Robinson-Trautman equations and Chandrasekhar's special perturbation of the Schwarzschild metric

A perturbation wave solution of the Robinson-Trautman equations is proved to be a perturbation of the Schwarzschild black hole which describes an outgoing axial gravitational wave and corresponds to a special case of Chandrasekhar's algebraically special perturbation of the Schwarzschild metric.     

An orthogonal form of the Schwarzschild metric

An orthogonal form of the Schwarzschild metric is given and the transformation relating this form to the standard one is found.     

An interior solution for the gamma metric

Interior solutions for a static, axially symmetric family of solutions of Einstein's equations are described. The interior solutions correspond to spatially bound matter and are properly matched to an exterior vacuum solution. The family of solutions discussed include the Schwarzschild solution as a special case. A general method is exhibited for transforming any spherically symmetric interior solution to an interior for the other members of the family of solutions. The energy density remains positive for at least a finite range of the parameter that describes the family of solutions. Two solutions are explicitly exhibited. One is transformed from the constant density Schwarzschild interior solution and one from the Adler interior solution. The first solution would be expected to be unstable under adiabatic perturbations of the matter, the second would be expected to be stable.Supported in part by The National Science Foundation under Grant No. INT 782-5663.Supported in part by Consejo Nacional de Investigaciones Cientificas y Technologicas (CONICIT), Venezuela.     

An improved radiation metric

An improved radiation metric is obtained in which light rays make a small nonzero angle with the radius, thus representing a source of finite size. Kaufmann's previous solution is criticized. The stabilization of a scatterer near a source of gravitational field and radiation is slightly enhanced for sources of finite size.     

Relation between the generalized Schwarzschild metric and the Tolman metric

In the present paper the relation between the generalized Schwarzschild metric (the Schwarzschild metric including the four-dimensional curvature tensor) [1] and the Tolman metric is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 29–36, October, 1977.     

The uniqueness of the Schwarzschild interior metric

It is shown that every conformally flat axisymmetric stationary space-time is necessarily static, and that if the source is a perfect fluid then the space-time metric is the usual Schwarzschild interior metric.     

Flowing liquid crystal simulating the Schwarzschild metric

We show how to simulate the equatorial section of the Schwarzschild metric through a flowing liquid crystal in its nematic phase. Inside a liquid crystal in the nematic phase, a traveling light ray feels an effective metric, whose properties are linked to perpendicular and parallel refractive indexes, n _o and n _e respectively, of the rod-like molecule of the liquid crystal. As these indexes depend on the scalar order parameter of the liquid crystal, the Beris-Edwards hydrodynamic theory is used to connect the order parameter with the velocity of a liquid crystal flow at each point. This way we calculate a radial velocity profile that simulates the equatorial section of the Schwarzschild metric, in the region outside of Schwarzschild’s radius, in the nematic phase of the liquid crystal. In our model, the higher flow velocity can be on the order of some meters per second.     

Use of the Schwarzschild metric in the Klein-Gordon equation

It is shown that substitution of the Schwarzschild metric sensor into the Klein-Gordon equation predicts the usual perihelion advance of classical general relativity.     

Circular orbits and relative strains in Schwarzschild space-time

The equation of the relative strain is analyzed in tetrad form with respect to a family of observers moving on spatially circular orbits, in the Schwarzschild space-time. We select a field of tetrads, which we term phase locking frames, and explicitly calculate how, in the equatorial plane, the orbital acceleration, its gradient and the Fermi drag add together to compensate the curvature and assure equilibrium among a set of comoving neighbouring particles. While equilibrium is achieved in the radial and azimuthal directions, in the direction orthogonal to the equatorial plane there is a residue of acceleration which pulls a particle towards that plane leading to a harmonic oscillation with a frequency equal to the proper frequency of the orbital revolution. This measurement, combined with those of the frequency shift of an incoming photon and the frequency of precession of the local compass of inertia, enables one to determine the relativistic ratio 2M/r, whereM is the gravitational mass of the source andr the coordinate radius of the circular orbits.     

An indefinite metric and perturbation theory. II

The present paper is a refinement and elaboration of the ideas of [1] on the applicability of Feynman procedure in the nonlinear spinor theory of elementary particles. The possibilities of probabilistic interpretation in a scheme with an indefinite metric and of constructing a unitary physical scattering matrix are analyzed. The modified perturbation theory proposed in [1] is used to compute meson masses and meson-nucleon coupling constants. The results turn out to be physically satisfactory.The author is grateful to Prof. D. Ivanenko.for his interest and useful comments, to Dr. V. Ya. Fainberg for valuable remarks, and to L. Dadashev for stimulating criticisms.     

