Nondegenerate space-time geometry

Updated 25/12/2001

Metric conception of geometry

At the metric conception of geometry (MCG) the geometry is determined on any point set M by setting real function G(P,P') with the properties

G(P,P')=G(P',P),        G(P,P)=0


The function G(P,P') is called the world function, and the geometry determined by the world function is called tubular geometry, or shortly T-geometry. All relations of T-geometry are derived as relations of proper Euclidean geometry written in terms of the world function.

Vector PQ of T-geometry is the ordered set of points PQ={P,Q}. Its length is |PQ|=G(P,Q). The scalar product (PQ.RS) of two vectors PQ   and  RS has the form


In the special case, when the points P and R coincide, one has



which means the cosine theorem for the triangle with vortices at the points P,Q,S.

Two vectors P_0P_1and Q_0Q_1 are parallel, provided the angle a between them is equal to zero, i.e.


 Taking into account that |P_0P_1||P_0P_1|=(P_0P_1.P_0P_1), this condition can be written in the form of the second order Gram's determinant

det||(P_0P_i .Q_0Q_k)||=0,       i,k=1,2


In general, the condition (4) is not so strong, as the condition (3), it means that vectors P_0P_1and Q_0Q_1 are linearly dependent (collinear P_0P_1||Q_0Q_1 , i.e. parallel, or antiparallel).

In general, in the proper Euclidean space n vectors P_0P_ii=1,2,...n are linearly dependent, if and only if the nth order Gram's determinant vanishes

F_n(P^n)=det||(P_0P_i .P_0P_k)||=0,       i,k=1,2,...n


Thus, in virtue of (2) the necessary and sufficient condition of the linear dependence of   n vectors P_0P_ii=1,2,...n is written in terms of  only world function, without referring to linear space. The formulation of the linear dependence condition contains only several points of the space and the world function between them.

The proper Euclidean geometry can be described completely in terms of the world function and finite sets of points P^n={P_0,P_1,...P_n}.  Such a description of geometry will be referred to as sigma-immanent description. At such a description of the Euclidean geometry the world function is to satisfy some sigma-immanent conditions I - III.

One can prove the following


The sigma-space V={M,G} is theEuclidean space, if and only if the conditions I - III are fulfilled.

This theorem means that information, concluded in the world function is sufficient for construction of the Euclidean geometry. This information is sufficient also for construction of a geometry, if conditions I  -  III, specific for Euclidiean geometry are not satisfied.  Any world function assiciates with some geometry (T-geometry).

In particular, in any geometry n points P^n={P_0,P_1,...P_n}, which do not satisfy the condition (5) determines the nth order natural geometric object (NGO). This object, called the nth order tube, is the set T(P^n) of points, defined by the relation


In the case of the proper Euclidean geometry the nth order tube T(P^n) is the n-dimensional plane. In the case of two points P_0, P_1 the first order tube T(P_0P_1 ) is the straight line, passing through points P_0, P_1. Changing the Euclidean world function G_E, one obtains destorted Euclidean space with the world function G_D. In this more general case, when conditions I - III are not fulfilled, and T-geometry is not the Euclidean one, the tube T(P_0P_1 ) is a hallow  (m-1)-dimensional tube (m is the dimension of the space), but not a line.  The T-geometry with world function


where D is some function, is nondegenerate T-geometry. In such a geometry the segment T([P_0P_1 ])of the tubeT(P_0P_1) between  points  P_0, P_1  has the cigarlike shape. This "cigar" is a set of ends R of the vector P_0R, parallel to the vector P_0P_1 . In the proper Euclidean space there is only one vector P_0R of fixed length |P_0R|, which is parallel to the vector P_0P_1. In the distorted space V_D with the world function G_D there is a set of  many such vectors. The ends of the vectors P_0R form the "cigar". The T-geometry, where there are many vectors P_0R of fixed length parallel to the vector P_0P_1 is nondegenerate geometry. Under some world functions the (m-1)-dimensional "cigars" T([P_0P_1 ]) may degenerate into a curves, and one obtains a degenerate T-geometry. The proper Euclidean is a degenerate geometry.

Natural geometrical objects (NGO) are sets of points determined by the geometry and several parameters (basic points of NGO). Points of the space are parameters of NGOs. Any geometry is described by their natural geometric objects (NGO). The order of NGO is determined by the number of basic points. In the proper Euclidean geometry the point P is the zeroth order NGO defined by one parameter (the same point P). The straight L(P,Q) is the first order NGO defined by two different points P,Q. The plane L(P,Q,R) is the second order NGO, defined by three different points P,Q,R which do not lee on one straight. In other geometries NGOs have another form, but they always are defined by corresponding geometry. For instance, in the Riemannian geometry the first order NGO is a geodesic. Description of a geometry in terms of its NGOs is the most adequate way of construction of the geometry. For instance, the proper Euclidean geometry is described conventionally in terms of points, straights, planes,....(i.e. in terms of its NGOs). In the proper Euclidean geometry NGOs are defined by their properties by means of axioms. On the other hand, if distances S=S(P,Q) between all pairs of points P,Q are given, the NGOs of Euclidean geometry can be defined via distances. In this case the axioms describing properties of NGOs turn to theorems, and geometry is defined by the form of the distance function S=S(P,Q) . Practically it is more convenient to use instead of S=S(P,Q) the so called world function

G(P,Q) = S(P,Q) S(P,Q)/2


which is always real, even so the geometry is the Minkowski geometry.

