Igor  Zlobin

 TO  THE  PROBLEM  OF  ORIENTATION  IN  TIME.  

1.  Introduction

According to local causality postulate [1] it is possible to send the signal from one point of the manifold  M  (we take that manifold  M  is coherent because the information concerning untied parts is inaccessible to us) to  another only in case when these points can be coherent by non-spacelike geodesic curve. The concept of  the manifold comes naturally with accordance to our idea about a continuity of space-time. As a matter of fact, the structure of space-time is manifold  M  given with Lorentz´s  metric and affine connectedness determined by it.

    Question on orientation in Time closely related to a condition of local causality. If we consider some area of space-time  U  on manifold  M (  U Ì  M )  on which the Time “arrow” [1] is tightly connected with increase of quasiinsulated thermodynamic systems entropy,  then we can expect that in each point of this area the  local Time“ arrow” exists   a  p r  i o r i   given

by non-asymptotic way.

    Today we do not have a quite clear idea what is the essence of   relations established  between the Time ”arrows” in different parts of space-time. Hawking S. and Ellis G. formulated this problem quite clearly [1]. From now and then we will name the initial task as a Hawking-Ellis problem for the reasons of simplicity of discussion.

This problem is formulated as: … it is not quite clear what is connection between separately taken the Time “arrow” and other the Time ”arrows”, which are defined by Universe expansion and a condition of the radiation in electrodynamics… .

2.  The analysis of the Hawking-Ellis problem

    This suggestion of the Hawking-Ellis problem will be built on a basis of some other conceptual proposals, than those which

take place in [1]. First of all we will be interested in the questions connected to chronological aspects. The Universe in whole might  be possibly considered as a global thermodynamic system evolving both in space and inTime. Asit was noted above the thermodynamic system of any sort  is characterized with entropy

and so with determinate direction in Time.

    Representing the Universe as multi connected topological manifold  M it is reasonable to assume that the inside part of this structural formation int M  is filled with the amount of events  å Q_s  ”inhabited" with material bodies. Any event  Q_s  as a physical phenomenon is characterized  by  its  taking place in Time. Consequent alternation of the given events is strictly determined and works according to causally-chronological principles, moreover, according to the proposition [1]  in physically realistic solutions the condition of causality and chronological conditions are equivalent.

    Taking in consideration all above mentioned it seems  physically reasonable to formulate following proposal                                                                                                             

Proposal 2.1

The beginning of inflating of the Universe [2]  and following that its development are directly coherent to function  Ŧ appear as global Time of the Universe.

    In order to the condition of steady causality [1] be complete it is necessary to enter function  Ŧ  and to connect it with the Universe (and so with manifold  M, too ) (Fig. 1).The condition of steady causality is determined as: steady causality is identified everywhere in  M ,  if and only if the function  Ŧ  is taking place and its  grad  is by timelike everywhere, i.e.

metric  g – negative

,                        ( 2.1 )

 where  g – Lorentz´s metric;  X – nonzero vector;  p – free point belonging  to  M  in which nonzero  X  by  timelike; 

g ( X, X ) – scalar square;  D_p is space, representing variety of all directions in  p and identified as vector space in  p because its tangent to  M. The function  Ŧ on manifold  M  is extrapolated as global Time of the Universe in sense that it increases along each by nonspacelike curve directed to the Future, herewith  Ŧ ∈ M   [1]. It is easy to notice that  Ŧ reflects such current of  Time (from the Past to the Future), in which all events  Q_s  along by  timelike curve  g are determined by

cause-and-effect relations. By the curve  g  we mean a curve of nonzero extent, herewith one point is not a curve.

    As it was noted earlier, the inside area of the Universe is forecasted as a fusion of massive numbers of material bodies and it is possible to give to each of them functional correspondence in a form of the local by  timelike Time ”arrows”–  j_i . This gives us possibility to say, that  there is  i-number of bodies in  int M  for those the condition  int  M:  i ® j_i  is complete. In this case we can see that in physical point of view the limits of global Time of the Universe is  possible to expand in the continuous manner, i.e. now function  Ŧ mainly consists of association of additive the local Time ”arrows”. Therefore, it can be found adequate the local Time ”arrow” –  j_i  to each material body of the Universe. Considering above mentioned we can say that function  Ŧ  is nothing else but association of the precise number of the local Time ”arrows” (Fig. 2)

Ŧ =  U  j_i ,                                                     ( 2.2 )

          ⁱ

    If we consider the local Time ”arrow” in more broad physical sense, it appears as a time interval   ∆t   which is defined by stable time of existence of some, freely selected body from the moment of its appearing  t^'  till the moment of its disintegration  t^" 

        j_i  ~  ∆t ,                                                            ( 2.3 )

 herewith the following conditions should be complete

    As follows out  of the general physical considerations, in case if we have two homogeneous systems in determined moment of Time, for example, two comparable by their nature bodies (physical, chemical, geometrical and etc. properties of them are identical), then they can have completely identical the local Time “arrows”. And, on the contrary , if two bodies are completely

different by all its characteristics, then nether their the local Time “arrows” can not be identical.

