Zero value of the schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: \(\partial _\kappa ( - gg^{0_k } ) = 0\) and \(\Gamma _{ik}^0 \partial _0 g^{ik} - \Gamma _{ik}^k \partial _0 g^{0_i } = 0\) . In aT-orthogonal metricds ² =g ₀₀ dt ² ?g _αβ dx ^α dx ^β these conditions take a simple form: ?₀(detg _αβ ) = 0 and (?₀ g ^αβ )(?₀ g _αβ ) = 0.     

On an integral over the surface ofN-dimensional unit sphere

An integral which occurs in the new matrix ensembles and the width fluctuation factor is evaluated using a transformation which changes a Gaussian into an exponential. It is expressed in the form of a series whose terms are found using a simple recursion relation. It is shown that the series can be summed in closed form for the two-dimensional case.     

On the plane gravitational condensor with the positive gravitational constant

The equations of motion for the plane shell consisting of an ideal fluid and creating the de Sitter solution on one side and the static plane symmetric solution of Einstein equations with positive cosmological constant on the other side are derived. Solution of the equation of motion for the case of “plane gravitational condensor” is found and the necessary agreement with the solution of the problem in the caseA = 0 is demonstrated.     

The equation of motion and gravitational radiation of an ideal fluid sphere moving in a gravitational background

On the basis of an approximation method developed in a previous paper the motion of an ideal fluid sphere in a weak gravitational background is investigated. The sphere is assumed to be small in the sense that its radius is small compared with the change of the background . Furthermore the deformations of the sphere when accelerated by the background are assumed to be small compared with the extension of the sphere in the absence of acceleration. In the lowest mixed order (mixed of the background and the retarded potentials of the sphere in lowest order) the equation of motion is yielded by integrating Einstein's conservation law of energy and momentum over the world-tube of the sphere. One obtains an equation of motion for the center of the sphere that is identical with the geodesic line linearized in . In the case of a static background of a localized matter distribution it is shown that Einstein's energy-momentum complex formed with the retarded potentials from the accelerated motion of the sphere in lowest order (lowest mixed order) leads to an outgoing radiation of gravitational energy. All radiation terms can be expressed in terms of the background and the world-line of the center of the sphere.     

Quasi-normal oscillations of a schwarzschild black hole

The modes of oscillation of a Schwarzschild black hole are determined within an analytic framework. These quasi-normal modes are related to the bound states of the inverted black hole potential which is approximated by the inverted Eckart potential. For a given angular momentum parameter j, the real part of the quasi-normal frequency decreases as the mode number n (or, equivalently, the damping factor) increases, in agreement with the results of numerical studies.     

Equations governing the oscillations of a self-gravitating elastic sphere under the influence of gravitational waves

We consider the influence of a plane gravitational wave on the following Newtonian system: a self-gravitating elastic sphere without rotation. We determine the equation governing the elastic oscillations within the framework of the general relativity. We include dissipative processes by using a simple relativistic model of viscoelasticity. If the wavelength is much larger than the dimension of the elastic sphere, we transform these equations into the form derived by Ashby and Dreitlein previously. We also determine the equations governing the elastic oscillations within the framework of the Brans-Dicke scalar-tensor theory of gravitation.     

On the exterior gravitational field of a mass with a multipole moment

Recently, Gutsunaev and Manko [6] presented a procedure for obtaining new static axisymmetric solutions of Einstein's vacuum field equations from a known one. We show that this procedure is based on the property that the derivatives of a harmonic function are harmonic. The special case of a metric with mass and quadrupole moment is investigated and compared with the Erez-Rosen metric.     

Exact partial solutions for the surface dynamics of a dielectric liquid with a charged surface in the gravitational field

We study the evolution of instability in the boundary of a perfect dielectric liquid with a free surface charge in an external electric field. Conformal variables are used to find exact partial solutions to the equations of motion for the case when the charge completely shields the field above the liquid, the electrostatic and gravitational forces being taken into account. The solutions describe the development of instability of the initially planar boundary until sharp dimples are formed on it.     

Radiation reaction for a rotating sphere with rigid surface charge

Radiation reaction in classical electrodynamics is studied using as a simple model a rotating sphere with rigid surface charge. Because of the transparent geometric nature of the problem, we obtain an exact integral expression for the “self-torque”, which depends causally and linearly on the angular speed. This leads to a linear equation of motion, which does not have any runaway solutions, as long as no negative mechanical moment-of-inertia is added to the system. The results are illustrated with a few examples corresponding to predetermined motions and external torques.     

Behavior of a charged particle in a schwarzschild field

The influence of the De Witt self-action force on the motion of and electromagnetic emission from a charged particle in a Schwarzschild field is considered. It is shown that a charged particle in a Schwarzschild field is equivalent to a neutral particle of the same mass in a certain Reissner-Nordstrom field. A relationship is found between the power of the electromagnetic emission from an accelerated charge and the power of the thermal emission generated in a reference frame with the same acceleration at the event horizon. The quantum-mechanical problem of the motion of and emission from a charge in the field of a minihole is considered. Wave functions, the energy spectrum, and the widths of quasi-stationary levels are found with allowance for the De Witt self-action force. It is shown that the latter is important for large charges, when the solution becomes oscillatory. "Brainstorm" Little Science and Technology Enterprise. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 75–82, July, 1998.     

On the way towards a generalized entropy maximization procedure

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptotically for q∈(0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.     

On the gravitational field of a massless particle

The gravitational field of a massless point particle is first calculated using the linearized field equations. The result is identical with the exact solution, obtained from the Schwarzschild metric by means of a singular Lorentz transformation. The gravitational field of the particle is nonvanishing only on a plane containing the particle and orthogonal to the direction of motion. On this plane the Riemann tensor has a -like singularity and is exactly of Petrov typeN.This work was supported in part by Fonds zur Förderung der Wissenschaftlichen Forschung.     

Interpolation of the schwarzschild and Bertotti-Robinson solutions

The two well-known solutions of general relativity, the Schwarzschild and the Bertotti-Robinson solutions, are shown to be described by a common metric. In a particular coordinate system we show the connection between this metric and the metric of Reissner-Nordström.