短程透镜问题的最优解

本文给出了带卷边短程透镜问题的一个最优解,透镜子午线光滑无奇点,在透镜边缘处卷边的曲率半径趋于无穷大,具有使该点弯曲损耗达到极小的最优卷边形状.最优解优于已有的短程透镜其他解析形式解,完善了设计短程透镜的解析法.     

On a physical interpretation of the Schwarzschild metric

A method is devised for giving a physical interpretation to the customary Schwarzschild coordinates in the vicinity of a charged or uncharged isolated mass. The construction is accomplished by introducing systems that are allowed to freely fall in toward the mass from infinity (drift-systems). It is demonstrated that the Schwarzschild spatial coordinates and their increments have a full physical significance in terms of rod and clock measurements performed in the drift-systems. The time coordinate and its increment are not so amenable to treatment and cannot be considered as having been given such physical significance. In the discussion the Schwarzschild metric about an uncharged and charged mass is derived, in part, by heuristic classical arguments employing conservation of energy. The arguments are then shown to be valid by consulting the Field Equations. In the derivation the gravitational singularity (at 2GM/C ²) takes on the significance of being the location at which a drift-system achieves the speed of light relative to a proper system at the same point.     

Influence of dark energy on time-like geodesic motion in Schwarzschild spacetime

In this paper we investigate the influence of the dark energy on the time-like geodesic motion of a particle in Schwarzschild spacetime by analysing the behaviour of the effective potential which appears in an equation of motion. For the non-radial time-like geodesics, we find a bound orbit when the particle energy is in an appropriate range, and also find another possible orbit, which is that the particle drops straightly into the singularity of a black hole or escapes to infinity. For the radial time-like geodesics, we find an unstable circular orbit when the particle energy is the critical value, in which case it is possible for the particle to escape to infinity.     

Time machine and geodesic motion in Kerr metric

The existence of nontrivial causality violation is a peculiar property of the space-time around a naked singularity. In the case of Kerr metric witha > m we have found that for a particular class of geodesies that could in principle violate causality, the conditions for causality violation are never satisfied.On leave of absence from Institute of Astronomy, Polish Academy of Sciences, Warsaw, Poland.     

Application of a singular perturbation expansion to the solution of certain Fokker-Planck equations

We show that a singular perturbation expansion for the solution of a parabolic equation can be applied to some Fokker-Planck equations arising in the analysis of the effects of noise on laser operations. A generalization to the approximate solution of the Smoluchowski equation, when diffusion is a small effect, is given.     

Fluctuation around horizon on a Schwarzschild black hole using the null geodesic method

Using the null geodesic method, Hawking radiation from the horizon of a Schwarzschild black hole is calculated. The thermodynamics can be built successfully on the horizon where the apparent horizon and event horizon are coincident with each other. When a relativistic perturbation is given to the horizon, the first law of thermodynamics can also be constructed at a new supersurface near the horizon successfully. The expressions of the characteristic position and temperature are consistent with the previous result while the thermodynamics was built on the event horizon in a Vaidya black hole. Therefore, the thermodynamics of a dynamical black hole should be constructed on the apparent horizon exactly, and the event horizon thermodynamics is just one of the perturbations near the apparent horizon.     

Circular orbits of an electron around a proton in Schwarzschild geometry

The energy of an electron in circular orbits around a proton in Schwarzschild geometry has been investigated and is found to be red shifted. The electrical dipole moment of such a system is also estimated.     

On the twin paradox in static spacetimes: I. Schwarzschild metric

Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider the twin paradox in static spacetimes. According to a well known theorem in Lorentzian geometry the longest timelike worldline between two given points is the unique geodesic line without points conjugate to the initial point on the segment joining the two points. We calculate the proper times for static twins, for twins moving on a circular orbit (if it is a geodesic) around a centre of symmetry and for twins travelling on outgoing and ingoing radial timelike geodesics. We show that the twins on the radial geodesic worldlines are always the oldest ones and we explicitly find the the conjugate points (if they exist) outside the relevant segments. As it is of its own mathematical interest, we find general Jacobi vector fields on the geodesic lines under consideration. In the first part of the work we investigate Schwarzschild geometry.