     Now,  when the Universe is in a dynamic condition, it is quite difficult to select two and more material bodies strictly equivalent to each other by all parameters. That leads to the fact that inside of global Time of Universe  Ŧ   there is no clear orientation between the local Time “arrows” j_i .

    As it will be shown bellow, there might really exist such a physical phenomena which gives us evident possibility to determinate the correlation between one separate the local Time “arrow” and other the local Time

“arrows”, i.e. hereby one of the probable solutions of the Hawking-Ellis problem is being  formulated.

    Let us assume that in some area of space-time  Υ  ( Υ Ì  M ) there were two free selected test bodies  U  and  V  which cooperate with each other in some way. These test bodies were taken so, that they did not belong to the same type, i.e. they had different properties and physical parameters. Because these test bodies influence each other, function-physical connection between  j_u  and  j_v  remains despite of different orientation of  their the local Time “arrows”. The question is how to find this connection, i.e. it is necessary to find such a calibrate parameter which would allow us to determinate the correlation between selected the local Time “arrows”. It has the solution, but to find it we have to make the following procedure:

1)      the local Time “arrows”  j_u  and   j_v   are situated regarding to each other so that their  start points  are coming together in point – 0. According to our plan this point appears as a pole so, that

{ 0 Î j_u  ∩  j_v }  «  { 0 Î j_u } Ù { 0 Î j_v },                                      (2.5)

2) we also assume that one of the local Time ”arrows”, for example  j_u   has a direction parallel to global Time of Universe           Ŧ ,  j_u ∥Ŧ.  Then the local Time “arrow” of  test body V  will be orientated to  j_u  at some freely taken angle (Fig. 3).

    From the physical point of view it is very important conclusion. Indeed, from what has been said above it is follows that reflection of the local Time “arrow” j_v   onto the local Time“ arrow” j_u  can be made by means of angle parameter. Let us name this parameter as  a  phase  angle  of  Time  and mark it  - Y_z . Then the following takes place:

Y_z :    j_v  ®  j_{u ,}                                                                      (2.6)

where  Y_z  reflection  j_v , to  Y_z (  j_v  )  ®   j_u  .

    The index  z  is necessary for us in order to select the given angle from all kinds of knowing geometrical angles. We shell give a definition of a phase angle of Time.

Definition 2.1

The phase angle of Time  Y_z   is  the angle between the local Time “arrows” j_i  (brought to the unified pole) which define coherence  in process of orientation    in internal areas of global Time of the Universe.

    Obviously the value of the phase angle of  Time  Y_z    is variable value, i.e. value  Y_z  of the concrete  separate freely taken pair of considered the local Time “arrows” is strictly individual and it is characterized only by initial pair of these local “arrows”. If  we analyze the situation when there is i-number of the local Time “arrows”, it is easy to determine (knowing what is the phase corner of Time) the functional coherence between this local Time “arrow” and other local “arrows”. In other words there is a possibility to observe how the local Time “arrows”  j_i  are orientated regarding to each other inside global Time of Universe  Ŧ.

    Thus, we can make the following conclusion: Hawking-Ellis problem can be solved correctly enough in case if it is reduced to the finding of values of a phase angle of Time  Y_z .                    

3. Conclusion

    Considered in this article scenario shows that in order to find the solution to the problem of coherence in process of orientation in Time it is necessary and enough to indicate the angular characteristic allocated in a rank of the phase angle of Time   Y_z . We have to do it in order to connect functionally  certain areas of Time (the local Time  “arrows”) in the inflated

Universe.  

References:

 1. S. W. Hawking, G. F. Ellis,  The Large Scale Structure of Space-Time, Cambridge University Press, 1973, 431 p.

 2. А. B. Gut,  P. R .Stienhard, In the  World Science (Moscow) 7, (1984), 56, p. 21-36