Letter: Gravitational Entropy of Constrained Instanton

The seeds for quantum creations of universes areconstrained gravitational instantons. For all compactconstrained instantons with a U(1) isometry, the period of the group parameter is identifiedas the reciprocal of the temperature. If remains a free parameter under the constraints, then theEuclidean action becomes the negative of the entropy. Asexamples, we perform the calculations forthe Taub-nut and Taub-Bolt-type models andstudy the quantum creation of the Taub-nutuniverse.     

Superenergy of Gravitational Waves in the Exact Theory

Gravitational waves and radiation in the exact theory are studied in a unique framework. The observer's point of view is introduced. Some results by Lichnerowicz are generalized and interpreted in terms of reference frames. This allows us to recognize the role played by the gravitational force field in the exact generalization of Bel's superenergy flux theorem. It is also possible to recover the usual concept of wave as energy transmission, by means of a suitable superenergy scalar.     

Weak Lensing, Shear and the Cosmic Virial Theorem in a Model with a Scale-Dependent Gravitational Coupling

It is argued that, in models where the gravitational coupling is scaledependent, predictions concerning weak gravitational lensing and shear are essentially similar to the ones derived from General Relativity. This is consistent with recent negative results of observations of the MS1224, CL2218 and A1689 systems aimimg to infer from those methods the presence of dark matter. It is shown, however, that the situation is quite different when an analysis based on the Cosmic Virial Theorem is concerned.     

The Investigation of the Model of Gravitational Repulsion in Einstein's General Theory of Relativity

The gravitational field of a static, sphericallysymmetric source of mass M and scalar charge q isconsidered. It is shown that the metric expression forthis source is considerably simplified in two limiting cases: a) for M2 4q²/G,that is when the mass of the source is the maincontributor in the gravitational field; b) forq² M²G/4, when theenergy-momentum tensor of the static, spherically symmetric scalar field is the main contributorin the gravitational field. In the limiting caseq² M²G/4, the geodesicsof the massive and massless particles are studied. It isshown that gravitational forces of repulsion act on a particle movingnon-radially in this field. As a result, voids should becreated in the region surrounding such sources in theUniverse. Moreover, the stars with considerable scalar charge q² M²G/4 will act not as convexgravitational lenses as in the case whenq² M²G/4, but as concavegravitational lenses for the electromagnetic rays oflarge impact parameter.     

Gravitational Waves and Discontinuous Motions

The mathematical theory of gravitational wave-fronts is revisited with the help of orthogonal decomposition techniques. Thus many important well known results about gravitational waves in empty space are readily obtained in the local geometry framework and in general coordinates. Then different kinds of motion in the presence of a wave are investigated. Our study shows that a discontinuity effect, in traversing the wave-front, arises in the motion. The importance of this effect increases with the complexity of the considered particle. No essential discontinuity is present in simple motions such as that of a point particle and, differently from what usually believed, that of a dust of particles governed by the geodesic deviation equation.     

Charged Gravitational Instantons in Five-Dimensional Einstein-Gauss-Bonnet-Maxwell Theory

We study a solution of the Einstein-Gauss-Bonnet theory coupled to a Maxwell field in five dimensions, whose euclidean continuation gives rise to an instanton describing black hole pair production. We also discuss the dual theory with a 3-form field coupled to gravity.     

Sparling Two-Forms, the Conformal Factor and the Gravitational Energy Density of the Teleparallel Equivalent of General Relativity

It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field in a unique way. The tegr amounts to an alternative formulation of Einstein's general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of spacetimes with distinct topologies, and the outcome of these analyses agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the spacetime manifold. In this article we establish a relationship between the expression of the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We will also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field.     

Simulating the Performance of Michelson- and Sagnac-based Laser Interferometric Gravitational Wave Detectors in the Presence of Mirror Tilt and Curvature Errors

We model the behaviour of single-bounce delay-line dual recycled Sagnac-based interferometer and compare its performance with that of single-bounce Michelson-based system. Geometric imperfections such as mirror tilt and curvature mismatch can strongly influence the performance of the Sagnac device due to the inherently smaller free spectral range (FSR) and higher order mode spacing. This leads to a greater number of higher order modes near, or within the signal band in the signal recycling cavity of the instrument. The important consequence of that is, in general, a greater sensitivity of a Sagnac system to nonideal parameters and imperfections of various kind affecting the performance of a real interferometer. A number of optical configurations have also been examined numerically to determine the best possible optical arrangement, in the presence of such geometric imperfections. We show that there is an optimum choice for the nominal radius of curvature of the end mirrors which results from balancing the loss due to mirror tilt against that due to curvature mismatch.     

