Properties of Perturbative Solutions of Unilateral Matrix Equations

A left-unilateral matrix equation is an algebraic equation of the form a ₀+a ₁ x+a ₂ x ²+·+a _n x ⁿ =0 where the coefficients a _r and the unknown x are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.     

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the papers style. This research was partially supported by NSF grant DMS-0305048.     

Field Equations due to a Constant Modification in General Relativity

In this article, by adding a constant to Einstein-Hilbert action, we derive field equations for a non-vacuum space. Also we derive a general solution for these field equations, considering a de Sitter like initial geometric constraint. It is shown that how this additional constant can affect usual gravitational field equations, which are derived from general relativity.     

Degenerate Solutions of General Relativity from Topological Field Theory

Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional $BF$ theory on any framed surface Σ embedded in M. In these solutions the cotetrad field e (and thus the metric) vanishes outside a neighborhood of Σ, while inside this neighborhood the connection A and the field satisfy the equations of 4-dimensional BF theory. Our construction works in any signature and with any value of the cosmological constant. If for some 3-manifold S, at fixed time our solutions typically describe “flux tubes of area”: the 3-metric vanishes outside a collection of thickened links embedded in S, while inside these thickened links it is nondegenerate only in the two transverse directions. We comment on the quantization of the theory of solutions of this form and its relation to the loop representation of quantum gravity. Received: 21 April 1997 / Accepted: 22 August 1997     

Charged Analogues of Henning Knutsen Type Solutions in General Relativity

In the present article, we have found charged analogues of Henning Knutsen’s interior solutions which join smoothly to the Reissner–Nordstrom metric at the pressure free interface. The solutions are singularity free and analyzed numerically with respect to pressure, energy-density and charge-density in details. The solutions so obtained also present the generalization of A.L. Mehra’s solutions.     

Exact Analytical Solution to Equations of Perihelion Advance in General Relativity

Previously the perihelion advance in binary system was computed approximately. We will present an exact analytical solution to nonlinear differential equation of perihelion advance by method of Jacobian elliptic function and the advanced angle between successive perihelions. Project supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars from Ministry of Education, China (Grant No. [2004]527) and the Natural Science Foundation of Hunan Province, China (Grant No. 06JJ2026)     

A Solution to Einstein's Equations for the Mixmaster Universe in Complex General Relativity

Starting from Einstein's equations of the Classical General Relativity, new kinds of solutions for the Mixmaster model are explored. By dispensing with the extension to the complex variable field, which is usual in problems such as the Laplace equation or the harmonic oscillator, in a similar manner to that of Quantum Mechanics, the equations appear to have solutions that belong to the complex General Relativity. A first integral is performed by establishing a separation of the first derivatives. Then a second integral is obtained once the respective equations with separate variables are found and whose integrals provide a family of complex solutions. However, reality conditions do not seem to be easily imposed at this stage. Above all, it is significant that the classical Einstein's equations for the debatably integrable Mixmaster model present complex solutions.     

On Singularities of General Relativity and Gravitational Equations of Fourth Order

Comparing Einstein's gravitational equations and equations containing fourth-order derivative corrections we discuss some aspects of singularities of spherically symmetric static vacuum solutions from the point of view, first, of the coupling to non-gravitational matter and, second, of the Einsteinian classical particle programme.     

The Quasi-Maxwellian Equations of General Relativity: Applications to Perturbation Theory

A comprehensive review of the equations of general relativity in the quasi-Maxwellian (QM) formalism introduced by Jordan, Ehlers and Kundt is presented. Our main interest concerns its applications to the analysis of the perturbation of standard cosmology in the Friedman-Lemaître-Robertson-Walker framework. The major achievement of the QM scheme is its use of completely gauge-independent quantities. We shall see that in the QM-scheme, we deal directly with observable quantities. This reveals its advantage over the old method introduced by Lifshitz that deals with perturbation in the standard framework. For completeness, we compare the QM-scheme to the gauge-independent method of Bardeen, a procedure consisting of particular choices of the perturbed variables as a combination of gauge-dependent quantities.     

Solutions of the Field Equations and Possible Meaning of the Hermitian Theory of Relativity

A procedure is stated, which allows to build solutions of the Hermitian theory of relativity from known solutions of the general theory of relativity. Solutions depending on three co-ordinates, built from Minkowski metric, as well as Hermitian generalizations of the Weyl-Levi Civita solution are shown. They suggest that the imaginary part of the fundamental tensor may encompass fields of different physical behaviour, like the electromagnetic field and a field responsible for forces which do not depend on the distance between charges which cannot exist as individuals. In the generalizations of the Weyl-Levi Civita solution these fields appear to be decoupled from gravitation in a peculiar way.     

On Charged Analogues of Matese and Whitman Interior Solutions in General Relativity

A set of three static fluid spheres once derived by Matese and Whitman (in Phys. Rev. D 22:1270, 1980) are charged by means of a particular electric intensity to obtain compact objects like white dwarfs, quark stars and neutron stars etc. The charged objects so obtain and their neutral counterparts are analysed on astrophysical grounds. This revealed that two of the charged spheres and one of the neutral seed solutions are well behaved. The corresponding red shift and adiabatic index together with other physical entities like pressure, density and velocity of sound through the star models are studied systematically.     

General Solutions of Reflection Equations for Bariev Model

The integrable general open-boundary conditions for the one-dimensional Bariev chain are considered. All kinds of solutions to the reflection equation （RE） and its dual are obtained.     

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.     

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.     

General Covariance in General Relativity?

The mathematical approach to General Relativity insists that all coordinate systems are equal. However physicists and astrophysicists in fact almost always use preferred coordinate systems not merely to simplify the calculations but also to help define quantities of physical interest. This suggests we should reconsider and perhaps refine the dogma of General Covariance.     

Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities

We present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments. Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.