4-Dimensional Einstein Gravity Extended by a 3-Dimensional Gravitational Chern-Simons Term

When 4-dimensional general relativity is extended by a 3-dimensional gravitational Chern–Simons term an apparent violation of diffeormorphism invariance is extinguished by the dynamical equations of motion for the modified theory. The physical predictions of this recently proposed model show little evidence of symmetry breaking, but require the vanishing of the Pontryagin density.     

The structure of the Newtonian limit

We consider a smooth one-parameter family of four-dimensional manifolds X,≥0, each one endowed with a covariant metric g. It is assumed that g is a Lorentz metric for each >0, i.e., the signature of g is (+,−,−,−) for >0, while the limit metric g₀ on X₀ is assumed to be degenerated of rank 1, i.e., the signature of g₀ is (+,0,0,0). We characterize when the limit manifold X₀ inherits the geometric structure of a Newtonian gravitation. The limit manifold X₀ is a Newtonian gravitation if and only if there exist the limits of the Levi-Civita connection , the curvature operator and the contravariant Einstein tensor G² as →0. Moreover, the existence of these limits is characterized in terms of the Taylor expansion of the family {g} with respect to the parameter .     

Connections Between the Riemann and the Lorentz Force Laws

It is known that for the magnetic force due to a closed circuit, the Weber force law can be identical to the Lorentz force law. In this investigation it is shown that for both the electric and the magnetic force of the quasi-static case, the Riemann force law can be identical to the Lorentz force law, while the former is based on a potential energy depending on a relative speed and is in accord with Newton's law of action and reaction.     

Thomas rotation and the parametrization of the Lorentz transformation group

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

SOME PROPERTIES OF Λω^p（X,μ）

In this paper, we shall give a necessary and sufficient condition for which the dual of Λ _ω ^p (X, M, μ) (0<p<∞) is zero, and a necessary and sufficient condition for which Λ _ω ^p (X, M, μ), (0<p<1) is normable. Supported by 973 project (G1999075105), RFDP (20030335019) and ZJNSF(RC97017).     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.     

A modified Lorentz theory as a test theory of special relativity

A modified Lorentz theory (MLT) based on the generalized Galilean transformation has recently received attention. In the framework of MLT, some explicit formulas dealing with the one-way velocity of light, slow-clock transport and the Doppler effect are derived in this paper. Several typical experiments are analyzed on this basis. The results show that the empirical equivalence between MLT and special relativity is still maintained to second order terms. We confirm recent findings of other works that predict the MLT might be distinguished from special relativity at the third order by Doppler centrifuge experiments capable of a fractional frequency detection threshold of 10^–15.