A model of a discrete pregeometry on a microscopic scale is introduced. This model is a finite network of finite elementary processes. The mathematical description is a d-graph that is a generalization of a graph. This is the particular case of a causal set. The aim of this study is to construct the particles as emergent structures. The particles in this model must be cyclic processes. The general dynamics and several examples are given. A simple dynamics generates a hierarchy of cyclic processes. An algebraic representation of this dynamics is given. It is based on the algebra of creation and destruction operators. Loops are described by bosonic operators and causal connections are described by fermionic operators.

Discrete mechanics: a sequential growth dynamics for causal sets, and a self-organization of particles

The Fermi transport law is commonly used to construct “non-rotating” reference frames. The basis for this interpretation is the bouncing photon experiment ([1], p. 123),and the vanishing of the Coriolis acceleration ([4]). We use the latter property to construct a transport law for spatial axes along an observer's world line in theories of gravitation more general than Weyl's (and Einstein's). Insistence on a transport law built on the first property precludes any generalization beyond Weyl's.

Fermi transport and the Coriolis acceleration

We proved in a previous paper that a generalized circulation theorem characterizes Einstein's theory of gravitation as a special case of a more general theory of gravitation, which is also based on the principle of equivalence. Here we pose the question of whether it is possible to weaken this circulation theorem in such ways that it would imply more general theories than Einstein's. This problem is solved. Principally, there are two possibilities. One of them is essentially Weyl's theory.

Weak Circulation Theorems as a Way of Distinguishing between Generalized Gravitation Theories

New constants of motion for systems of free gravitating particles in Newton's and Einstein's theories of gravitation

Differential constants of motion for systems of freely gravitating particles in general relativity are first defined and then determined. It is shown that they are all consequences of the local simultaneity conservation property of general relativity. It is proved, further, that the restriction to vacuum conditions does not change the set of differential constants of motion, excluding the nonphysical cases of space-time of dimension 2 or 3. Another consequence is that nothing can be inferred from local (in space and time) measurements about the orientation of a laboratory in free fall relative to Fermi transported axes. A similar property exists in Newton's theory.

A Kovetz

Papers

Characterization of gravitational theories by cumulative effects in free particles' motion and in the behaviour of clocks