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 (P,P')/2. On the other hand, the proper Euclidean geometry is a construction built on the basis of some axiomatics. The proper Euclidean can be modified in two directions

• 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 OE in the proper Euclidean geometry and all relations RE between them can be expressed in terms of the world function sE of the Euclidean space and only in its terms.

OE = OE (sE ), RE = RE (sE )

The world function sE 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 sE and only via it.

After deformation of the Euclidean space, when the world function sE of the Euclidean geometry GE is replaced by another world function sD , all geometrical objects OE = OE (sE ), and all relations RE = RE (sE ) between them are transformed to another geometrical objects and another relations between them.

OE = OE (sE )® OD = OE (sD ),

RE = RE (sE )® RD = RE (sD )

Geometrical objects OD = OE (sD ) and relations between them RD = RE (sD ) are geometrical objects and relations between them of another physical geometry GD , which is described completely by the world function sD. Geometrical object OD = OE (sD ) in geometry GD correspond to the geometrical object OE = OE (sE ) in the Euclidean geometry GE. In other words, the physical geometry GD is as pithy, as the Euclidean geometry, because the geometry GD 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 GD is as consistent as the proper Euclidean geometry GE, because in the physical geometry GD 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 sE 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 TPP' in the proper Euclidean space is defined by two its points P and P'. The relation, defining the straight line TPP', has not to depend on dimension of the Euclidean space. The straight line is defined by the relation

TPP'={R| PP'||PR},

where R is the running point of the set TPP' and condition PP'||PR means that vectors PP' and PR are collinear, i.e. the scalar product (PP'.PR) of these two vectors satisfies the relation

(1)

(PP'.PR)2=(PP'.PP') (PR.PR)º |PP'|2|PR|2.

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

(PP'.PR)= 0.5(|PP'|2+|PR|2-|P'R|2) º s (P,P')+s (P,R)-s (P',R)

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 P3 ={P,P1,P2,P3} and forming three basic vectors PPa , a =1,2,3. Then the collinearity condition can be written in the form of three equations, with two of them being independent.

(2)

(PPa .PP')=a (PPa .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 PPa , a =1,2,3. Then equations (1), (2), and

T(P,P',P1,P2,P3)={R| PP'||PR},

determine the geometrical object which depends on five points P,P',P1,P2,P3. This geometrical object describes a complex, consisting of the straight line and the coordinate system, represented by four points P3 ={P,P1,P2,P3}. In the Euclidean space dependence on the choice of the coordinate system and three points P1,P2,P3 determining this system, is fictitious. The geometrical object T(P,P',P1,P2,P3) depends only on two points P,P' and coincides with the straight line TPP'. But at deformations of the Euclidean space the geometrical objects T(P,P',P1,P2,P3) and TPP' are deformed differently. The points P1,P2,P3 cease to be fictitious in definition of T(P,P',P1,P2,P3), and objects T(P,P',P1,P2,P3) and TPP' become to be different geometric objects, in general.

Which of two geometrical objects should be interpreted as the straight line, passing through the points P,P' in the deformed geometry GD? Of course, the straight line is TPP' , because its definition does not contain a reference to a coordinate system, whereas definition of T(P,P',P1,P2,P3) depends on the choice of the coordinate system. In general, definition of geometric objects and relations between them may not refer to the means of description.

But in the given case the geometrical object TPP' is, in general, two-dimensional surface, whereas T(P,P',P1,P2,P3) is intersection of two two-dimensional surfaces, i.e. T(P,P',P1,P2,P3) is, in general, a one-dimensional curve. The one-dimensional curve T(P,P',P1,P2,P3) corresponds better to our ideas on the straight line, than the two-dimensional surface TPP'. Nevertheless, it is TPP', that is analog of the Euclidean straight line in geometry GD.

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 TPP' and T(P,P',P1,P2,P3) coincide.

(3)

TPP' =T(P,P',P1,P2,P3)

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)ikdxidxk is deformed into the Riemannian interval dS(R)2=g(R)ikdxidxk. Deformation is chosen in such a way that any Euclidean straight line T(E)PP' , passing through the point P, collinear to the vector PP' coincide with the geodesic T(R)PP' , passing through the point P, collinear to the vector PP' in the Riemannian space. Condition (3) of coincidence of objects TPP' and T(P,P',P1,P2,P3) restricts class of possible physical geometries, reducing it to the class of Riemannian geometries.

Note that in physical geometries, satisfying the condition (3), the straight line TQ;P''P', passing through the point Q collinear to the vector PP', is not a one-dimensional line, in general. If the Riemannian geometries be strictly physical geometries, then they would contain non-one-dimensional geodesics (straight lines). But the Riemannian gemetries are not strictly physical geometries, because at their construction one uses not only the deformation principle, but some other methods, containing a reference to the means of description. In particular, in the Riemannian geometries absolute parallelism is absent, and one cannot to define a straight line, passing through the point Q collinear to the vector PP', provided points P and Q do not coincide. On one hand, lack of absolute parallelism allows one to go around the problem of non-one-dimensional straight lines. On the other hand, it makes the Riemannian geometries to be inconsistent, because they cease to be physical geometries.