An Approach to Colliding Plane Waves in General Relativity

For many years after Einstein proposed his general theory of relativity, only a few exact solutions were known. Today the situation is completely different, and we now have a vast number of such solutions. However, very few are well understood in the sense that they can be clearly interpreted as the fields of real physical sources. The obvious exceptions are the Schwarzschild and Kerr solutions. These have been very thoroughly analysed, and clearly describe the gravitational fields surrounding static and rotating black holes respectively. In practice, one of the great difficulties of relating the particular features of general relativity to real physical problems, arises from the high degree of non-linearity of the field equations. Although the linearized theory has been used in some applications, its use is severely limited. Many of the most interesting properties of space-time, such as the occurrence of singularities, are consequences of the non-linearity of the equations. Keywords: General Relativity , Space-Time, Singularities, Non-linearity of the Equations.

Slow stationary sources

The linearized theory is applied to sources such as ordinary stars whose speed is small compared to the speed of light. This yields the “gravitoelectromagnetic” theory. The gravitoelectromagnetic field equations are obtained, along with their general solution via scalar and vector potentials. It is shown how to calculate the metric perturbation, and hence the field, due to a rotating ring or a ball, and thus how to calculate orbits, timing, and the Lense-Thirring precession.

Numerical Simulation and Linearized Theory of Vortex Waves in a Viscoelastic, Polymeric Fluid

In a high viscosity, polymeric fluid initially at rest, the release of elastic energy produces vorticity in the form of coherent motions (vortex rings). Such behavior may enhance mixing in the low Reynolds number flows encountered in microfluidic applications. In this work, we develop a theory for such flows by linearizing the governing equations of motion. The linear theory predicts that when elastic energy is released in a symmetric manner, a wave of vorticity is produced with two distinct periods of wave motion: (1) a period of wave expansion and growth extending over a transition time scale, followed by (2) a period of wave translation and viscous decay. The vortex wave speeds are predicted to be proportional to the square root of the initial fluid tension, and the fluid tension itself scales as the viscosity. Besides verifying the predictions of the linearized theory, numerical solutions of the equations of motion for the velocity field, obtained using a pseudo-spectral method, show that the flow is composed of right- and left-traveling columnar vortex pairs, called vortex waves for short. Wave speeds obtained from the numerical simulations are within 1.5% of those from the linear theory when the assumption of linearity holds. Vortex waves are found to decay on a time scale of the order of the vortex size divided by the solution viscosity, in reasonable agreement with the analytical solution of the linearized model for damped vortex waves. When the viscoelastic fluid is governed by a nonlinear spring model, as represented by the Peterlin function, wave speeds are found to be larger than the predictions of the linear theory for small polymer extension lengths.

The Rayleigh–Taylor instability in a porous medium

The classical Rayleigh–Taylor instability occurs when a heavy fluid overlies a lighter one, and the two fluids are separated by a horizontal interface. The configuration is unstable, and a small perturbation to the interface grows with time. Here, we consider such an arrangement for planar flow, but in a porous medium governed by Darcy’s law. First, the fully saturated situation is considered, where the two horizontal fluids are separated by a sharp interface. A classical linearized theory is reviewed, and the nonlinear model is solved numerically. It is shown that the solution is ultimately limited in time by the formation of a curvature singularity at the interface. A partially saturated Boussinesq theory is then presented, and its linearized approximation predicts a stable interface that merely diffuses. Nonlinear Boussinesq theory, however, allows the growth of drips and bubbles at the interface. These structures develop with no apparent overturning at their heads, unlike the corresponding flow for two free fluids.

Dispersion of generalized Rayleigh waves in the half-plane covered with pre-stretched two layers under complete contact

In this paper the dispersion of the generalized Rayleigh wave propagation in the non-prestressed half-plane covered with pre-stretched two layers under complete contact conditions is investigated by 3-D linearized theory of elasticity. The layers and the half-plane are assumed that elastic, homogeneous, isotropic, and the complete contact conditions are existed. The inter phase zone between the upper layer and half-plane is modeled by this second layer. The purpose of the investigation is the determination on the effect of the existence of the second layer to the considered generalized Rayleigh wave propagation velocity. For this purpose, firstly the same materials were selected for both layers and the results obtained in previous studies for a single layer in the literature were verified, the accuracy of the modeling was shown, and then the effect of the second layer on the considered problem was shown by selecting the different materials and applying different initial pre-stresses. Consequently, the present study can be considered as the investigation of the existence of the inter phase zone which is characteristic one for the composite materials to the dispersion of the generalized Rayleigh wave propagation. Numerical results obtained and discussed.

Algebraic Approach to Bose–Einstein Condensation in Relativistic Quantum Field Theory: Spontaneous Symmetry Breaking and the Goldstone Theorem

We construct states describing Bose–Einstein condensates at finite temperature for a relativistic massive complex scalar field with $$|\varphi |^4$$ | φ | 4 -interaction. We start with the linearized theory over a classical condensate and construct interacting fields by perturbation theory. Using the concept of thermal masses, equilibrium states at finite temperature can be constructed by the methods developed in Fredenhagen and Lindner (Commun Math Phys 332:895, 2014) and Drago et al. (Ann Henri Poincaré 18:807, 2017). Here, the principle of perturbative agreement plays a crucial role. The apparent conflict with Goldstone’s theorem is resolved by the fact that the linearized theory breaks the U(1) symmetry; hence, the theorem applies only to the full series but not to the truncations at finite order which therefore can be free of infrared divergences.

Gravitational Interaction of Cosmic String with Spinless Particle

We consider the gravitational interaction of spinless relativistic particle and infinitely thin cosmic string within the classical linearized-theory framework. We compute the particle’s motion in the transverse (to the unperturbed string) plane. The reciprocal action of the particle on the cosmic string is also investigated. We derive the retarded solution which includes the longitudinal (with respect to the unperturbed-particle motion) and totally-transverse string perturbations.

Plane contact problem on the interaction of a pre-stressed strip with an infinite inhomogeneous stringer

The article is devoted to the study of problems of contact interaction of an infinite elastic inhomogeneous stringer with a prestressed strip clamped along one edge. As a result of the research, we have obtained a resolving system of recurrent systems of integro-differential equations. In general, the studies were carried out for the theory of large initial and various versions of the theory of small initial deformations within the framework of the linearized theory of elasticity with an elastic potential of an arbitrary structure. Integral integer differential equations are obtained using the integral Fourier transform. Their solution is presented in the form of quasiregular infinite systems of algebraic equations. The article also investigates the influence of the initial (residual) stresses in strips on the distribution law of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed form using the Fourier transform. The stress expressions are represented by Fourier integrals with a fairly simple structure. The influence of the initial stress on the distribution of contact stresses has been studied and mechanical effects have been found under the action of concentrated loads.