Towards a Probabilistic Paradigm for Gravitational Collapse, Black Hole Creation and Singularity Smearing

A prototype probability interpretation ispresented for the Oppenheimer-Snyder model ofspherically symmetric, gravitational collapse of apressureless ensemble of n point particles. A transitionprobability P(R(t), t; R₁, t₁) isderived for an initial sphere or fluid star of radius Rat comoving time t, collapsing smoothly andhomogeneously to any finite radii R(t, r) < R atcomoving t > t₁ and R(t) = 0 at t = t_f. The transitionprobability is evaluated in two cases. In the firstcase, Planck's constant is assumed zero and smoothdifferential limits exist for space and matter on alllength scales down to zero. The probability for singularityformation converges smoothly to unity as R 0 ort t_f: the collapse is deterministic atall scales. There is also a finite, nonzero probabilityof event horizon formation at R = R_h = 2GM, but the starcontinues to collapse through this radius since there isalways a higher probability of reaching any smallerradius R < R_h. An event horizon forms sothe collapsed state is still a black hole. In the classical limit(as 0) the singularity returns with unitprobability. Finally, we briefly discuss how the final,fuzzy, collapsed state may be related to aspects ofstring theory. The emphasis of the paper is on theconceptual ideas and general possibilities which couldarise when incorporating stochastic mechanics andanalysis into general relativistic collapse.     

New Classes of Toda Soliton Solutions

We provide a detailed investigation of limits of N–soliton solutions of the Toda lattice as N tends to infinity. Our principal results yield new classes of Toda solutions including, in particular, new kinds of soliton–like (i.e., reflectionless) solutions. As a byproduct we solve an inverse spectral problem for one–dimensional Jacobi operators and explicitly construct tri–diagonal matrices that yield a purely absolutely continuous spectrum in (-1,1) and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of . Received: 16 October 1995/Accepted: 23 July 1996     

Soliton Solutions for Nonisospectral BKP Equation

The soliton solutions for the nonisospeetral BKP equation are derived through Hirota method and Pfaffian technique. We also derive the bilinear Baeklund transformations for the isospectral and nonisospeetral BKP equation and find solutions with the help of the obtained bilinear Baeklund transformations.     

Soliton Solutions in Three-Component Bose-Einstein Condensates

We obtain soliton and plane wave solutions for the coupled nonlinear Schrotinger equations, which describe the dynamics of the three-component Bose-Einstein condensates by using the Hirota method. Meanwhile we find that the system which has attractive atomic interaction will only possess a shape changing （inelastic） collision property due to intensity redistribution in the absence of the spin-exchange interaction. As a discussed example, we investigate the one-soliton, two-soliton solutions and collisional effects between bright two-soliotn solution, which lead to the intensity redistribu tion.     

Soliton Solutions to the Coupled Gerdjikov-Ivanov Equation with Rogue-Wave-Like Phenomena

Bilinear forms of the coupled Gerdjikov-Ivanov equation are derived. The N-soliton solutions to the equation are obtained by Hirota's method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov-Ivanov equation as well as its bilinear forms and its solutions.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless (CID) equations is derived. It will be shown that this bilinear form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions are derived.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.     

Soliton Asymptotics of Solutions of the Sine-Gordon Equation

An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.     

Soliton Solutions of Integrable Hierarchies and Coulomb Plasmas

Some direct relations are given between soliton solutions of integrable hierarchies and thermodynamic quantities of the Coulomb plasmas on the plane. We find that certain soliton solutions of the Kadomtsev–Petviashvili (KP) and B-type KP (BKP) hierarchies describe 2D one- or two-component lattice plasmas at special boundary conditions and fixed temperatures. It is shown that different reductions of integrable hierarchies describe pure or dipole Coulomb gases on 1D submanifolds embedded in the 2D space.     

Soliton Solutions for a Negative Order AKNS Equation

In this short paper, bilinear form of a negative order AKNS equation is given. The N-soliton solutions are obtained through Hiorta's direct method.