Classical mechanics is acting (or is geometrically interpreted) in a three-dimensional conventional Euclidean space  which fully corresponds to physiological ability of thinking creatures from planet Earth to geometrical imagination. The STR, having rejected the classical idea of absolute (surgeometrical) time t, has immediately been "immersed” into the four-dimensional space  each point of which is characterised by coordinates . Here the constant  m/s may be regarded as a universal (i.e. independent of the choice of a frame of reference) "transforming constant", which transforms quantities of x, y, z type, possessing the dimension of length (m) into quantities , each having the dimension of the first coordinate t. This version has no principal distinction from a four-dimensional space with four coordinates x, y, z, ct in use.

But the four-dimensional world of STR is not a conventional Euclidean space  (possessing index k=0) but, actually, it is pseudoeuclidean space possessing negative index k=3 (or an isomorphic to it space  possessing index k=1). This conclusion follows from the fact, that the only numerical invariant  (space-time interval) in R₄, k=3 is determined by a contravariant tensor x^s, where s=0, 1, 2, 3 (x⁰=t, x¹=t_x, x²=t_y, x³=t_z) and by a diagonal metric tensor of the second rank g_ss through expression

. (2.1)

Note that in the space  all the three numbers g_ss have magnitudes 1, and there exist two independent numerical invariants:

1) the contraction  or the contraction of the type

;

2) surgeometrical invariant t (absolute time).

But "our" space  is characterised not only by its being a tensor space but also by its being a vector space in the sense that in space  there exist vectorial invariants of the  type. These invariants , , ... are existing in the space even before we introduced there the concept of basis unit vectors , i.e. even before our introducing there the concept of a tensor of the first rank.

Together with a linear transformation of the three coordinates of the type

in the vector space  there takes place the analogous co-transformation of the corresponding unit vectors : if the column of three coordinates is multiplied by the orthogonal matrix of rotation  then the string of the three unit vectors is multiplied by the inverse matrix . As a result, nothing changes in the resulting expression for the vector :

(2.2)

So we ohght to regard vector  in (2.2) as a special invariant contraction of one “numerical tensor”  and one “basis tensor” , i.e. the contraction .

In this case one must not speak about the transformation only of the coordinate tensor x^s – y^s , if the new coordinates  are set off upon the same unit vectors  . Even more prohibited is to say, for instance, that a two component vector  in plane , regarded as “a part of the tensor x^s - y^s, s=1, 2, 3“, when this “part” is transferred by transformation into "another vector"  referred to the same unit vectors  and . Actually, the vector  is a quite different from the vector : in the system of unit vectors () it is a two-component one, but it becomes a three-component vector in the new system of unit vectors (), so as the contraction (convolution)  remains invariant. .

But just these incorrect actions are employed in STR while transforming all the four components of the tensor  using the Lorentz transformation and dealing afterwards with the so changed three-component "vector"  set off upon unchanged unit vectors . This new "vector" , that depends on the transformation of a four-component tensor , is regarded in a new mooving inertial frame of reference  possessing its own time  and the axes  parallel to the axes Oxyz of the former "immovable" frame of reference K possessing its own time t.

The tensor space R₄, k=3, in which STR is realised, may be turned into a vector space R₄, k=3 in which vectors  exist even before we introduce the concept of a usual four-component numerical (coordinate) tensor. To do so one has to generalise the concept of a scalar product of two arbitrary vectors  and  from the space R₄, k=3: the scalar product of two vectors  and  is such a symmetrical and bilinear function  which to each pair of vector arguments  and  puts into correspondence the real number s which is not necessarily positive in case of  =.

In this pseudoeuclidian vector space R₄, k=3 consisting of real fourdimensional vectors with real coordinates, the index k=3 indicates that there exist three linearly independent fourdimensional vectors  , such that . This is why in R₄,k=3 as a system of basis unit vectors it is convenient to choose the system of vectors  such that  and  and with condition , when . By this, only real quantities (here ) are permitted as multipliers for unit vectors .

