symmetries are lost, new symmetries arise. This is evidence for a dynamical evolution of local symmetries. We formulate a new theorem on dynamical symmetry evolution. The proposed new classical model can be useful for better understanding anomalies in quantum field theories.     

Positivity of solutions to inhomogeneous mixed-type equations of higher orders

We study the solution to inhomogeneous mixed-type elliptic-hyperbolic equations of higher orders. We prove that the solution sign is constant and depends on the sign of the right-hand side of the equation.