Non-Riemannian generalizations of the Born–Infeld model and the meaning of the cosmological term

Theory of gravitation based on a non-Riemannian geometry with dynamical torsion field is geometrically analyzed. To this end, the simplest Lagrangian density is introduced as a measure (reminiscent of a sigma model) and the dynamical equations are derived. Our goal is to rewrite this generalized affine action in a suitable form similar to the standard Born–Infeld (BI) Lagrangian. As soon as the functional action is rewritten in the BI form, the dynamical equations lead the trace-free GR-type equation and the field equations for the torsion, respectively: both equations emerge from the model in a sharp contrast with other attempts where additional assumptions were heuristically introduced. In this theoretical context, the Einstein [Formula: see text], Newton [Formula: see text] and the analog to the absolute [Formula: see text]-field into the standard BI theory all arise from the same geometry through geometrical invariant quantities (as from the curvature [Formula: see text]). They can be clearly identified and correctly interpreted both physical and geometrically. Interesting theoretical and physical aspects of the proposed theory are given as clear examples that show the viability of this approach to explain several problems of actual interest. Some of them are the dynamo effect and geometrical origin of [Formula: see text] term, origin of primordial magnetic fields and the role of the torsion in the actual symmetry of the standard model. The relation with gauge theories, conserved currents, and other problems of astrophysical character is discussed with some detail.

THE BARE DIFFUSION COEFFICIENT AND THE PECULIAR VELOCITY AUTO-CORRELATION FUNCTION

The bare diffusion coefficient is given as the time integral of the peculiar velocity autocorrelation function or PVACF and this result is different from the well known Green-Kubo formula. The bare diffusion coefficient characterizes the diffusion process on a length scale lambda. The PVACF is given here for the first time in terms of the positions and velocities of the N particles of the system so the PVACF is in a form suitable for evaluation by molecular dynamics simulations. The computer simulations show that for the two dimensional hard disk system, the PVACF decays increasingly rapidly in time as lambda is reduced and this is probably a general characteristic. Finally, the Einstein formula for Brownian motion is a bit different for the bare diffusion coefficient.

Conformal transformations and kinematical relativity

The Einstein relative-velocity formula. Milne (3) has remarked that in kinematical relativity, whereas the Lorentz transformation is valid in t-time but fails in τ-time, the Einstein relative-velocity formula is valid both in t-time and τ-time, thus implying the validity of the latter in a more general system. It was thought that some interest might be attracted to a description of the most general system under which the Einstein formula holds.

Orbits and rays in the gravitational field of a finite sphere according to the theory of A. N. Whitehead

The relativity theory of A. N. Whitehead permits one to calculate directly the gravitational field of a set of particles of assigned masses and arbitrary motions, and to investigate the orbits of test-particles and the paths of light rays in such a field. In this paper the hypothesis of Whitehead is extended to cover the case of a continuous distribution of matter; the field of a fixed sphere with a spherically symmetric distribution of matter is calculated and orbits and light rays discussed. Explicit formulae are obtained for advance of perihelion, angular velocity in a circular orbit, and deflexion of a light ray. The results differ only slightly from those of Einstein’s general theory of relativity by terms involving the distribution of matter in the sphere, except in the case of the deflexion of light, for which precisely the Einstein formula (depending only on total mass) is obtained.