Inertial Frame Transformation Based on Lobachevsky Function and Some
Optical Phenomena (Michelson-Morley experiment and Doppler Effect) 

 © N.G.Fadeev 1,2

1 Joint Institute for Nuclear Research, Dubna, Russia 
2 Email:  Nikolay.Fadeev@sunse.jinr.ru

Abstract: The inertial frame transformation based on Lobachevsky function  and some of its properties are reminded. The results of  introducing them into a physical consideration of the Michelson and standard Doppler effect in optics, are presented. It is that the new transformation also approves the negative result obtained in the Michelson experiment as the Lorentz one. But  the crucial result is expected for the measurements  performed with unequal lengths of the interferometer optical arms  and it will be in favour of the new one. In contrast to Lorentz contractions the relativity requirement of the new transformation  again leads to two different formulas for the Doppler effect in optics as in acoustics

Introduction: two possible ways to express the speed of light constancy principle

   It is used to consider that the invariance of an interval serves as mathematical tool to express the principle of the speed of light constancy and this is the reason of great importance of the interval invariance in the special theory of relativity and generally in the theoretical physics. The world famous Lorentz transformation can be obtained by the requirement for the interval to have the same form in any two inertial frames. In its turn the Lorentz transformation leads to the relativistic velocity summation law and  the relativistic mechanics (see Fig.1).                                                        

                
Fig.1 Two possible ways to express mathematically the principle of the speed of light constancy.

   A close relation between the special theory of relativity and the theory of Lobachevsky parallel lines (LPL) was noticed soon after A.Einstein's publication [1]. Further development of high energy physics made this relation more evident because the full correspondence between relativistic kinematics and Lobacvhevsky geometry was established [2]. Due to that the so called Lobachevsky  velocity space is widely used to present particle scattering processes in high energy physics. The two-photon pion decay was claimed as the physical equivalent for the main Lobachevsky geometrical axiom violating Euclidean V-th postulate[3]. This decay being represented in the velocity space due to its kinematics leads to the LPL in the plane of this space and visually demonstrates the principle of the speed of light constancy.

     Further investigations in the physical foundation of LPL revealed full      identity of the both theories. On the base of Lobachevsky function  the synchronous process of the particle and light beams propagation has been found. It has turned out that one of the main features of this process is quite similar to the optical model used by I.M.Frank in 1942 for Doppler effect consideration in refracting medium [4]. This synchronous process can be considered, on the one hand, as a new physical equivalent for the LPL and, on the other hand, as a physical foundation for solving the main relativity problems including the time relativity in a new way [5]. In this investigation Lobachesky function expresses the principle of the  speed of  light constancy instead of the interval invariancy and a new inertial frame transformation in the form of shifts has been obtained. It leads to the same relativistic velocity summation law and to the same kinematics and mechanics(see Fig.1).
 
  In framework of the new approach the Lorentz formulas can be also derived, but here they loose previous meaning as coordinates of the given event in the moving frame. They look as coordinates relative to the point, which is called the center of projectivity in the projective geometry [6], that does not coincide with the origin of the moving frame (and due to that they could be hardly considered as coordinates of an event). But the both sets of values are related in a simple way through the factor.

 Thus, there are two possible ways to express the principle of the speed of the light constancy and, respectively, there are  two sets of values: the well known Lorentz transformation and the new one - shifts. Some physically interesting properties of the transformation based on Lobachevsky function published in [5] are reminded here and the results of introducing them into the physical consideration of some optical phenomena, namely, the Michelson-Morley experiment and standard Doppler effect in optics, are presented in this paper.

  2. Some properties of transformation based on Lobachevsky function 

 Lorentz transformation [7]: 

           |               
          |                                                                                         (1)
                                  |            	
                                   |             

and the inertial frame transformation based on Lobachevsky function [5]: 
  
               |             
              |                                                                                  (2)
                            |             	
                             |              
can be written through parallel angle , defined by Lobachrvsky function:
,  , ,    (3)    , ,  ,                                     (4)

here  are coordinates of an event in a rest frame,  and   are coordinates of the same event in a frame moving with velocity  along  -axes (the upper sign)  or opposite -axes (the lower sign) , is the velocity  in units of the light velocity ,  is a value of rapidity  in units of   ( is the Lobachevsky  constant):

                                                                                                                                        (5)
     
It is seen from (3) that for any rapidity (and its velocity) there is a definite angle .. For the negative argument of the Lobachevsky function the parallel angle changes to [2], which corresponds to the same velocity but for the opposite direction.

  Let us consider some obvious properties of a new inertial frame transformation. 

First of all, one can see from (1) and (2) that:
 ,    ,    ,                                                                 (6)

i.e., the new transformation leads to the same relativistic velocity summation
law known from the Lorentz transformation.

II. This velocity summation law leads to the relativistic particle energy 
 and momentum definition:

,   ,   ,           (7)

and to the Lorentz transformation for them (for simplicity it is better to use 
units ). 

III. By direct substitution of (2) into the square of interval definition one can find the following:

 ,                                                                                  (8)

i.e.,   is not invariant.