Then in R₄,k=3 by analogy with (2.2) one obtains invariant vectors (invariant convolutions) in space R₄,k=3 of the form

, (2.3)

Here  are four new basis unit orthogonal vectors which are obtained as the result of linear and homogenous transformation of the four former unit vectors . The coefficients of these four equalities (realizing the transformation) make some symmetrical matrix  , which is the reverse matrix to the well known matrix  , describing the transformation of the four coordinates and using the most general expressions of Lorentz transformation in the Gerglotz version.

If  denote an element of the  - matrix  then new basis  in (2.3) we have as a four contractions of type . So, define by , two 4-columns from basis vectors , one will have , which denotes four homogeneous equalities. One can satisfy himself that  if  and  if , but . It means that new basis  (contrary to STR!) is not deformed, but only “pseudorotated in ”. And more: the Poincore group is representate by means of matrixes , where , because for two matrixes  we have:  and .

If all the orthogonal transformations of type  to combine afterwards with conventional rotations of the last three unit vectors  (under the condition that ) by using the orthogonal matrix , then the group of Poincare transformations (all matrixes of [D]_4,4 type) for four basis unit vectors in case when the origin O of this basis does not change will be obtained. The matrixes [D]_4,4 of co-transformations of four coordinates also belong to this group.

In the real vector space  as independent on each other there exist three types of "abstract things": 1) points, 2) vectors, 3)real numbers. These "things" are interconnected by a system of axioms. By this, the coordinates  have principally changed their sense comparatively with the space , because now each of the vectors , observing that , where s = x, y, z , has no length or no modulus. The same may be said about the vector . The concept of a distance between points A and B, belonging to , is valid not for all the points A and B is space  but only for those, for which the vector

(here  and  , ) possesses the modulus, i.e. if . No other approach to the concept of a distance in the vector space  is existing. If  then one reaches the cone of isotropy. Each vector belonging to this cone is orthogonal to itself or it has the zero length.

This is how the metric of  looks like. Naturally, we will consider only those physical directions of motion in , which correspond to vectors  satisfying the condition , or to vectors  if . Let us emphasize: above mentioned representation of the fourdimensional vector  is a more general case of contraction (convolution), which gives in space , a new and more general (vectorial) invariant than (2.1). Only such invariant – vectors in space  one can name as “vector”.

That is why considering the three-component part of the vector  of the form  in the given fixed frame of reference  is permitted only in that sense, that in another frame of reference  which is pseudorotated in the space  the same vector , will posses four components. Of course, the negative numerical invariant  must be observed. Prohibited is, though, to speak about the three-component vector  as of the transformed former three-component vector , because the vector  is just another vector in both frames of reference, actually being a four-component vector in the first frame of reference. Quite analogous is the situation with a one-component vector  which in the new system will become the same but now a four-component invariant vector  with the numerical invariant  kept.

On the basis of this we can state that geometrical interpretation of the Lorentz transformation  in the space  is not identical to purely mechanical presentation of this transformation in the space  with its two classical three-dimensional frames of reference K and K' moving relatively to each other with the velocity  and each having their own times t and t'. Yes, indeed, each of the abovesaid matrixes  or  descibing the transformation  is completely determined if three arbitrary numbers v_x, v_y, v_z are given under the condition , but all matrix operations of mutual multiplying of such matrixes  (altogether with all matrixes of pure rotation ) cannot be reduced to vectorial algebra in the space  even by adding the concept of different times t and t' in each system K and K` (New wine should not be poured into old wine skins). The very idea of three-dimensional system K mooving in “our” space , is the “old wine skin” from classical mechanics.

This is why we are going to deal with a neorelativistic inertial frame of reference  which is only pseudorotated (without usual “mooving”!) in relation to another system  (pure rotation of the three last unit vectors  with the vector  remaining unchanged may be here included). In this case the sense of a position vector  fixing the point A in the space , is that the arrow end of this vector is drawing a four-dimensional space curve and that the four coordinates  of this vector  are functions of the invariant argument t , which denotes "the time length" of an arc of this curve, laid off from the initial point , under the condition  , i.e.

