T-geometry and Its Application to SpaceTime Model

Updated October 10, 1999

T-geometry (nondegenerate geometry) is the most general fundamental conception of modern geometry. Applied for constructing the spacetime model, T-geometry explains freely absence of superlight motion of particles (taxyons) and stochastic motion of microparticles which are explained in the Minkowski spacetime model only by means of special additional suppositions. T-geometry allows one to explain quantum effects as natural geometric effects with the quantum constant being an attribute of spacetime model.

Abstract

The monograph contains two new conceptual points:

- Nondegenerate geometry (T-geometry).
- Dynamical conception of statistical description.

T-geometry is **not** a kind of Minkowski or Riemannian geometry equipped with enigmatic suppositon on quantum or stochastic nature of geometry. T-geometry is the most natural and simplest generalization of the Euclidean geometry. This kind of generalization was by chance overlooked in the course of the modern geometry developement. Instead one used the Riemannian geometry which is a more complicated generalization of the Euclidean geometry, because it is based on a use of two structures (topological structure and metric one). The T-geometry uses only metric structure, and it is more general than the Riemannian geometry, including it as a special case. If it is necessary, the topological structure may be also introduced in T-geometry on the the base of the metric structure.

Essentially, T-geometry is the metric geometry equipped with NGO-technique which admits to construct different geometrical objects. The metric geometry contains only one sort of geometric objects: the so-called shortest. This is a poor geometry. For constructing the shortest the metric geometry needs the triangle axiom. The same metric geometry equipped with the NGO-technique does not need the triangle axiom, because now it contains many different geometrical objects, constructed on the basis of the NGO-technique. T-geometry (i.e. the metric geometry equipped with NGO-technique) is a rich geometry. T-geometry is as rich and pithy as the Riemannian geometry, because the pithiness of geometry depends mainly on the number sorts of geometrical objects described by the geometry and, hence, on the mathematical technique used. The Riemannian geometry uses manifold as a mathematical tool for description of geometrical objects, whereas the T-geometry uses the NGO-technigue. NGO-technique is more general and adequate mathematical tool than the manifold.

The difference between T-geometry and Riemannian geometry can be explained as follows. Any geometry can be described in terms of **natural geometric objects **(NGO). By definition NGO is a set of points determined by geometry. NGOs depend on parameters (points). The zeroth order NGO is determined by one point *P*, the first order NGO is determined by two points *P,Q*, the second order NGO is determined by three points *P,Q,R,* etc.. In the proper Euclidean geometry the zeroth order NGO is the point *P *, the first order NGO is the straight, passing through the points *P,Q** *, the second order NGO is the plane, passing through the points *P,Q,R,* etc.*. *NGOs of other geometries has another shape. For instance, in the Minkowski geometry the zeroth order NGO is the light cone (but not a point). In the Riemannian geometry the first order NGO is geodesic. Any geometry deterimines its NGOs, and any geometry may be described by the set of all its NGOs. Besides, NGOs of the spacetime geometry are important, because they are connected directly with real phenomena which happen in the spacetime.

For instance, the proper Euclidean geometry can be formulated in terms of points, straights and planes (i.e. in terms of NGOs) with NGOs being determined by axioms which describe the NGO's properties. In the Riemannian geometry the first order NGO (geodesic) is defined via metric, and one does not need axioms for a construction of geometry. In T-geometry the NGOs are also defined via metric, but one uses another method of definition of the first order NGO, and other than Riemannian geometry appears. In both cases some method of definition of the first order NGO (Euclidean straight, passing through given points *P,Q*,) via Euclidean metric is used. If the metric changes, the shape of NGOs changes also. Changing Euclidean metric, one obtains the geometry other than Euclidean one (the Riemannian geometry, or T-geometry).

To obtain Riemannian geometry, the Euclidean straight, passing through points *P,Q, *is defined as the shortest **line, **connecting points *P,Q*, i.e. the Euclidean straight is defined by its extremal property. This definition contains a reference to the concept of **line** which is a complicated many-point structure. To introduce the concept of line (or curve), one needs to introduce some topological structure (topology) in the space. Thus, to construct Riemannian geometry, one needs two structures (topological and metric).

To obtain T-geometry, the Euclidean straight, passing through points *P,Q, *is defined via its collimetric property. The term "collimetric" is formed from two terms "collinear" and "metric". The collimetric property describes the collinearity condition of two Euclidean vectors **RP **and **RQ **via mutual Euclidean distances between points *P,Q,R*. The collimetric property can be described by the relation

V(P,Q,R) = 0,

where *V(P,Q,R) *is the area of Euclidean triangle with vertces at points *P,Q,R*, expressed by means of the Hero's formula via mutual distances between points *P,Q,R*. Indeed, vectors **RP **and **RQ **are collinear, and three points *P,Q,R *lie on the same straight, if and only if the area of the Euclidean triangle *PQR *vanishes. Then the Euclidean straight *T**_PQ*, passing through points *P,Q, *is defined as a set of points *R *

*T_PQ = *{*R|V(P,Q,R) = 0*}.

