Foundation of physical geometry

November 17, 2003

The Euclidean geometry is the basic geometry. All other geometries are obtained as a modification of the proper Euclidean geometry. On one hand, the proper Euclidean geometry is a science on mutual disposition of points and geometric objects in the space. This disposition is described by the metric *r* *(P,P')* (distance) between two points *P* and *P'* of the space, or by the world function *s* *(P,P')=r ^{2}*

- One can modify the proper Euclidean geometry, changing world function
*s*and using the fact that the proper Euclidean geometry is described completely by means of the world function. Any such a change of the world function is a deformation of the space. Geometry, appearing as a result of such a deformation, will be referred to as the physical geometry. - One can modify the proper Euclidean geometry, changing axiomatics of the Euclidean geometry. Geometry, appearing as a result of this modification, will be referred to as the mathematical geometry.

Maybe, terms 'mathematical geometry' and 'physical geometry' are not completely apposite, but it is necessary to distinguish between the two kinds of geometries and not to confuse them.

Examples of mathematical geometries are the geometry of Lobachevsky, the projective geometry, the symplectic geometry, etc. These geometries are not relevant to the space and the space-time, or their relation to the space is indirect. Mathematical geometries are not interesting for physicists, and we shall not consider them.

The physical geometries describe mutual disposition of geometrical objects in the space, or in the space-time. They are very interesting for physicists, because all physical phenomena evolve in the space-time, and configuration of the space-time appears to be very important for description of physical phenomena.

The remarkable property of the proper Euclidean geometry is that the proper Euclidean geometry can be completely described in terms of the world function, or in terms of the metric. It is the crucial point for construction of physical geometries. It means that all geometrical objects *O*_{E} in the proper Euclidean geometry and all relations *R*_{E }between them can be expressed in terms of the world function *s*_{E} of the Euclidean space and only in its terms.

*O*_{E }=* O*_{E }*(s*_{E }*), R*_{E }=* R*_{E }*(s*_{E }*) *

The world function *s*_{E} of the Euclidean space have special properties and satisfies a set of __conditions__, written in terms of the world function. These conditions contains the integer parameter *n*, which can be interpreted as the dimension of the proper Euclidean space. There is a theorem which states that these conditions are necessary and sufficient conditions of the Euclideaness. According to this theorem all parameters of the Euclidean space (dimension, collinearity condition of two vectors, metric tensor, scalar product, coordinate system, etc.) can be expressed via the world function *s*_{E}* *and only via it.

After deformation of the Euclidean space, when the world function *s*_{E} of the Euclidean geometry *G*_{E} is replaced by another world function *s*_{D} , all geometrical objects *O*_{E }=* O*_{E }*(s*_{E }*), *and all relations *R*_{E }=* R*_{E }*(s*_{E }*)* between them are transformed to another geometrical objects and another relations between them.

*O*_{E }=* O*_{E }*(s*_{E }*)® O*_{D }=* O*_{E }*(s*_{D }*), *

*R*_{E }=* R*_{E }*(s*_{E }*)® R*_{D }=* R*_{E }*(s*_{D }*)*

Geometrical objects *O*_{D }=* O*_{E }*(s*_{D }*)* and relations between them *R*_{D }=* R*_{E }*(s*_{D }*) *are geometrical objects and relations between them of another physical geometry *G*_{D }, which is described completely by the world function *s*_{D}. Geometrical object *O*_{D }=* O*_{E }*(s*_{D }*)*_{ }in geometry *G*_{D} correspond to the geometrical object *O*_{E }=* O*_{E }*(s*_{E }*)*_{ }in the Euclidean geometry *G*_{E}. In other words, the physical geometry *G*_{D }is as pithy, as the Euclidean geometry, because the geometry *G*_{D} contains all geometrical objects, which are contained in the Euclidean geometry._{ }

Thus, using deformation of the Euclidean geometry and the fact that the world function describes the Euclidean geometry completely, we can construct a physical geometry with any metric structure (with any distances between its points). The obtained deformed geometry *G*_{D} is as consistent as the proper Euclidean geometry *G*_{E}, because in the physical geometry* G*_{D} there are no its own axioms, theorems and statements. All relations between geometrical objects are taken from the Euclidean geometry in the deformed form. The only real problem of constructing a physical geometry is a writing of the Euclidean relations in the *s*-immanent form, i.e. in terms and only in terms of the world function *s* .

There are some subtleties in such a writing in the *s*-immanent form. The fact is that, that the world function *s*_{E }of Euclidean space has its own specific Euclidean properties, and these properties must not be used at writing in the *s*-immanent form. These __specific Euclidean properties__ are written for *n*-dimensional Euclidean space. They contain a reference to the dimension *n* of the Euclidean space. If we use them for definition of the geometric object *O*, the definition of the object *O* will contain parameter *n*, which has nothing to do with the geometrical object *O*.

For instance, the straight line *T _{PP'} *in the proper Euclidean space is defined by two its points

*T _{PP'}={R|*

where *R* is the running point of the set *T _{PP'}* and condition

(1)

*(***PP'***.***PR***) ^{2}=(*

The scalar product is defined via the world function by the relation

*(***PP'***.***PR***)= 0.5(|***PP'***| ^{2}+|*

Thus, the straight line is defined *s*-immanently, i.e. in terms of the world function *s* .

In the Euclidean geometry one can use another definition of collinearity. Condition of collinearity is satisfied, if components of vectors **PP'** and **PR** in some coordinate system** **are proportional. For instance, in the 3-dimensional Euclidean space one can introduce rectangular coordinate system, choosing four points *P ^{3}* ={

(2)

*(***PP**_{a}** ***.***PP'***)=a (***PP**_{a}** ***.***PR***), a =1,2,3,*

Here *a *is some constant. Relations (2) are relations for covariant components of vectors **PP' **and **PR** in the considered coordinate system with basic vectors vectors **PP**_{a}** ***, a =1,2,3*. Then equations (1), (2), and

*T(P,P',P _{1},P_{2},P_{3})={R| *

determine the geometrical object which depends on five points *P,P',P _{1},P_{2},P_{3}*. This geometrical object describes a complex, consisting of the straight line and the coordinate system, represented by four points

Which of two geometrical objects should be interpreted as the straight line, passing through the points *P,P'* in the deformed geometry *G*_{D}? Of course, the straight line is *T _{PP' }*, because its definition does not contain a reference to a coordinate system, whereas definition of

But in the given case the geometrical object *T _{PP' }*is, in general, two-dimensional surface, whereas

It is very difficult to overcome our conventianal idea that the Euclidean straight line cannot be deformed into many-dimensional surface, and **this idea has been prevent for years from construction of the physical geometries**.

Practically one chooses such physical geometries, where deformation of the Euclidean space transform the Euclidean straight lines into one-dimensional lines. It means that one choose such a geometries, where geometrical objects *T _{PP'}* and

(3)

*T _{PP'}*

Riemannian geometries satisfy this condition. The Riemannian geometry is a kind of physical geometry which is constructed on the basis of the deformation principle, when the infinitesimal Euclidean interval *dS*_{(E)}^{2}=g_{(E)ik}*dx ^{i}dx^{k}* is deformed into the Riemannian interval

Note that in physical geometries, satisfying the condition (3), the straight line *T _{Q;P''P'}*, passing through the point