,  (2.4)

If the point A is a free material point then all the four coordinates are the linear functions of the argument t . It is very important to realize here as follows: in a geometrical approach to the space  (and no other point of view we must have), having eliminated the parameter t from the four parametrically given functions , we will get, for instance, three functions x=f₁(t), y=f₂(t), z=f₃(t) or three other functions , which represent (in space !) the path (trajectory) of the point A . Now the information about how the point A is moving along this path is lost. For example, if all the three functions x=f₁(t), y=f₂(t), z=f₃(t) have happened to be linear, then it is not, though, an evidence of uniform motion of the point A in , but it only characterises the path in  as a rectilinear one. Along this rectilinear path the point A may perform non-linear motion when related to the argument t .

So, we have come to the conclusion about the relative character of the very idea of the uniformity: the motion is uniform but in relation to what? In relation to t, or in relation to t ? In STR every local time t or  is satisfying the evident conditions , i.e. time t or  changes “iniformly in relation to itself“. But noninvariant coordinate t (or ) of the invariant vector  may be changing nonlinearly in relation to the invariant argument t . And let us remind that any process  take place as a uniform relatively to itself, because . The idea of “absolutely uniform” time t (changing only its “tempo” ), -this idea is a relic of the classical mechanics.

If the point A is moving nonuniformly with respect to argument t along a rectilinear path  in the space  then it is the evidence that in  there is a field interacting with the point A. This motion of the point A must be “immersed”, though, not into a pseudoeucledean rectilinear space  but into some pseudoriemannean space V₄, for which every tangent in its point M space  is a pseudoeucledean one with k=3. Such a space V₄ has natural metrics which compensates,or replaces, the existence of a field in .

And only in that case when all the four coordinates of the point A are linear functions of t , we may select such a neorelativistic inertial frame of reference  in which the vector  is parallel to , i.e.  and . Nevertheless, it doesn’t mean that the point A is resting in this frame of reference: it continues to move along the invariant vector  exactly as it did earlier when the vector  possessed nonzero pseudoprojections  upon the unit vectors .

This way we have reached the sort of agreement (upon the neo-relativistic level) between the Friedmann - Hubble idea of the spreading Universe and the classical idea of the Newton first low. So the free material point is mooving uniformly (to invariant time ) along some invariant vector  in all neorelativistic inertial frames of reference. As one can see now it would be more correct to speak not about the spreading Universe but about natural four-dimensional flow of time tin the space . The flow of pseudotime t (first coordinat of ) appears to us to be different in different neorelativistic frames of reference  and . This is explained by the fact, that inhabitants of our planet posses senses, which distort the real picture of the Universe, because these our senses "reflect" the three coordinates x = ct_x , y = ct_y , z = =ct_z from space , upon “our” basis unit vectors  in the space , and the coordinate t is the same way reflected upon our "clock"; at the same time the invariant magnitude t is left unnoticed as an extrasensoric one, and it can only theoretically be computed on the grounds of our direct measurements of each of the four coordinates of the vector

in every moment t . Of course, it is very difficult to agree with such “fantastical idea”, that “in reality” (what is “reality”?) we live in pseudoeuclidean space, but let us try to believe more the most important sensation of formal logic, then we believe our usual physiological sensations, because it is a new ordinary sacrifice in the name of the Great Idea of Relativity. Let us remember that our physiological sensations had put us in an awkward situation. In the whole history of science, begining from Euclid, Ptolemaios, Kopernik, Bruno, Galilei, Newton, Lobachevsky, Einstein, Heisenberg the humanity had this perpetual question: what is “reality” and what is only a theoretical, mathematical (“wild and fantastical”) formal scheme. Outstanding academician Leo Landau was shure, that the reason of crisis in some modern problems in pure physics and mechanics is just the deficiency of humane fantasy, because real Universe is more fantastical and more intricate than any our fantasy and imagination.