This definition of the Eulidean straight contains only relatively simple three-point structures. It defines the Euclidean straight as a **set of points **and does not refer to any structures other than metric one (distance). In the proper Euclidean geometry this set of points *T**_PQ *is a line. In general, it is not so, if one deforms the proper Euclidean space, changing distances between points. If the deformation is small, the *T**_PQ *becomes a thin hallow tube. The obtained geometry, where straight lines are substituted by tubes, may be referred to as tubular geometry (T-geometry), or as nondegenerate geometry (the tubes are not degenerate into lines). Imposing on T-geometry additional constraints of extremality, which demand that tubes *T**_PQ* turn to lines, one returns to the special case of Riemannian geometry. The extremality constraint, formulated in terms of T-geometry, looks as a complicated condition including many-point structures. As one can see from this definition, the T-geometry is more fundamental, more general and simpler, than the Riemannian geometry.

- For instance, in the field of Riemannian geometry there is only one flat uniform isotropic spacetime (Minkowski spacetime), whereas in the field of T-geometry there is a class of flat uniform isotropic spacetimes which differ by the form of distortion function
*D*which describes the shape of tubes (in the given case it is a function of one argument). The Minkowski spacetime is a member of this class (in this case the distortion function*D*vanishes, and timelike tubes degenerate into timelike straights). - Applying T-geometry to a construction of spacetime model, one obtains surprising results which could not be obtained in the degenerate geometry of Minkowski. Spacelike straights are NGOs of the Minkowski geometry, considered as a kind of Riemannian geometry, but they are not NGOs of T-geometry. This fact is interpreted in the sense that T-geometry forbids taxyons, whereas the Riemannian geometry uses special additional constraints for forbidding taxyons. In application to spacetime model the T-geometry have the following unexpected properties:

- The particle mass is geometrized and can be concluded from the shape of particle world tube.
- World tubes of particles are stochastic, what agrees with existence of observable quantum stochasticity of microparticles.
- Elementary length (thickness of world tubes) appears in the spacetime model based on nondegenerate geometry.
- The four-velocity and momentum of a particle are not parallel in the Minkowski spacetime associated with real spacetime constructed on the basis of T-geometry.
- Coupling the elementary length (distorion) with the quantum constant, one succeeds to choose the optimal uniform isotropic spacetime model in such a way that the statistical description of stochastic world tubes coincides with their quantum description by means of the Schroedinger equation.
- After such a choice of the spacetime model the quantum constant
*h*becomes an attribute of the spacetime, and quantum effects become effects conditioned by natural geometric stochsticity. Quantum principles become unnecessary.

Dynamical conception of statistical description is introduced, because a consequent statistical description of stochastic world lines (tubes) on the basis of the probability theory is impossible. Dynamical conception of statistical description may be explained as follows. The result of an experiment with a single stochastic particle *S**_st* is irreproducible. But distributions of results of similar experiments with *N* independent stochastic particles are reproducible (if *N* is very large). Projecting many independent identical stochastic particles *S**_st* to the same spacetime region, one obtains a cloud *E*[*S_st*] of *N* independent identical particles *S**_st* moving randomly. If the number *N *of particles is very large, this cloud *E*[*S_st*] may be considered to be a continuous medium, or a fluid. This fluid is a deterministic dynamic system, because experiments with the fluid *E*[*S_st*] are reproducible. Besides, any reproducible series of experiments with the stochastic particle can be described in terms of the fluid *E*[*S_st*] without reference to any probabilistic construction (i.e. without a reference to the statistical ensemble's property of being a tool for calculation of average values). Coupling of such statistical description with the quantum description appears, if one takes into account that any nondissipative fluid (and, in particular, the fluid *E*[*S_st*]) has __wave function and spin__ as its attributes. T-geometrical spacetime model equipped with dynamical conception of statistical description admits one to consider quantum effects as effects generated by natural geometrical stochasticity.

There is __preface in English__ and detailed __author's review in English with formulas __(about 23 pp in PostScript format (205 KB)).

The monogrph is written in Russian (about 180 pp in __PostScript format__ (2100 KB)), Three figures: __fig1__, __fig2__, __fig3__.

It can be downloaded also in the form of pkzipped PostScript file __indg.zip__ (997 KB). Figures: __indgf.zip__.

Updated 10/10/99