Stars creating a gravitational repulsion

In the framework of the Theory of General Relativity, models of stars with an unusual equation of state $\rho c^20$ where $\rho$ is the mass density and $P$ is the pressure, are constructed. These objects create outside themselves the forces of gravitational repulsion. The equilibrium of such stars is ensured by a non-standard balance of forces. Negative mass density, acting gravitationally on itself, creates an acceleration of the negative mass, directed from the center. Therefore in the absence of pressure such an object tends to expand. At the same time, the positive pressure, which falls just like in ordinary stars from the center to the surface, creates a force directed from the center. This force acts on the negative mass density, which causes acceleration directed the opposite of the acting force, that is to the center of the star. This acceleration balances the gravitational repulsion produced by the negative mass. Thus, in our models gravity and pressure change roles: the negative mass tends to create a gravitational repulsion, while the gradient of the pressure acting on the negative mass tends to compress the star. In this paper, we construct several models of such a star with various equations of state.Comment: 6 pages, 4 figure

Topology of quasiperiodic functions on the plane

The article describes a topological theory of quasiperiodic functions on the plane. The development of this theory was started (in different terminology) by the Moscow topology group in early 1980s. It was motivated by the needs of solid state physics, as a partial (nongeneric) case of Hamiltonian foliations of Fermi surfaces with multivalued Hamiltonian function. The unexpected discoveries of their topological properties that were made in 1980s and 1990s have finally led to nontrivial physical conclusions along the lines of the so-called geometric strong magnetic field limit. A very fruitful new point of view comes from the reformulation of that problem in terms of quasiperiodic functions and an extension to higher dimensions made in 1999. One may say that, for single crystal normal metals put in a magnetic field, the semiclassical trajectories of electrons in the space of quasimomenta are exactly the level lines of the quasiperiodic function with three quasiperiods that is the dispersion relation restricted to a plane orthogonal to the magnetic field. General studies of the topological properties of levels of quasiperiodic functions on the plane with any number of quasiperiods were started in 1999 when certain ideas were formulated for the case of four quasiperiods. The last section of this work contains a complete proof of these results. Some new physical applications of the general problem were found recently.Comment: latex2e, 27 pages, 7 figure

Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle

We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too many solutions). Our results further extend the recent claim by Novikov et al. that the `Principle of self-consistency' is a natural consequence of the `Principle of minimal action.'Comment: 39 pages, latex fil