And something else: when we speak about the velocity of the point A in the space , we only mean a four-coordinate vector  having components . The magnitude of scalar product  is always equals to one, and the components of  are transformed by one of the transformations belonging to Poincare group.

If the point A is a free one and if in some inertial frame of reference  the three of its coordinates  do not change, then the three numbers , which are obtained as a result of division of the last three components of the vector  (in system ) by its first component, fully characterise the transformation matrix  transforming the system into the system . But these three numbers  make neither a vector nor even a formal tensor, as they are transformed by non-linear formulae and this is in contradiction with the rules of tensor algebra. The same case we have in “our” space : three coordinates , of the vector  make true tensor but two “fractions” , give not a tensor and not a vector, when relatively to usual rotation  in  (because two unit vectors  in expression  are not co-transformed relatively to non-linear transformation of numbers ). Naturally, the same three numbers  do make a vector in the space  by a compulsory condition that  is absolute time. From the point of view of the space , though, these three numbers  characterise the direction of the vector . By this, the condition  leads to the condition  which, actually, is a restriction set upon the choice of the direction in the space , or, in other words, the vector  must not coincide with an isotropic vector.

Consequently, in the space  there exist such directions  of motion from the point A to the point B for which the length of the vector , i.e. the neorelativistic distance from A to B , tends to zero, though the noninvariant (in space !) magnitude  may be at the same time fantastically great. This is how metrics of the space  looks like. Let us emphasize once more: the fourdimensional Lorentz’s transformations in space  are in fact formulae of the analitical geometry in this space; but classical STR is attempt to use the analitical geometry from  for generalization of the kinematics in quite another space ; but in , there exist its own fourdimensional kinematics, which we use in our neorelativistic mechanics.

And, finally, there is the last and maybe the most important question about the urgent necessity of making the relativistic principle of relativity more accurate. In STR the definition of this principle sounds approximately as follows: all the inertial frames of reference (classical three-dimentional frames of reference in the space  provided by their own local "clocks" are meant here) cannot be told from one another on the grounds of their physical properties ( no experiment can distinguish any of them as a preferable one).

But the point is that side by side to physical measuring devices in different inertial frames of reference there exist our theoretical equations of motion together with their solutions having, for example, the form , in which neither a function d_AB nor the argument t are invariant ones. These solutions, or these functions, we "measure", or analyze, using a "device" called a human eye, and this device does finds a difference! Now, is it possible indeed to construct an invariant function, describing some invariant physical or mechanical process, using noninvariant magnitudes d_AB and t. More of that, for each invariant vector considered in the space  there exists only one numerical invariant which can not simultaneously be a function and an argument of some function as well.

Let two material particles A and B be connected by an ideal elastic

"spring" and let them be vibrating in the space  along the vector . It is known that the equations of motion of these two points, derived by using methods of STR, first in the frame of reference K moving in the direction perpendicular to  and then in the frame of reference K' moving along , are different. And here anyone can say: "Let me analyze the both sets of equations of motion (in system K and in system K'), using the device called a human eye, and I will get quite a concrete information about the inertial frames of reference used". This statement is in deep contradiction with the whole idea of relativity.

Now we mean that the relativistic principle of relativity should not be declared at the very beginning of this theory but it must logically flow out from the solution of corresponding equations of motion as a consequence, i.e. everything must be approximately that way as it was with the classical Galileian principle of relativity until when physicists have discovered "forces" depending on velocities.

Such a neorelativistic principle of relativity may be made real only in the case, when we will utilize a more general method of "immersing” mechanics into geometry, using more general orthogonal transformations of coordinates and basis unit vectors in the more general pseudoeuclidean space  when n>4. In such space R_n there may exist a number (not less than two) of independent numerical invariants, giving us a chance to construct invariant functional relationships between independent invariants, which will describe real invariant physical processes.