Geometrical Meaning of Time and the Theory of Relativity

New geometrical model of time is suggested where time of body’s motion is defined as the length of its trajectories in four-dimension space-time. Within suggested approach periodical motions in clocks correspond to definite length of four-dimension trajectories that is clocks appear to be standards for measurements of length in four-dimension space analogously as hard sticks are standards for measurements of length in three-dimension space. This means that space and time are entities of the same geometrical nature. A suggested interpretation of time leads to necessity of changes in general theory of relativity. These changes are unessential for body’s motion in weak gravitational field.


Introduction
This work is a continuation of author's investigations on geometrisation of quantum mechanics [1] [2] [3] [4] [5]. As "geometrisation" we understand the investigation of such models of physical phenomena when these phenomena are expressed through understandable geometrical patterns and notions. We suppose that such model representations play an important role in physical investigations-the known example of use of just geometrical model for obtaining new physical results is the general theory of relativity-GTR (review of literature on geomerization in physics see, e.g., [6]) In this work the new geometrical representation of tine leads to necessity of changes in equations of GTR for motions in gravitational field. Preliminary results are published in [7] [8] [9].
To clarify what we mean speaking about "new" geometrical representation of time, we remind shortly about geometrical representation of time within existing special theory of relativity-STR [10] [11] [12] [13]. STR considers space and time as unique space-time continuum which points are named as "events." The set of four numbers ( ) , , , t x x x corresponds to each event, where t is the time when this event happened and 2 2 3 , , x x x are coordinates of the place of event. This four-dimension space-time continuum is named as "space of events". It represents a so called four-dimension Cartesian space where there is no initial geometry [11]

Geometrical Meaning of Time and New Geometrical Model of the Space of Events
Let us consider the space of events for a case when event is a displacement of free particle (material point) from the origin of some inertial coordinate system (it is considered as immovable) to the point with coordinates 1 2 3 , , x x x for the time t. The new isomorphism of such space of events in the four-dimension space with definite geometry is based on following suggestions: 1) Coordinates 1 2 3 , , x x x are the particle coordinates in three dimensional affine subspace of four-dimension affine space 1 2 3 4 , , , x x x x 2) Time of the particle displacement at distance 3 L in the point with coordi- x x x , multiplied by light velocity, equals to g L -the length of the particle straight geodesic trajectory in the above four-dimension affine space We place here coordinate' indexes upwards to stress the distinction between covariant and contravariant coordinates in affine space [10]. Two dimensional analogies of Minkowski space and suggested space are represented for comparison in Figure 1.
The Lorentz transformations have here, as in Minkowski model, the meaning of transformations transferring the space into itself. The transformations (1) express in STR the invariance of quadratic form respectively to transfer from one inertial frame to another (here , , This relation means, that  , , , Secondly, in contrast to Minrowski model surrounding three dimensional space is supposed to be not Euclidean, but affine space. Within such approach the known effects of STR (retarding of time, reduction of length) appear to usual proprieties of affine space, where form and volume of objects are changing under coordinate transformations [10]. At not relativistic velocities three dimensional space may be considered as Euclidean one and four-dimension affine space, considered above, may be considered as four-dimension pseudo Euclidean Minkowski space. This follows from the fact that the Lorentz transformations (1) and (6)

Geometrical Meaning of Time and General Theory of Relativity
The geometrical interpretation of time was suggested in previous Section for the free particl's motion, which is for motion in space without of the force fields.
New equations have formally the same form as the existing equations, but they principally differs from them because the time variable enters in the new equations in different way. In existing theory time, multiplied by light velocity, enters to the left side of Equation (8) as the fourth coordinate 4 x , but within suggested interpretation of time, we suppose that time of the particle's motion, multiplied by light velocity, is defined to be the length of the particle's geodesic trajectory. Choosing the length of geodesic trajectory as canonic parameter, we obtain, that time enters to the right side of Equation (8)

Conclusion
The geometrical interpretation of time is suggested as the