FRW Universe Models in Conformally Flat Spacetime Coordinates. III: Universe models with positive spatial curvature

We deduce general expressions for the line element of universe models with positive spatial curvature described by conformally flat spacetime coordinates. Models with dust, radiation and vacuum energy are exhibited. Discussing the existence of particle horizons we show that there is continual annihilation of space, matter and energy in a dust and radiation dominated universe, and continual creation in a LIVE domined universe when conformal time is used in Friedmann-Robertson-Walker models with positive spatial curvature. A general procedure is given for finding coordinates to be used in Penrose diagrams. We also calculate the age and the redshift of some universe models using conformal time.Comment: 22 pages, 9 figure

Relativistic contraction and related effects in noninertial frames

Although there is no relative motion among different points on a rotating disc, each point belongs to a different noninertial frame. This fact, not recognized in previous approaches to the Ehrenfest paradox and related problems, is exploited to give a correct treatment of a rotating ring and a rotating disc. Tensile stresses are recovered, but, contrary to the prediction of the standard approach, it is found that an observer on the rim of the disc will see equal lengths of other differently moving objects as an inertial observer whose instantaneous position and velocity are equal to that of the observer on the rim. The rate of clocks at various positions, as seen by various observers, is also discussed. Some results are generalized for observers arbitrarily moving in a flat or a curved spacetime. The generally accepted formula for the space line element in a non-time-orthogonal frame is found inappropriate in some cases. Use of Fermi coordinates leads to the result that for any observer the velocity of light is isotropic and is equal to $c$, providing that it is measured by propagating a light beam in a small neighborhood of the observer.Comment: 15 pages, significantly revised version, title changed, to appear in Phys. Rev.

FRW Universe Models in Conformally Flat Spacetime Coordinates. II: Universe models with negative and vanishing spatial curvature

We deduce general expressions for the line element of universe models with negative and vanishing spatial curvature described by conformally flat spacetime coordinates. The empty Milne universe model and models with dust, radiation and vacuum energy are exhibited. Discussing the existence of particle horizons we show that there is continual creation of space, matter and energy when conformal time is used in Friedmann-Robertson-Walker models with negative spatial curvature.Comment: 25 pages, 12 figure

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 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.Comment: 18 pages, paper accepted for publication in Gen. Rel. Gra

FRW Universe Models in Conformally Flat Spacetime Coordinates. I: General Formalism

The 3-space of a universe model is defined at a certain simultaneity. Hence space depends on which time is used. We find a general formula generating all known and also some new transformations to conformally flat spacetime coordinates. A general formula for the recession velocity is deduced.Comment: 12 page

Magnetohydrodynamics in the Inflationary Universe

Magnetohydrodynamic (MHD) waves are analysed in the early Universe, in the inflationary era, assuming the Universe to be filled with a nonviscous fluid of the Zel'dovich type ($p=\rho$) in a metric of the de Sitter form. A spatially uniform, time dependent, magnetic field ${\bf B_0}$ is assumed to be present. The Einstein equations are first solved to give the time dependence of the scale factor, assuming that the matter density, but not the magnetic field, contribute as source terms. The various modes are thereafter analysed; they turn out to be essentially of the same kind as those encountered in conventional nongravitational MHD, although the longitudinal magnetosonic wave is not interpretable as a physical energy-transporting wave as the group velocity becomes superluminal. We determine the phase speed of the various modes; they turn out to be scale factor independent. The Alfv\'{e}n velocity of the transverse magnetohydrodynamic wave becomes extremely small in the inflationary era, showing that the wave is in practice 'frozen in'.Comment: 19 pages, LaTeX, no figures. Minor additions to the Summary section and Acknowledgments section. Two new references. Version to appear in Phys. Rev.

Randall-Sundrum Model in the Presence of a Brane Bulk Viscosity

The presence of a bulk viscosity for the cosmic fluid on a single Randall-Sundrum brane is considered. The spatial curvature is assumed to be zero. The five-dimensional Friedmann equation is derived, together with the energy conservation equation for the viscous fluid. These governing equations are solved for some special cases: (i) in the low-energy limit when the matter energy density is small compared with brane tension; (ii) for a matter-dominated universe, and (iii) for a radiation-dominated universe. Rough numerical estimates, for the extreme case when the universe is at its Planck time, indicate that the viscous effect can be significant.Comment: 18 pages, RevTeX4, no figures. Discussion in Sec. III expanded; new references. To appear in Phys. Rev.

The Relative Space: Space Measurements on a Rotating Platform

We introduce here the concept of relative space, an extended 3-space which is recognized as the only space having an operational meaning in the study of the space geometry of a rotating disk. Accordingly, we illustrate how space measurements are performed in the relative space, and we show that an old-aged puzzling problem, that is the Ehrenfest's paradox, is explained in this purely relativistic context. Furthermore, we illustrate the kinematical origin of the tangential dilation which is responsible for the solution of the Ehrenfest's paradox.Comment: 14 pages, 2 EPS figures, LaTeX, to appear in the European Journal of Physic

Evolution of the Bianchi I, the Bianchi III and the Kantowski-Sachs Universe: Isotropization and Inflation

We study the Einstein-Klein-Gordon equations for a convex positive potential in a Bianchi I, a Bianchi III and a Kantowski-Sachs universe. After analysing the inherent properties of the system of differential equations, the study of the asymptotic behaviors of the solutions and their stability is done for an exponential potential. The results are compared with those of Burd and Barrow. In contrast with their results, we show that for the BI case isotropy can be reached without inflation and we find new critical points which lead to new exact solutions. On the other hand we recover the result of Burd and Barrow that if inflation occurs then isotropy is always reached. The numerical integration is also done and all the asymptotical behaviors are confirmed.Comment: 22 pages, 12 figures, Self-consistent Latex2e File. To be published in Phys. Rev.

Magnetic Surfaces in Stationary Axisymmetric General Relativity

In this paper a new method is derived for constructing electromagnetic surface sources for stationary axisymmetric electrovac spacetimes endowed with non-smooth or even discontinuous Ernst potentials. This can be viewed as a generalization of some classical potential theory results, since lack of continuity of the potential is related to dipole density and lack of smoothness, to monopole density. In particular this approach is useful for constructing the dipole source for the magnetic field. This formalism involves solving a linear elliptic differential equation with boundary conditions at infinity. As an example, two different models of surface densities for the Kerr-Newman electrovac spacetime are derived.Comment: 15 page