Exact fields of charge and mass distributions in general relativity.
Einstein field equations solutions generated via static spherically symmetric mass and charge distributions
Cohen, J. M.↗
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The radial heat polynomials and related functions.
Generalized heat equation solution expansion in terms of radial heat polynomials, Appel transforms and Laguerre polynomials
RADIAL DISTRIBUTION↗
Gravitational radiation in the limit of high frequency. I - The linear approximation and geometrical optics.
WKB approximation HF gravitational wave solutions to general relativity vacuum Einstein equations with strong gravitational fields, discussing light waves analogy
Isaacson, R. A.↗
An exact solution for a thick domain wall in general relativity
An exact solution of the Einstein equations for a static, planar domain wall with finite thickness is presented. At infinity, density and pressure vanish and the space-time tends to the Minkowski vacuum on one side of the wall and to the Taub vacuum on the other side. A surprising feature of this solution is that the density and pressure distribution are symmetric about the central plane of the wall whereas the space-time metric and therefore also the gravitational field experienced by a test particle is asymmetric.
Goetz, Guenter↗
Static, axially symmetric, interior solution in general relativity.
Static axisymmetric interior solution of Einstein field equations which matches smoothly to one of Weyl exterior solutions
EINSTEIN EQUATION↗
Effect of temperature-dependent electrical conductivity on transport processes in magnetosolidmechanics
The effect of temperature-dependent electrical conductivity on transport processes for a solid block is analyzed on the basis of a one-dimensional steady-state model under specified thermal boundary conditions. Assumptions are that the solid has an infinitely segmented electrode configuration, the magnetic field (By) may be resolved into a constant applied field and an induced field, the gradient of the electrochemical potential is equal to the electrostatic potential, a constant potential difference is applied externally across each pair of opposite electrodes, and all material properties except electrical conductivity are constant. Conductivity is expressed in normalized form in terms of a baseline conductivity and a constant for the material. The application of the assumptions of the model to the general phenomenological relations yields the governing equations. Solution of these equations gives the distribution of temperature, electric current density, and magnetic field strength along the length of the solid. It is shown that significant differences exist between the case for constant electrical conductivity and the case where electrical conductivity is temperature dependent.
Craig, G. T.↗
Gamma Ray Bursts and the Birth of Black Holes
Black holes have been predicted since the 1940's from solutions of Einstein's general relativity field equation. There is strong evidence of their existence from astronomical observations, but their origin has remained an open question of great interest. Gamma-ray bursts may the clue. They are powerful explosions, visible to high redshift, and appear to be the birth cries of black holes. The Swift and Fermi missions are two powerful NASA observatories currently in orbit that are discovering how gamma-ray bursts work. Evidence is building that the long and short duration subcategories of GRBs have very different origins: massive star core collapse to a black hole for long bursts and binary neutron star coalescence to a black hole for short bursts. The similarity to Type II and Ia supernovae originating from young and old stellar progenitors is striking. Bursts are tremendously luminous and are providing a new tool to study the high redshift universe. One Swift burst at z=8.3 is the most distant object known in the universe. The talk will present the latest gamma-ray burst results from Swift and Fermi and will highlight what they are teaching us about black holes and jet outflows.
Gehrels, Neil↗
A generalized Reissner-Nordstroem solution
Einstein-Maxwell equations solutions relation to Robinson and Robinson metrics compared with Reissner-Nordstrom solution relation to Schwarzschild
Robinson, I.↗
Theory of biaxial graded-index optical fiber
A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.
Kawalko, Stephen F.↗
A coupled mode formulation by reciprocity and a variational principle
A coupled mode formulation for parallel dielectric waveguides is presented via two methods: a reciprocity theorem and a variational principle. In the first method, a generalized reciprocity relation for two sets of field solutions satisfying Maxwell's equations and the boundary conditions in two different media, respectively, is derived. Based on the generalized reciprocity theorem, the coupled mode equations can then be formulated. The second method using a variational principle is also presented for a general waveguide system which can be lossy. The results of the variational principle can also be shown to be identical to those from the reciprocity theorem. The exact relations governing the 'conventional' and the new coupling coefficients are derived. It is shown analytically that the present formulation satisfies the reciprocity theorem and power conservation exactly, while the conventional theory violates the power conservation and reciprocity theorem by as much as 55 percent and the Hardy-Streifer (1985, 1986) theory by 0.033 percent, for example.
Chuang, Shun-Lien↗
Geometry of Chaos in the Two-Center Problem in General Relativity
The two-center problem in General Relativity is the problem of geodesic motion in the two black-hole spacetime defined by the Majumdar-Papapetrou solution of the Einstein-Maxwell equations.
general relativity↗
The effect of closed boundary conditions on a stationary dynamo
One of two boundary conditions generally assumed in solutions of the dynamo equation is related to the disappearance of the azimuthal field at the boundary. Parker (1984) points out that for the realization of this condition the field must escape freely through the surface. Escape requires that the field be detached from the gas in which it is embedded. In the case of the sun, this can be accomplished only through reconnection in the tenuous gas above the visible surface. Parker concludes that the observed magnetic activity on the solar surface permits at most three percent of the emerging flux to escape. He arrives at the conclusion that, instead of B(phi) = 0, the partial derivative of B(phi) to r is equal to zero. The present investigation is concerned with the effect of changing the boundary condition according to Parker's conclusion. Implications for the solar convection zone are discussed.