IV. The same way one can obtain noninvariance of the scalar product :

 ,                                          (9)

where are - components of a particle momentum in the  rest frame.

V. The light ray  from the origin of the rest frame  under the parallel angle to the -axes allows one to introduce  the conception of the projectivity  (or the projective correspondence between  the 1-, 2- and 3-dimensions of multitudes) and its main invariant :

  ,                                                        (10)

the complex fraction of any four corresponding points [6]. 

VI. Let us also remind the length (at ) and the time duration  (at ) contractions due to the new transformation and the Lorentz one:

 ,   .                                  (11)                                                

It is important to note  that coordinates marked by  ('shifted' coordinates) and the 'primed' values are related by the -factor: , , , (it is seen from here and also from comparison of (1) and (2)). Since the is a constant for the both sets of values (1)-(2), then this relation means that Maxwell's equations are invariant with respect to the new  coordinate transformation in the same way as with respect to the Lorentz one.
  This contractions and transverse coordinates transformation will be used in the  sections below.

3. Michelson-Morley optical experiment 

   It is well known that the negative result of Michelson-Morley experiment was
understood on the base of the length contraction followed from the Lorentz transformation. The introduction of the relativity theory into the consideration 
of this experiment takes place at the last step [8]:
,  ,                                         (12)

where  is the total time for light beam propagation (there and back) in the first interferometer's arm  oriented parallel to the Earth's velocity , is the total time for light beam propagation (there and back) in the second arm  oriented perpendicular to the first one,  and  are lengths of the corresponding optical arms and is . 
   
    For equal lengths of arms the value for time difference was expected to be as follows:
                                   (13)

but the experiment has shown the so called 'negative' result corresponding to 
.
   
     To obtain agreement with the experimental result, one should correct the length of the arm (oriented according to the Eath's velocity) due to the relativity requirement. In the case of Lorentz contraction instead of one should use value, i.e.:

,                                              (14)

In this case for  one has: 

   ,                                 (15)

and  for (as it was in the experiment) the , as it was observed.

   In the case of using Lobachevsky transformation instead of one should use the value and instead of the transverse length (coordinate) should use , i.e., for one has:

  ,                                         (16)
 
and for has again .

   It is important to note that the result (15) is still a function on instead of the new result (16). The both transformations are adequate to the observed value due to the equal lengths of the interferometer arms . 

  It is interesting to compare the result of this experiment obtained for for the both predictions (15) and (16).

  3. Doppler effect in optics 

   As in the previous section the introduction of the relativity theory to  consider the standard Doppler effect in optics, can be also done at the last step of deriving formulas for the frequency shift in acoustics [8]: 

a) for the case when the source is moving and the receiver/observer is still:
 ,  , ,                                    (17)
b) for the case when the observer is moving and the source of waves is still:
 ,    ,   ,                               (18)
where  is the number of waves emitted by the source during the time ,and  are emitted and registered frequencies (of light here), respectively,,  is the relative velocity between the source and the device (observer), is the speed of the signal,   is the time of waves registering, the upper sign refers to the object moving away, the lower sign refers to the object moving in the opposite direction.

    Now one should take into account the time contraction in the moving frame and , . Let us make it firstly for Lorentz transformation () and use  from (17) and (18), respectively: 

a)      ,                   (19)
b)      .                     (20)  

The same can be done for the new transformation (one should use ): 

a)      ,                                        (21)
b)     .                                           (22)    
   It is seen from (19) and (20) that Lorentz contraction leads to the same result for the registered frequency in a) and  b) cases in optics (at ). But the relativity requirements from the new transformation leads again to two different formulas as in acoustics, but with  a)  b).  
  Thus, for the same relative velocity there are three different formulas for the registered frequency (19) - (22). 

4. Conclusions

 A new inertial frame transformation has been obtained by using Lobachevsky function as a mathematical tool to express the principle of the speed of light constancy.
  A pair of light beams chosen according to the Lobachevsky relation between the given velocity and parallel angle, makes the concept of the time relativity to be natural and transparent.
  A new transformation leads to the same relativistic kinematics as the Lorentz one, but it violates the invariance of both - the interval and the scalar product .
  A new transformation leads to the length and time contractions by times more than that for the Lorentz one. It also leads to transverse coordinate shifts.
 New length contractions and transverse coordinate shifts in a moving frame completely explain the negative result of the Michelson-Morley experiment. 
  Moreover, if this experiment is performed with nonequal lengths of the
interferometer optical arms, then the results will be crucial for one of the two transformations.  The predictions based on the both transformations differ in principle and the results are in  favour of the new one. 
 Michelson-Morley experiment with should be performed. 
 A new time contraction leads to two formulas for the standard Doppler   frequency shift in optics (as in acoustics). 
 The experiment for accurate measurements Doppler effect should be repeated. 

   The author expresses his gratitude to A.P. Cheplakov, O.V. Rogachevsky and V.V. Koukhtine for useful discussions and to S.V. Chubakova for the help in prepearing the English version of the paper.

References

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   1942,v6,pages 3-31.
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   P2-2004-048(rus.);                                 
   Proceedings of gAlbert Einstein Century Int.Conf.hParis,18-22July,2005 
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