Geometry of any space may be considered as a result of a deformation of the proper Euclidean geometry, when one changes distances S=S(P,Q) between the space points P, Q. Any such deformation changes the shape of NGOs. On the other hand, any geometry is some kind of generalization of the proper Euclidean geometry. The way of generalization of the Euclidean geometry depends essentially on the way of definition of the Euclidean straight (the first order NGO). If the Euclidean straight L(P,Q) is defined as shortest line between the points P,Q , determining L(P,Q), one obtains the Riemannian geometry. If the Euclidean straight L(P,Q) is defined by its collimetric property as a set of points, one obtains the T-geometry (or tubular geometry, or geometry of tubes). The term "collimetric" is constructed of two terms "collinear" and "metric". Collimetric property means the collinearity of two vectors RP and RQ expressed via distances between points P,Q,R. Mathematically collinearity RP||RQ is expressed by means of the relation


where according to the cosine theorem the scalar product (RP.RQ) of vectors RP and RQ can be expressed via their lengths |RP| and |RQ|, or via world function G(P,Q) = S(P,Q) S(P,Q)/2 by the relation

(RP.RQ) = G(R,P) + G(R,Q) -- G(P,Q)


The Riemannian geometry is such a generalization of the proper Euclidean geometry which uses definition of the first order NGO as the shortest line. This generalization to the Euclidean geometry is based on two structures: the topological structure and the metric one. The fact is that the concept of a line needs a definition, and this definiton is carried out via topological properties which are introduced independently of the metric properties. As a result the Riemannian geometry is a geometry based on two structures.

The T-geometry is a generalization of the proper Euclidean geometry which uses the definition of the first order NGO as a set of points having the collimetric property. The set of points does not need an additional definition. As a result the T-geometry is a geometry based on only metric structure. The T-geometry is more general and simpler than the Riemannian geometry. T-geometry contains the Riemannian geometry as a special case. On the other hand the first order NGO defined as a set of points is not a line in general. The first order NGO in T-geometry appears to be a hallow tube. In some cases these tubes can degenerate into one-dimensional lines. Then one obtains degenerate geometries (Riemannian and Euclidean). Nondegenerate geometry is another name for the T-geometry.

In the T-geometry the first order tube T_{P,Q} (the first order NGO) is determined by the points P and Q as the set of running points R

T_{PQ} = (R|RP||RQ ) = (R|(RP.RQ)(RP.RQ) = (RP.RP)(RQ.RQ))

where all scalar products can be expressed via the world function by means of the relation

(RP.RQ) = G(R,P) + G(R,Q) -- G(P,Q).


It is clear intuitively that the whole geometry can be constructed, if one can construct NGO. It is clear in the case of the proper Euclidean geometry. It is so in the case of a general geometry defined by an arbitrary world function G(P,Q).

For degenerate geometries (Euclidean, Riemannian, Minkowski) the first order tube T_{P,Q} degenerates into one-dimensional line L_{P,Q} due to extremal properties of the world function. In this case it is possible to construct a manifold, to determine the dimensionality of the space and introduce a coordinate system. (see the monograph and corresponding reference in the list of papers). Construction of the manifold on the basis of the metric means that in some cases it is possible to construct the topological structure on the basis of the metric one.

In the general case it is impossible to construct a manifold. A mathematical technique of work with a nondegenerate geometry is not yet developed. But in the case, when the real world function is distinguished slightly from that of the Minkowski space, one can use approximately the relation



where G_M is the world function of the Minkowski space and D is a distortion function describing a degree of nondegeneracy of the space-time.

In this case it is possible to construct a manifold on the base of G_M, to introduce a coordinate system and to consider the distortion function D=D(P,Q) as some two-point field in the Minkowski space-time. If the real space-time is uniform and isotropic, the distortion function is a function of only G_M


In the space-time with the nonvanishing distortion D a world tube of a real particle of the mass m is defined as a broken tube consisting of segments T_[P,P'] of the same length. The length of the segments is proportional to the mass m of the particle. Thus, the mass of a particle becomes a geometrical quantity. As a result of the nonvanishing distortion the relative position of two adjoining segments is not fixed, (although the hyperbolic cosine of the angle between them for a free particle is equal to 1). In this sense the world tube of a real particle is stochastic. The stochasticity is the more intensive the less the mass of the particle.

Statistical description

General way of statistical description (SET)

By definition a statistical description of a stochastic system S_st is a replacement of a description of a single stochastic system S_st by a description of N independent identical systems S_st (with N tending to infinity). As a result of such a replacement a deterministic dynamic system E[S_st] arises. E[S_st] is known as a statistical ensemble of stochastic systems S_st.

Experiments with a single S_st are irreproducible, whereas experiments with E[S_st] are reproducible. In other words, a result of a single experiment is irreproducible, but distributions of results are reproducible,

The principal goal of the statistical description is a construction of E[S_st] as a dynamic system. If such a dynamic system E[S_st] has been constructed, then, investigating this dynamic system, one can calculate and explain all results of reproducible experiments with S_st (in reality with ensembles of S_st ). There is no general way of the construction of the dynamic system E[S_st], corresponding to the stochastic system S_st.

The statistical description in itself does not need any probabilistic constructions. These probabilistic constructions (the statistical ensemble as a tool for calculation of average values) are needed only for interpretation of the E[S_st] dynamcis and experiments with E[S_st] in terms of the mean behaviour of the stochastic system S_st. Such an interpretaition has also a heuristic meaning.

If the distortion function has asymptotic value D = h/b, where h is the quantum constant and b is some universal constant connecting the usual mass of the particle with the length of the world tube segment T_[P,P'], the statistical description of stochastic world tubes coincides with the quantum description (Schroedinger equation).

Updated 25/12/2001