Choudhuri, A. R.↗
Unsteady supersonic aerodynamic theory for interfering surfaces by the method of potential gradient
A generalized solution of the hyperbolic wave equation was further developed to relate the velocity components at a field point to the potential gradient distribution in the dependence domain. Singular integrals were evaluated in closed form, with numerical integration methods for more complex but analytic functions. Idealization of the lifting surfaces by trapezoidal elements with two sides parallel to the streamlines is computationally efficient. Streamwise integrals were performed analytically, and spanwise integrals were neccessary only on element leading and trailing sides. All integrands vanish on the Mach cone. Pressure distribution on a double delta wing and generalized aerodynamic coefficients for three AGARD planforms were calculated and compared with available results.
Jones, W. P.↗
(abstract) Spacecraft Doppler Tracking with the Deep Space Network in the Search for Gravitational Waves
The Deep Space Network generates accurate radio science data observables for investigators who use radio links between spacecraft and the Earth to examine small changes in the phase and/or amplitude of the signal to study a wide variety of structures and phenomena in space. Several such studies are directed at aspects of the theory of general relativity such as gravitational redshift and gravitational waves. A gravitational wave is a propagating, polarized gravitational field, a ripple in the curvature of space-time. In Einstein's theory of general relativity, the waves are propagating solutions of the Einstein field equations. Their amplitudes are dimensionless strain amplitudes that change the fractional difference in distance between test masses and the rates at which separated clocks keep time. Predicted by all relativistic theories of gravity, they are extremely weak (the ratio of gravitational forces to electrical forces is about 10(sup -40)) and are generated at detectable levels only by astrophysical sources - very massive sources under violent dynamical conditions. The waves have never been detected but searches in the low-frequency band using Doppler tracking of many spacecraft have been conducted and others are being planned. Upper limits have been placed on the gravitational wave strength with the best sensitivities to date are for periodic waves being 7 x 10(sup -15).
gravitational waves theory of general relativity D↗
Finite difference solution for a generalized Reynolds equation with homogeneous two-phase flow
An attempt is made to relate elements of two-phase flow and kinetic theory to the modified generalized Reynolds equation and to the energy equation, in order to arrive at a unified model simulating the pressure and flows in journal bearings, hydrostatic journal bearings, or squeeze film dampers when a two-phase situation occurs due to sudden fluid depressurization and heat generation. The numerical examples presented furnish a test of the algorithm for constant properties, and give insight into the effect of the shaft fluid heat transfer coefficient on the temperature profiles. The different level of pressures achievable for a given angular velocity depends on whether the bearing is thermal or nonisothermal; upwind differencing is noted to be essential for the derivation of a realistic profile.
Braun, M. J.↗
Integration-free interval doubling for Riccati equation solutions
Starting with certain identities obtained by Reid (1972) and Redheffer (1962) for general matrix Riccati equations (RE's), we give various algorithms for the case of constant coefficients. The algorithms are based on two ideas - first, relate the RE solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant, the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration-free, superlinearly convergent iterative solution to the algebraic RE. Preliminary numerical experiments show that our algorithms, arranged in square-root form, provide a method that is numerically stable and appears to be competitive with other methods of solving the algebraic RE.
Sidhu, G. S.↗
Existence of frozen-in coordinate systems
The 'frozen-in' coordinate systems were first introduced in the works on 'reconnection' and 'magnetic barrier' theories (see review by M.l.Pudovkin and V.S.Semenov, Space Sci. Rev. 41,1 1985). The idea was to utilize the mathematical apparatus developed for 'general relativity' theory to simplify obtaining solutions to the ideal MHD equations set. Magnetic field (B), plasma velocity (v), and their vector product were used as coordinate vectors. But there exist no stationary solutions of ideal MHD set that satisfies the required boundary conditions at infinity (A.D.Chertkov, Solar Wind Seven Conf.,Pergamon Press,1992,165) having non-zero vector product of v and B where v and B originate from the same sphere. The existence of a solution is the hidden mine of the mentioned theories. The solution is constructed in the coordinate system, which is unknown and indeterminate before obtaining this solution. A substitution of the final solution must be done directly into the initial MHD set in order to check the method. One can demonstrate that 'solutions' of Petschek's problem, obtained by 'frozen-in' coordinate systems, does not satisfy just the 'frozen-in' equation, i.e. induction equation. It stems from the fact that Petschek's 're-connection' model, treated as a boundary problem, is over determined. This problem was incorrectly formulated.
Chertkov, A. D.↗
Linearization of Schwarzschild's line element - Application to the clock paradox.
This article studies the relativistic theory of the motion of a particle in the presence of a uniform acceleration field. The problem is introduced as a linearization of the fundamental line element of general relativity. The linearized line element is a solution of Einstein's field equations. The equations of geodesics corresponding to this line element are solved and applied to the clock paradox problem.-
Broucke, R.↗